a ™ ye RAE Say cf 3 LOU Ee oho T Se SILT ETE et ee hehe cetntakenctenshelake es Set basse +t ane Hi Tene! . as 7 Senta gocie et oe Se AER RRR NS ss THE Grauelers Insurance Cumpany ACTUARIAL DEPARTMENT f LIBRARY FA, Pe ERR tte . NUMBER D e a Ris EMR pe Ath Pe at ere ee Sones See PRICE —ia SSSI ~) > 3 =| ~ é w 0 q > i) & i=] —————— ann beds rt Ca Laaeat Piya sf gt i y : < } ' : / ; } = a = _ j ! y 5 THIS BOOK IS NOT TO BE TAKEN FROM THE LIBRARY|| WITHOUT THE ui ‘ # ¢ CONSENT OF THB LIBRARIAN ij "3 f, re q S32 EES SS oe : te i | WATICG } rity t <3 4, : : é 4 \ LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN ——————— ti RE APRICOT TREATISE ON PROBABILITY: FORMING THE ARTICLE UNDER THAT HEAD IN THE SEVENTH EDITION OF THE ENCYCLOPADIA BRITANNICA. BY THOMAS GALLOWAY, M.A. F.R.S. SECRETARY OF THE ROYAL ASTRONOMICAL SOCIETY. EDINBURGH: ADAM AND CHARLES BLACK, NORTH BRIDGE, BOOKSELLERS TO HER MAJESTY FOR SCOTLAND. 1839. ' rar a taste , aees SS — i SS. S ioe (a apa ene th Re IN Thema | an 8 OE oA i, Gee tency. UR nite emer on eer ner 7 ae aes Fa, OS HE a Li EDINBURGH: S$ALFOUR & JACK, Printers, Niddry Street. CONTENTS. INTRODUCTION. ..s+essessesseessessessseerssessesssssessessssseeesesses PARE I. SECTION I. GENERAL PRINCIPLES OF THE THEORY OF PROBABILITY..........P. 16, ArTicLes.—1l-4, Definition of the terms Chance and Proba- bility. 5,6. Measure of mathematical probability. 7. Probability of compound events. 8. Example in numbers. 9. Probability of an event which may happen in different ways. 10. Identity of the formule for simultaneous and successive events, SECTION II. OF THE PROBABILITY OF EVENTS DEPENDING ON A REPETITION OF TRIALS, OR COMPOUNDED OF ANY NUMBER OF SIMPLE EVENTS, THE CHANCES IN RESPECT OF WHICH ARE KNOWN 4 PRIORI AND CON- STANT. eeetes BSS Lie LE 6 Ce iGO: 6 Rasa WSF 68 ste 1c Hele e160 6a U6.6s «5 86.60 s re'e béleln'e o's Pr: AP le ARTICLES.—11,12, Probabilities of the different combinations that may happen in a series of trials. 18. Application of the binomial theorem. 14, Example in a particular case. 15. Extension to the case of any n:mber of simple events. 16—21. Examples. 22, 23. Artifice for abbreviating the calculation. 24,25. Determination of lv CONTENTS. the number of trials required to render the probability of an event equal to an assigned fraction. \SECTION III. OF THE PROBABILITY OF EVENTS DEPENDING ON A REPETITION OF TRIALS, OR COMPOUNDED OF ANY NUMBER OF SIMPLE EVENTS, THE CHANCES IN RESPECT OF WHICH ARE KNOWN 4 PRIORI, AND VARY IN THE DIFFERENT TRIALS. eocccescetoce SOS Soe ser oes oversee setoes ARTICLES. —26. Expression for the probability of the different pos- sible results in particular cases. 27, 28. Extension to the general case. 29. Examples. 30. Solution of a question proposed by Huygens. SECTION IV. OF MATHEMATICAL AND MORAL EXPECTATION. coscspescceccvecsiossks aus ARTICLES,—81. Definition of the term expectation. 32. Example of mathematical expectation. $3. Mathematical expectation not ap- plicable in particular cases. 34. Hypothesis of Bernoulli. 35, 36. Formule to express the value of a moral expectation. 37—39. Con- sequences of the formule. 40. Application of the theory of Ber- noulli to the subject of insurances. 41. Petersburg problem. SECTION V. OF THE PROBABILITY OF FUTURE EVENTS DEDUCED FROM EXPERI- eeerer mer oe 1D. ENCE .ccccocccces COC n ee FESO SEHHET HST STE HHEHEHHAH GET HEH OHHEEe ArticLEs.—42. Hypotheses respecting the causes of an event. 43. Determination of the probabilities of the different hypotheses. 44. Probability of a future event deduced from the probabilities of the hypotheses. 45. Extension of the formule to any finite number of causes or hypotheses. 46. Sense in which the term cause is used in i sits... 58 Ne ae eae Sele Care —mipntaainediis OE AE TEE SET Neth GEL ART AMMO iM iN i eat CONTENTS, v this theory. 47,48. Examples of the formule. 49. Case in which the formule to physical or moral events. 51, 52. Extension of the formulz to the case of an infinite number of different causes. SECTION VI. OF BENEFITS DEPENDING ON THE PROBABLE DURATION OF HUMAN LIFE eeeces CROSSE OR ESSE HHHET EET ERE SESE He SORES BES Hoe8 seoeetuasinesecen ire 90, ARTICLES.—53, Principles on which the probability of life is com- puted. 54,55. Method of computing the value of an annuity on a single life. 56. Of a life annuity for terms of years. 57. Of an- nuities on joint lives. 58. On the survivor of any number of lives. 59—63. Methods of computing the values of assurances on lives. SECTION VII. OF THE APPLICATION OF THE THEORY OF PROBABILITY TO TESTIMONY, AND TO THE DECISIONS OF JURIES AND TRIBUNALS...ee+....- »P.100. ArTICLES.—64, 65. Expression for the probability of an event at- tested by a single witness on an assumed hypothesis. 66. Case in which the event attested is extremely improbable. 67. Case in which the character of the witness is altogether unknown. 68. Ex- pression of the probability of an attested event, regard being had to the a priort probability of the event. 69, 70. Probability of events attested by several witnesses. 71. Formule for the case of conflicting testimony. 72—75. Successive testimony, or tradition. 76. Application to the verdicts of juries. 77—81. Probability of acquittal and condemnation under different hypotheses. 82, 83. Probability of a verdict being correct when pronounced by a given majority. 84. Numerical expression for the error of a verdict when arbitrary values are given to the constants. 85, 86. Values of the con- stants, deduced from the records of the criminal courts in France. the different causes are not equally probable. 50. Application of me 8 The Reader is requested'to substitute the following List of Errata for that at the end of the volume. ERRATA. Page 40, note, for Essai sur les Probabilités, read Théorie Analytique des Probabilités. 43, line 7 from foot, for adopt read adapt. 49, line 13, read beginning with. In the note, read Traité Elz- mentaire. 54, line 7, for a—m, read a—m’. 81, first line, for (1+2+3...+)n, read (1+2+3...+n). 89, line 9, for m—m, read m+m’. 135, last line, for (n+x)—”+2—2, read (n+x)~"—-?—?. 188 line 8, for e—?, read ee 193, line 8, delete products of the. 198, line 9, for e’, e”, read e’e”. 203, line 19, for Ac’, read Ac. — line 3 from foot, for (1), read (2). 209, last line, for les read des. 210, line 2 from foot, for 1833, 1834, and 1835, read 1834, 1835, and 1836. 211, last line, for Abhandlungen, read Nachrichten. at 0 tests Net EAN Ni MNO oO ae Nl a,” hl ht a —— a PREFACE. In drawing up the following treatise for the Ency- clopzedia Britannica, my design has been, to pre- sent a general view of the principles, applications, and more important results of the mathematical theory of Probability, as laid down in the best and most recent works on the subject; particu- larly those of Laplace and Poisson: and, without entering into the details of mathematical difficul- ties, to explain the methods of applying analysis to the solution of the principal questions, as fully as the limits of the space which could be appropriated to the article would permit. In the prosecution of this design, questions con- nected with lotteries and games of chance, the sub- jects to which the earlier writers on Probability chiefly confined themselves, and which are frequent- ly supposed to form a principal part, if not the whole of the science, occupy but a small portion of the work ; indeed they are only introduced as furnishing exam- cuity, the general principles of the theory, and to give an outline of the manner in which these are applied to some cf the more important questions which have been investigated by Laplace and Poisson. The examples will be selected with a view to shew the nature of the principal results of the mathematical theory, as well as the peculiar methods of anz- lysis which are of most general application. a ee ee SS es SVS ae —S a ae ne ae — fan ae PROBABILITY. SECTION I. GENERAL PRINCIPLES OF THE THEORY OF PROBABILITY. 1. The term probable, in its popular.acceptation, is used in reference to any unknown or future event, to denote that in our judgment the event is more likely to be true than not, or more likely to happen than not to happen. With- out attempting to make an accurate enumeration of the va- rious circumstances which are favourable or unfavourable to its occurrence, or to balance their respective influences, we suppose there is a preponderance on.one side, and ac- cordingly pronounce it to be probable that the event has occurred, or will occur, or the contrary. 2. If we can see no reason why an event is more likely to happen than not to happen, we say it is a chance whe- ther the event will happen or not; or if it may happen ir more ways than one, and we have no reason for suppos- ing it will happen in any one of these ways rather than in another, we say it is a chance whether it will happen in any assigned way or in any other. Suppose, for example, an unknown number of balls of different colours to be placed in an urn, from which a ball is about to be extracted by a person blindfold. Here we have no reason for supposing that the ball about to be drawn will be of one colour rather than another, that it will be white rather than black, GENERAL PRINCIPLES. 17 or.red ; and accordingly say it is a chance whether the ball will come out of a particular colour, or a different. In this instance, then, the term chance denotes, simply, the absence of a known cause. If, however, we are made acquainted with the number of balls in the urn, and the number there are of each of the different colours, the term is used ina definite sense. For instance, suppose the urn to contain ten balls, of which nine are white, and the remaining one black, we say there are nine chances in favour of drawing a white ball, and one chance only in favour of drawing the black ball. Chance, in this sense, denotes a way of hap- pening, or a particular case or combination that may arise out of a number of other possible cases or combinations ; and an event becomes probable or improbable according as the number of chances in its favour is greater or less than the number against it. Chance and presumption are also fre- quently used synonymously with probability. 3. The mathematical probability of any event is the ratio of the number of ways in which that event may happen to the whole number of ways in which it may either hap- pen or fail. Thus, recurring to the previous example, the event, namely, the drawing of a ball from an urn con- taining 9 white balls and 1 black, may happen in 10 dif- ferent ways, inasmuch as any one of the 10 balls may be drawn ; but in one only of those ways will the event be a black ball; and therefore the probability of drawing the black ball is. In like manner, as there are 9 differ- ent ways in which a white ball may be drawn, or 9 chances of drawing a white ball, and ten chances in all, the probability of drawing a white ball at the first trial is 5° It follows immediately from this definition, that the proba- bility of drawing a ball of either colour will remain the same, AN, MIT Saw tL INE nag PTI Re MIEN RT DAA RE ee a Te i8 ; PROBABILITY. however the number of balls in the urn may be increased, provided those of each colour are increased in the same pro- portion. For instance, suppose the number of white balls to be 45, and the number of black balls to be 5; the num- ber of chances in favour of drawing a black ball is 5, while there are 50 chances in all, Conse the pro- bability of a black ball being drawn is 5;=74,. In the same manner, the probability of drawing a white ball is 48=,% the same as before. Generally, let E and F be two con- trary events, that is to say, such that the one or the other of them must necessarily happen, and both cannot happen together ; and let a be the number of chances or combina- tions which produce the event E, and 6 be the number of combinations which produce the event F, or cause the fail- ure of E;; then the probability that E will happen is G8 a+b’ and the probability that F will happen, or that E will noé happen is . In future, the term probability will be us- b ba ed only to signify mathematical probability. SN arene wR TT 4. It is to be carefully remarked, that the different .¢f2p%e0n «chances or combinations which form. the-elenyents-of Rae be QA ef bability are supposed to be perfectly equal,. If this equa- 3 lity does not hold, and there is any circumstance respect- 4: Ny _ing the event under consideration which renders one com- bination or set of combinations more likely to occur than, = another, the different combinations must be “ snultiplied ee nunabers ~proportional--to: their respective facilities, afters which the units in each multiplier may be regarded as so! many distinct chances, from which the probability of the event will be found by the above formula. This is equiva- lent to saying that a combination or chance which is twice y ~ ta 7 hy Mos tif ih f , GENERAL PRINCIPLES. 1g as likely to happen as another, must be regarded as two equal and similar combinations in comparison of that other ; a proposition which is sufficiently obvious. 5. It follows from the above definition, that the probabi- lity of any contingent event is measured by a fraction less than unity, and may have any value between 0 and 1. It follows, also, that the sum of the two fractions which mea- sure the probabilities of two contrary events is equal te unit, which is the measure of certainty, inasmuch as either the one or the other necessarily occurs. Thus, in the last i, iad A example, the probability of the event E is a+b a 6b Perey ea pang Hence if p denote the probability of any event E, and g the pro- e P the contrary event F is ——, and a-+-b bability of the contrary event F, we have g=1—p. This consequence of the definition is of great importance in the calculation of probabilities. 6. We have here supposed the result of a trial to be ne- cessarily one or other of two events E and F ; but it is easy to imagine the trial to be of such a kind that it may give rise to any one of a number of events E, F, G, H, &c. each having a given number of chances in its favour. This case is represented by supposing an urn to contain balls of as many different colours or sorts as there are different events. Let the urn be conceived to contain @ balls of the sort which produces the event E, 6 of the sort which produces F’, e of the sort which produces G, and so on; and let a4 b+e+d, &c. =k, so that & is the whole number of balls in the urn. The probabilities of the different events E, F, G, H, &e. are then, respectively, by the definition, ,and that of PROBABILITY. a b c d A icih oak eke the sum of which =1. In fact, if a ball be drawn at all, it must be of one or other of the different sorts contained in the urn; and consequently the sum of all the probabilities amounts to, unit or certainty. &c. 7. When an event is compounded of two or more simple events independent of each other, the probability of the compound event is equal to the product of the probabilities of the several simple events of which it is compounded. Let us imagine two urns, A and B, of which A contains a white balls and 6 black, and B contains a’ white and 6’ black. Make a--b=c, and a’+b’=c’, and let the com- pound event whose probability is to be determined be the drawing of a white ball from both urns. Now, as each of the ¢ balls in A may be drawn with any one of the c’ balls in B, the whole number of ways in which the balls in A may be differently combined by pairs with the balls in B, or the whole number of possible cases is ce’. But the num- ber of cases favourable to the compound event is evidently the number of different ways in which a white ball may be drawn from A with a white ball from B, and therefore equal to aa’. Hence by the definition (4), the probability that a j . aa’ fe white ball will be drawn from both urns is oars Now, if c p denote the probability of drawing a white ball from A, and p’ that of drawing a white ball from B, we have by the mi a a’ aa’ , definition p= ay and p/= 73 whence ool EP In general, let p denote the probability of an event E, p’ that of another event E’, p” that of a third E”, and so on; then the probability of the concourse of the events E, GENERAL PRINCIPLES. 21 EB’, E”, &c., or the probability that they will all happen, is PXp’ Xp”, &c.; that is to say, the probability of an event compounded of any number of simple and independent events, is the product of the respective probabilities of the several simple events. The probabilities that the several simple events E, E’, E,” &c., will nxo¢ all happen, or that some of them will hap- pen and others fail, are easily determined in the same man- ner ; it will be sufficient to indicate their several expres- sions. Suppose there are only three simple events, of which the probabilities are respectively p, p’, and p’’; and let q=\—p, ¢=1—y’, ¢ =1—p". The product paq" ex- presses the probability of the compound event which con- sists in E happening and E’ and E” both failing; qp’q’’ is the probability that E’ will happen, and that E and E’” will both fail; pp‘p” is the probability they will all three happen ; 1—pp’p’"is the probability they will mot all three happen, or that one of them at least will fail; gq’q’’ is the probability they will all fail; and 1—gq’q’’ is the probability they will not all three fail, or that one at least of them will happen. 8. As an example of the application of this rule, suppose it were required to assign the probability of throwing aces, at one throw, with two common dice. As a common die has six symmetrical faces, there are in respect of each die six ways equally possible, in which the simple event may "happen. The probability therefore of throwing ace with one die is 3, that is, p=}. In respect.of the second die, we have also p’=4 ; hence the probability of the compound event, or that aces will be thrown is pp'=ix4=,,. The probability that aces will mo¢ be thrown at any assigned trial is therefore (5) 1—3,=3{; and the odds against throwing aces at any given trial are 36 to 1. | —~ Ven ec eee ae a epee ae ema 2 “= oe Lit Ralaeeee cena iper eam ia a 22 PROBABILITY. Again, suppose two numbers, each consisting of 7 di- gits, to be taken at random, (for instance from a table of logarithms), and let it be proposed to assign the proba- bility that the substraction of the one from the other will be performed without its being necessary, in any Case, to increase the upper figure. Here, as each digit may have any one of the ten values from 0 to 9 both inclusive, and as each of those values in the upper line may be com- bined with any one of them in the lower line, there are 100 different combinations or equally possible cases for each par- tial substraction. Now, if the upper figure be 0, there is only one of those cases favourable to the event, or which will admit of the substraction being performed, namely, when the figure below is also 0.’ If the upper figure be 1, there are two cases favourable, namely, those in which the under figure is 0 or 1. If the upper figure be 2, there are three favourable cases, namely, when the under figure is 0, 1, or 2. Proceeding in this way through all the di- gits, the whole number of favourable cases is found to be 14243444546474849+410=55. Hence, for each partial substraction there are 55 favourable cases out of 100 possible cases ; therefore (4) the probabi- lity that any one of the figures in the upper line is not less than the corresponding figure in the under line is 3% ; and 100 of the seven simple events or partial substractions, whence, by (7), the probability of the compound event is we have p=p’=p’’=&c.= 5,5, for the probability of each pXp' Xp" X &= (sar) "=(-55)7= 0152243, ay. ; Sony 4 which is less than z£, and greater than ¢. 9. When an event may happen in several different ways, GENERAL PRINCIPLES. 23 each independent of the others, the probability of the event is the sum of all the partial Re eliat ae taken in respect of each of the different ways. 4 Gg d Suppose there are x different urns A,, he inside hace each containing balls of two colours, white and black, and let the whole number of balls in each urn respectively, be Cry G5, Cs Cc 59° esee n> and the number of white balls in each be Dy Maes, . tes ecthgs and let the event E be the extraction of a white ball in drawing a ball from any urn at random. In this case there are n different ways, all equally probable, in which the event may happen, for it may be drawn with equal facility from any one of the urns. The probability that the ball will be . ‘ 1 drawn from any given urn, A,, is therefore —; and if it n be drawn from this urn, the probability of its being white is =; therefore, by (7), the probability of a white ball being ( a WaT : drawn from A ms ae —.. In like manner the probabi- Fae 1 lity of a white ball being drawn from A, is shewn to be ba | as i —.—; from A, -—, and so on. Denoting nN Co 17 Cs therefore by p the whole probability of the event E, the proposition affirms that a le ae ~ 7” a, a To prove this, let the fractions —!, —2 &c. be reduced to a common denominator, aiid suppose the equivalent frac- tions to be a f i SPUR RE pitt Bel gr PROBABILITY. “We may now conceive the urns A, A,,A;...A, to be re- placed by others, each containing the same number, y, of balls, and of which the first contains a, white balls, the se- cond a,,and so on; and itis evident that the chance of a white ball being drawn from this new system of urns will be pre- cisely the same as it was for a white ball being drawn from the first system. Now the probability of drawing a white ball from the new system will not be altered by placing the whole of the my balls in a single urn, for they may still be conceived as arranged in -groups, disposed in any manner whatever, each group containing the same number of balls, and the same proportion of white to black as were in the separate urns ; and as each group contains the same num- ber of balls, the chance of laying the hand on any one group is the same as that of laying it on any other. The probabi- lity of drawing a white ball from the single urn, is therefore the same as for drawing it from the group of separate urns which contain each the same number of balls. But the pro- bability of drawing it from the single urn is the ratio of the number of white balls contained in the urn to the number of both colours, therefore (this probability being p) we have ] feo A + a,-+-4,......-a,) 5 of mas a a, whence, substituting for, —*, &c., their respective ey Y a: a values, —, —*, &c., we have Spee l sa a a a 1 2 3 n pai (42 coe tS). Pita deca a ace C, As a particular case suppose three urns A, B, Ctobe placed ee CT GENERAL PRINCIPLES. aa together, of which A contains 2 white balls and 1 black ; B 3 white balls and 2 black, and C 4 white and 3 black, and let it be required to determine the probability p of a white ball being drawn from the group by a person who is ignorant of the contents of the different urns. As there is no reason for selecting one urnin preference to another, the pro- bability that he will put his hand into the urn A is 4; andif he draw from this urn the probability that a white ball will be drawn is 2, there being 2 cases favourable to that event, and 3 cases inall. The probability of both events is there- fore} x = 2. In like manner, the probability of the ball being drawn from B is }; and if drawn from B the proba- bility of its being white is 33 therefore, the probability of this compound event is 3 X $= 5. Lastly, the probability of the ball being drawn from C is 4; and if drawn from C the probability of its being white is 4; therefore, the pro- >)” ! bability of this compound event is} x 4= 4. Hence, ; oat by the proposition now demonstrated, the complete pro- bability of the event E is P=s5+s + ot = 372. If all the balls had been placed in a single urn, the proba- bility of drawing a white ball would have been 7, for there are 3-+- 5 + 7 = 15 balls in all, of which 2 +3+4+4=9 are white. But 7% = 189; a fraction which differs sensibly from $23, the measure of the probability of the same event when the balls are distributed in the manner above supposed amongst the different urns. The distinction between the two cases is important. An. ¥ cee 10. The rule laid down in (7) for finding the probability of a compound event applies alike whether the simple events are determined simultaneously or in succession. In fact, when the simple events are entirely independent of each C | ia i ee 5 - iE zg f prere 26 PROBABILITY. other, the chances which determine the compound event are not influenced in any way by the intervention of time. Sup- pose, for example, the compound event to be the throwing of a certain number of points with a given number m of dice; the chances for and against the event are obviously the same whether the m dice are thrown at once, or a single die is thrown m times successively. But as the determin- ation of the probability of a compound event is in general facilitated by supposing the simple events to be decided one after the other, it will be convenient to view the subject in this light in explaining the method of forming the differ- ent combinations of the chances by which the probabilities of compound events are determined. EVENTS DEPENDING ON REPETITION. SECTION II. OF THE PROBABILITY OF EVENTS DEPENDING ON A REPE= TITION OF TRIALS, OR COMPOUNDED OF ANY NUMBER OF SIMPLE EVENTS, THE CHANCES IN RESPECT OF WHICH ARE KNOWN 4 PRIORI, AND CONSTANT. 11. Suppose an urn to contain a + b balls, a white and 6 black, and let a ball be successively drawn, and replaced in the urn after each drawing, in order that the chances in favour of drawing a ball of either colour may be the same in every trial, and let it be required to find the respective probabilities of the different possible results of any number of drawings. a Let us first suppose the number of trials to be two. The event may happen in anyof these four different ways: first white, second white; first white, second black; first black, se- cond white ; first black, second black. Assuming W to re- present the simple event which consists in the drawing of a white ball, and B that of a black ball, and supposing the or- der of the arrangement of the two letters to correspond with the order of succession of the simple events, the four pos- sible cases or combinations will be represented thus :— WW, WB, BW, BB. | Now let the probability of drawing a white ball in any trial be p, and that of drawing a black ball be g, (whence, et ee ee ee i ee 28 " EVENTS DEPENDING ON REPETITION. ize the probabilities of the four possible atin Is 3) compound events are by (7) respectively as under : probability of WW =p X p= p? of WB a= 7x: 9 sing | of BW 971% p — 99, i) of BB =qxq= 9? T= ee le ee oe ent are SS Se If we disregard the order of succession, and consider the two bi arrangements WB and BW, which are equally probable, as t forming the same compound event, namely, a ball of both i Hi colours in the two trials, the probability of this event, by ) ui (9), becomes 2 pg. The sum of the probabilities of all the } i possible arrangements is therefore 1 Pe +2pq9+ 7 = (P+? whence it appears that the probabilities of the different ar- rangements in two trials are respectively the terms of the development of the binomial, (p+q)2. Let us next suppose the number of trials to be three. The different arrangements that may be formed of the NEE TRIE AP LONE CITI NO EPI simple events in three trials, with the probability of each peewee respectively, are as follows :— | (a WWW, probability of which =ppp=p3 } INV. WB, ss: cebewen o,eeeeenoens =ppq=p'¢ | WY, BOW 3 ise bee cb ste geet be a aeeeie os =pqp=p"q a3 Walgid 155k codebase cbachoncno ei aerss =pqq=py" : DW By tigate leh Oe ee es =9pq=py’ hi BED ANS eles isa tosh aa eztecae fos =9gqp=pq’ bit “BOL OH OSA ery Pre ers ip 97005 It thus appears that the probability of obtaining two events cu of one kind, and one of the other, is the same in whatever order they succeed each other, and, in fact, is independent = — CHANCES KNOWN A PRIORI, AND CONSTANT. 29 of the order. Disregarding, then, the order of succession, and considering the combination of two white balls with one black, in whatever order they may be arranged, as the same compound event, the probability of its occurrence in any order whatever, being the sum of its probabilities in each particular order (9), is 3p?g. In like manner, regard- ing the combination of two black balls with one white, in any order of arrangement, as the same compound event, its pro- bability is 3pq*. The compound event resulting from three trials must then happen in one of four different ways, namely, 3 white balls; 2 white, combined with] black, in any order ; 2 black, combined with one white, in any order; or, lastly, 3 black; and the sum of the probabilities of these different cases IS pe +3p°¢+3p7 +P =(P+q)’. Hence the probabilities of all the different possible combi- nations in three trials are respectively given by the deve- lopment of the binomial (p+ q)°. 2. In general, let p denote the probability of any simple event KE, then the probability of E happening twice in two trials is p*, of happening thrice in 3 trials p*, and of hap- pening m times in m successive trials, py”. In like manner, the probability of the contrary event F being g (p+q=1), the probability of F happening z times in 7 successive trials is g’. Hence (7) the probability of E happening m times, and then F happening x times in succession, in m 4-7 trials, is p"q". But the probability of these events happening in any assigned order is the same as that of their happening in any other assigned order 3 therefore p”g” is the measure of \ the probability that Ewill, occur, me times, and F will occur V3 yaa n times in edefermjnafe ord ee Vow; sf Wye neh, and let U be the number of different w ays ty hich’ m events E, and PA / t Le} O° AG IRITENT Sie BE AIT MSE ni I + Pana tei igs oat pts SE i ee re # y eg he tc hs Bias el ane ea ace ¢ i Diep ae mH ‘ RA a F Gg tire Le Bk Pilg By paws ae a which shew that the probability P, or the product Up”g EVENTS DEPENDING ON REPETITION. se WMA Bae nm events F’, can be-combined i in h Soy gnd P be the proba- bility of any one of these Cobra ns W hatever, or the proba- bility of E occurring m times, and F occurring times in trials, without regard to the order in which they succeed each other, we have then PS=Upr a. In order to determine the 2s. of U, we may suppose the events in question to be. 80 mAARY- -differ ent things repre- ¢ sented by the letters/AyB, ‘cop, Ey-&e. of which -therecare m ae one kind, and v of another, and make m-+-n=h; then by the span theory of combinations, we have 4 Oe a on SDR eye Nea Uy OD ney abe This value is U is symmetrical in respect of m and 2, aod may be otherwise written in either of the two following “Us; _ forms, : fe galid | amr amc ha a me EOE. Soe See Veeen tien tis h(h—1)(A—2).....000 cagablat | Bahia 1 iy, Weds ROM ce tmaspene MLE mm is the (m-++-1)th term of the development of the bino- mial (p+gq)" arranged according to the increasing powers of p, or the (n+1)th term of the same development ar- ranged according to the increasing powers of g. Hence we conclude that when p and q remain constant, the pro- babilities of all the different compound events which can be formed by the combination of the simple events E and F in & trials, are expressed by the different terms of the formula (p+q)" expanded by the binomial theorem. The whole number of possible cases is evidently /+1, for in h experiments, E may occur / times, A—1 times, a Sees Fs . m™ ha NF \ — ; . LF \ ; aS aed CHANCES KNOWN A PRIORI, AND CONSTANT. 31 h—2 times......h—/A times ; this last being the case in which the contrary event F occurs in all the trials. The different cases are unequally probable, both by reason of the greater or smaller number of combinations by which ue may be produced, and which in reference to each case is represented by U, and by reason of the inequality BES pand gq. It will be shewn afterwards, that when p=q, and h is a whole number, the most probable case is that in which the occ Reenices of E and F are equal; and if / is an odd number, the two most probable cases are those in which the difference in the number of occurrences of E and the number of occurrences of F is unity. 4+ 13. In order to place the proposition now demonstrated in a clearer light, let us consider separately the different terms of the development of (p+), namely, l 2 php’ 9g + a HO ph a¢8 g’ eoeeeeeeesersetesreoee Ms \A—2)i+++h—n+1 Phi ae eeesen ve ae eee ae | beso The fiyst term p” expresses the probability that the event E will” every one of the / trials. The second term hp"—!q expresses the probability that E will occur A—1 times, and F once, without distinction of order; that is to say F may happen at the first or last or any intermediate trial. If a determinate succession is proposed, for example, that of h—1 times the event E in succession, and F in the next trial, the probability of the event in the assigned order is found by suppressing the coefficient A, and is consequently p’—'¢ The third term ————p’—°g? expresses the probability cas ‘calle - Stina BAST Le a ae the different terms of the binomial (p + ¢)', on sup- pressing the coefficients, become all equal; so that a parti- cular order being assigned in each of the possible cases or combinations, all the cases become equally probable. Thus, suppose a shilling to be tossed 100 times in succession, the probability of head turning up in every trial is(4)1°°. The probability of 50 heads and 50 tails in any assigned order is (3)°° x (4)5°=(4)1 °°; if m+n=100, the probability of m heads and x tails is also ($)"(4)"=(4)"t" =(4) 108, Hence the probability of any compound event formed by the combination of two simple contrary events succeeding each other in an assigned order, and each having the same probability, is independent of the ratio of the simple events, and depends only on the number of trials. Before the trials, it is an-even wager that head will be turned up in succession 100 times, and that the result of 100 trials will be 50 heads and 50 tails in a given order of succession, or any proportion of heads to tails in an order arbitrarily chosen. This con- sideration is frequently lost sight of in reasoning about those CHANCES KNOWN A PRIORI, AND CONSTANT. events of the natural world, which are termed extraordinary and miraculous. If in tossing a shilling 100 times into the air, the number of heads turned up is found nearly equal to the number of tails, the event excites no surprise ; some- thing like it was expected, On the contrary, if the diffe- rence between the number of heads and the number of tails is considerable, the event is termed extraordinary ; and if head turned up in every trial without exception, we should “@*<«: scarcely be persuaded that such an event was entirely the re- /.. sult of chance, and independent of a special cause. Never- theless, the @ priori probability that every trial will give head, is precisely the same as the probability of throwing any given number of heads and tails in an assigned order of succession. It will, however, be proved afterwards, that if such an event as throwing head 100 times in succession were actually observed, the probability of a special cause having intervened, would approach very nearly to certainty. 15. Hitherto we have supposed the compound event to be formed by the combination of two simple events only, us now suppose there are any number of simple events, EX E., E,, &c. of which the respective probabilities are DP e Ps, &c. and such that one or other of them necessarily hap- pens in each trial, so that p, + p,+p;+, &c.= 1, and de- termine the probability of any assigned combination of them in a given number of trials. This case may be represented by supposing an urn to contain a number of balls of as many different colours as there are distinct events; the event ay will be the drawing of a ball of the colour ¢, and its proba- bility p, will be the fraction whose numerator is equal to the number of balls of the colour 7, and denominator the whole number of balls in the urn. Now the probability of the o E and F, one of which necessarily excludes the other. Let - Pai? SARI Sin Ab aR. ied te ge sae a ee ee Lela TPT ae TaD ee ‘ t | re tee: ig ti ANS RRA PALIT GTA OBIE AT ute st a MOE ey prtiocame ss Benak ao eee Sets, 34 EVENTS DEPENDING ON REPETITION. event E, happening m times in succession is p? by Eras that of E, happening 7 times in succession is p",; that of E., happening v times in succession pZ ; and so on. There- fore (7) the probability of the compound event which is formed by the occurrence of m times E,, 2 times E,, 7 times E,, and so on, these events succeeding each other in or- der, is the product pt p? p%, &c. But the probability of the simple events succeeding each other in any particular order is the same as that of their succeeding in any other assigned order (12); consequently, if U’ denote the num- ber of different ways in which m events E,, 2 events E,, 7 events E%, &c. can be combined, or succeed each other, and P’ be the probability of the compound event in any order whatever, we have, Pe ar i ce Assuming h=m+n-+r-+, &c. we have also by the theory of combinations, LRG areas ore 15.3. Bocce SLC? s Bic) Oe 2a teenie the factor U’ being the coefficient of the term which has for ee its multiplier p? p2 pz, &c. in the expansion of the mul- tinomial (p,+po+p;+ &c-)", whence 1 Da ee ae ee ae = — pn" 7, &e. 1.2.3..m x 1.2:3..WX1.2.8...7 Kw. 1h ete Pp’ We shall now proceed to give some examples of the ap- plications of the preceding formule. 16. Let it be proposed to assign the probability P, of throwing ace once, and not oftener, in four successive throws of the same die.—Simpson, p. 15. Here, the chance of throwing ace in a single trial being i, we have p=}, and consequently g= 8, and also A=4. Now the compound event being the occurrence of the sim- vit aaeheahsiinoeeeieneme aie tenia eee CHANCES KNOWN A PRIORI, AND CONSTANT. 35 ple event E, whose probability is p, once, and of the con- trary event F three times, the probability of the compound event is that term of the development of"(p+-g)* which is multiplied by pg®. If, therefore, in the formula, one Bee nee © Be: sc ws Dene oo we make p=}, g=8, h=4, m=1, n=3, we shall have pe ee eve! 4) =a Es eee A eas 6 324 which is the probability required, and the same as that of throwing one ace, and not more than one, at a single throw with 4 dice. The probability of the contrary event, that is to say, the probability of either not throwing an ace at all, or of throw- 3 ing more aces than one is 1—}25 = 199; and therefore the odds against throwing one ace and no more in 4 throws of a La, ws common die are 199 to 125, or 8 to 5 very nearly. ae pare nL 17. If in this example it had been proposed to assign the probability of throwing ace once at least, instead of once aan tae: and not more, it would have been necessary to have includ- ce ee eect ola | Af ed those cases in which the ace occurs twice, or three times | ALD ? Sa eae Ea or in each of the four trials. he binomial (p-+-q)* gives P* + 4p? q+ Op?q? + 4p’ +94, the first term of which expresses the probability of throw- — et ee ing ace four times in succession ; the second that of throw- ing ace three times, and another number once; the third s that of throwing ace twice, and a different face twice; the fourth that of throwing ace once, and a different face three times ; and the fifth that of throwing a different face in each of the four trials. But as every one of these compound events, excepting the last, satisfies the condition of ace be- ing thrown once at least, the whole probability of that event j eee med} Lt ey ee 36 EVENTS DEPENDING ON REPETITION. ‘must be the sum of the probabilities of the different events by which it may be produced (9) and is consequently | 4 1 \3 5 ] Q 5 2 ] 5 3 671 Cs) +4) 3 +5(a) Ce) +4) G) = i558 In general, the sum of the first -+- 1 terms of (p+q)" ex- presses the probability of obtaining not less than h—x events, the probability of each of which is p, or not more than con- trary events, the probability of each of which is g. Since p+g=1, the sum of all the terms of the series pro- duced by the expansion of (p+q)" is equal to unit, and therefore the sum of any number of the terms is equal to unit diminished by the sum of the remaining terms. This consideration frequently gives the means of abridging the calculations. Thus, in the preceding example, instead of expanding the binomial (4 + 3)* in order to find the proba- bilities of throwing 4 aces, 3 aces, 2 aces, and 1 ace only, in a series of 4 trials, we might have sought the probability of not throwing aceat all. The probability of not throwing ace in a single trial is 8, and therefore (7) that of not throw- ing it in 4 trials is (§)*=5,45;. Hence the probability of the contrary event, namely, that ace will be thrown once or oftener, is 16 7o°s = 7£s0'5 3. the same as before. 18. Leta shilling be tossed; what is the probability that more than 3 heads will turn up in the first 10 trials? In this case, p=3, g=3, A=10; therefore (p4q)* =(4+1) =($)1°(14+1)1!°. Now the last term of this development expresses the probability that head will not turn up in any one of the ten trials ; the last but one, the probability that it will turn up once ; the last but'two, the probability that it will turn up twice; and the last but three, the probabil- ity that it will turn up three times; therefore the four last CHANCES KNOWN A PRIORI, AND CONSTANT. 5 i terms include all the different ways in which the ten trials give not more than three heads ; and their sum consequently expresses the probability that mot more than 3 heads will be thrown. Now the last four (or first four) terms of the expancion of (14+ 1)1° are Ts 10.9.8 Le. ai sae and their sum is 176, which multiplied by (3)!°= > 4,, gives 7/55, for the probability that not more than 3 heads will turn up ; whence the probability of the contrar y event or that more than 3 heads will be throw n, is 1176 — 1o24—#%5 and the odds in favour of throwing heads more than three times in 10 trials are 53 to 11. 19. A and B engage in play; the probability of A’s winning a game is p, and the probability of B’s winning a game is g; required the probability P, of A’s winning m games before B wins z games, the play being supposed to terminate when either of those events has occurred. It is evident that the question must be decided at the latest, by the (m+-n—1)th game ; for supposing m+ n—2 games to have been played, there is only one combination according to which the match can remain undecided, name- ly, that in which A has won m—1, and B n—] games; and in this case the next game necessarily decides the match. Suppose m-+ x games to have been played. The proba- bility that of these games m have been won by A, and w by B, is represented by the term of the binomial (p+q)mts in which the factor p”q* occurs (13) ; which term is ie Geccstcso Ee . z : P ™G & ‘ | uals Bape ata Wie mice: ana But A cannot win m games out of m-+- x exactly unless he wins the last game, for otherwise he must have won jn games a ea a Dc ee : ee f p hint Hib bs ie § & @.i3¢ £4 38 EVENTS DEPENDING ON REPETITION. out of m-+-a—l, if not out of a smaller number. In order therefore that A may win m games out of m-+-ax exactly, it is necessary in the first place that he wins m—1 out of m-+.x—1 in any order, and then that he wins also the next game. Now the probability of his winning m—1 games out of m-+x#—1 in any order (13) is 0 2 B.eeceeee M+2—] DSO el ee 2 ce nd the probability of his winning the following game is p, ae the probability of both events is (7) ap AES. Bin ear et 2 pg, i ahs Py) ees Fo dl bahia Aiba ee which, therefore, expresses the probability of A’s winning m games out of m-+-x exactly. If we suppose x=0, this formula becomes p”, which is the probability of A’s winning m games in succession. If w= 1, it becomes mpg, the probability that A wins m games ae ch) fae, out of m+1. If x=2, it becomes — pq’, the pro- bability that A wins m games out of oie If x=3, it be- m(m-1)(m+2 comes ( + us ms ) o"q°, the probability that A wins m games out of m+3; and soon. Continuing this process till we arrive at the term multiplied by p”g*, the sum of the probabilities of all the different compound events is mim+-l1l). mm---1)... fie py l+mg+ ae _ a + - ap ) Aas which expresses the probability of A’s winning out of a number noé greater than m+-x. bf es Now it has been shewn, that the match is necessarily decided by (m-+-n—1) games; consequently the solution ee Pore te CHANCES KNOWN A PRIORI, AND’ CONSTANT. 39 of the question is obtained by substituting »—1 for # in the last formula, which will then express the probability of A’s winning m games in any order, out of a number not greater than m-+-n—l. On making this substitution, we obtain ae L-- mg + een, Graver cunts: epeeerteoneeoernee 4. m(m-+-1)...m+-n—2 ge \ le 2 ae) The probability Q that the match will be decided in fa- vour of B, or that B will win 2 games out of a number not greater than m+-n—1, is found by changing m into 2, and p into g, and is therefore Uap Oe Q=q" l+np-+ va 1).++2-m+n—2 1) ST Bae ae te see eeoeerereesee ] 2 5 ar WR ; ] As an example, let us suppose p= 3 C ey m==4, and n=2. The probability of A’s winning the match, or the value of P, becomes oN ] 112 at 4 Bo ee (5) {i+ 5} 518 and the probability of B’s winning the match, or the value of Q, 1\2 4 2 8 5 131 (5) {it54i 5(5) + 33(5) \ aS In this example the skill of A is supposed to be twice as great as that of B, and the number of games that must be won by him in order to gain the match is also twice as great as the number required to bey won by B in order that B may SS | . recta OES GPT ale gah Prag Ate Ce a iaaiceeeiiniialtiane 4 Sia +4 : P “r a F “ een ee ee ee a 40 EVENTS DEPENDING ON REPETITION. gain; one might therefore suppose, that when they begin to play the chances in favour of each are equal. But the result shews that the chances in favour of A are fewer than those in favour of B in the proportion of 112 to 131; whence it appears that it would be unsafe to wager that a player Se ee ee eer eer ae een ney who has two chances in his favour while his adversary has ell tice ea a ae ME only one, will gain four games before his adversary shall have gained two. Suppose A and B, engaged in play, agree to leave off be- fore the match is decided, it is evident that the stakes ought ieee to be shared between them in proportion to their respective srobabilities of winning, and consequently the share of each I 2, is found from either of the above expressions for P and Q. — Nie et “il This was one of the questions proposed by the Chevalier de Méré to the celebrated Pascal, to which allusion has al- ready been made. 20. An urn contains +1 balls, marked with the num- bers 0, 1, 2,.3......23 a ball is successively drawn and re- placed in the urn, so that the chance of drawing any given pol”, number remains the same in each trial, whatis the probability LALLA OT a PR en eH Tan! om pal Fan that in A trials the sum of the numbers drawn will be equal tos?! ms 64. The solution of this problem depends on the number of i a ways in which the number s can be formed by the addition of Bit h different numbers, each of which may have any value from eet Oto. Ifwe suppose the numbers marked on the balls to ¥ a be indexes ofa certain quantity a, and develope the expres- hae sion (°-+ a! +-2”......-0")", the coefficient of any term of the development will indicate the number of different ways ARE! thi in which the balls may bedrawn, so that thesum of the num- bers drawn in A trials shall be equal to the sum of the in- Aas IR gt ee ag * Demoivre, Miscellanea Analytica, p. 196 ; Laplace, Essai sur les Probabilités, p. 253, et seq. 41 CHANCES KNOWN A PRIORI, AND CONSTANT. dexes of x in that term. If, therefore, we denvte by N the coefficient of that term of the development in which the sum of the indexes is s, then N will be the number of cases fa- vourable to the event. But the whole number of possible cases is (7-4-1); therefore the probability of the event is N+(n+1)*. On account of the particular form of the polynomial in question, the value of N is found without difficulty. —yrtril Because 2°--x!-x?....., 2" = fy therefore ae (wo tavlte?,...., + 2")'’=(1—a"t")*(1—2)—*. Now, ex- pressing these two factors in series, we have (1—antl) tnt? AV sent MADO—2) a 3(n-+1) ae pees geen, 1) 2 —e hih+-1)(h+2) Cin 1 he ee and the coefficients of the several terms of the product of these two series in which the sum of the indexes is s will be found as follows :— (1.) Multiply the first term of the first series by that term of the second series of which the argument is 2°; the coeffi- h(h+1)(h+-2)...... h-+-s—l PUA Ai ged hh sues’ s (2.) Multiply the second term of the first series by that cient of the product will be term of the second series which has for its argument a—”—!; the coefficient of the product will be h(h+1)(h42)...... h + s—_n—2 —h Xx : : ees eet s—n—] (3.) Multiply the third term of the first series by that term of the second series which has for its argument a*—*(n+1); the coefficient of the product will be nace pe te NT 8 IT AGC PRA TNE IEIAE EE Tem MN MOE 6 ag GLI ew a Eee Vat Sabai eects caveats eed Faces wanperonae a el nes ae 7 Ne 8 eS i oa €28 ce 4 £ t e | Eid 42 EVENTS DEPENDING ON REPETITION. h(h—1) _ h(hA+1)(h4+-2) h+-s—2n—3 a RUT BREN PES Paks (4.) Proceed in the same manner with the fourth term of the first series, and so on with the others, advancing at each new multiplication one term to the right in the first series, and 2-4-1 terms to the left in the second series, until a term is reached in the first series, the exponent of x in which is equal to, or greater than s. The sum of the several products thus obtained will be the value of N. We have therefore wa h(h+- 1)(A-+ 2) 322083 h . h(h+-1)(h+2) 1 Deas AG ae | : 2 x Fk oh + The series now found for N may be changed into another, having a more elegant form, by reducing all the terms to others having the common denominator 1.2.3......A—1. This will be accomplished by leaving out of the numerator and denominator of the first term all the numbers after A—1 to s, (including s), when s is greater than h—1, or by in- serting the numbers between s and A—1 (the last included), when s. as [S.. = ROS Gee eee 60 MATHEMATICAL AND MORAL EXPECTATION. 32. Suppose A and B to engage in play ; let p be the pro- bability of A’s winning a game, q the probability of B’s win- ning it, and sa sum of money staked on the issue of the game. By the definition, the mathematical expectation of A is ps, and that of Bis gs. Now if we suppose these expec- tations to be purchased by A and B; the sums they ought respectively to pay for them, or in other words to stake on the issue of the game, must be proportional to their respec- tive expectations, in order that they may play on equal terms. Let therefore a be the sum staked by A, and 6-the sum staked by B, we have then ps: gs :: a: 6, and consequently pb=qa. Now suppose a-+-b=s, or that the sum played for is the amount of the stakes; then, since 6 is the sum A ex- pects to gain, and pis the probability of his gaining it, pb is the mathematical value of A’s expectation of gain. In like manner ga is the mathematical value of B’s expectation of gain. Hence it follows, that when the sum staked by each is proportional to his probability of winning, the mathematical expectations of the two players are equal ;.so that after the stakes have been placed, and before the event is decided, they might exchange places without advantage or disadvan- tage to either. It follows likewise, that since the sum which the one must gain is just that which the other must lose, the product ga, which is B’s expectation of gain, may be regard- ed as A’s expectation of loss; or (if taken with a negative sign)as part of A’s whole expectation, which then becomes pb—gqa. But pb—qa=0; whence the condition of A before the event is decided is. not altered by the circumstance of his having staked on the issue of the play. ' 33. This conclusion at first sight appears paradoxical ; for it is certain, that after the stakes are placed, A must either gain the sum 8 or lose a, and therefore his fortune will of fa MATHEMATICAL AND MORAL EXPECTATION. 61 necessity either be increased by the gain of his adversary’s stake, or diminished by the loss of his own. The explana- tion depends on theorems which will afterwards be demon- strated relative to the repetition of trials, from which it re- sults, that though in a single trial the player must either lose or gain, yet on multiplying sufficiently the number of games, a probability will at length be obtained, approaching as nearly to certamty as we please, that the sum gained or lost in the long run will not exceed a certain given fraction (which may be as small as we please) of the whole sum staked, provided the play is undertaken on terms of mathe- matical equality. But this indefinite repetition of the ha- zard is practically impossible ; and innumerable cases may easily be imagined, in which an individual will be guided by other considerations than the mere mathematical value of the expectation. in undertaking or declining a risk. A person of moderate fortune would scarcely be persuaded to risk L.500 for the expectation of gaining L.5, though the chances. might be 100 to 1 in favour of the event which would produce that sum; but numbers would be found wil- ling enough to pay L.65 for the expectation of gaining L.500, the chances being 100 to | against them. In both cases, however, the expectation would be purchased at its real abstract value. According to the formula of mathematical expectation, the man whose sole fortune consists of a lot- tery ticket which has an equal chance of turning up a prize of L.20,000 or a blank, is in an equally advantageous posi- tion as_ he who is in possession of L.10,000; yet no man of ordinary prudence, if offered his choice of the two states, would hesitate as to which he ought to give the preference. Common sense will prevent a man from risking a sum, the loss of which would be attended with great privations, even a 4 5 A i 4 & | 62 MATHEMATICAL AND MORAL EXPECTATION. when, mathematically speaking, the chances are consider- ably in his favour. It is also obvious that two individuals whose fortunes are very unequal cannot engage in play with the same advantage, although the chances in favour of each, in respect of a single game, are precisely the same. The one who has a large fortune can repeat the hazard so often ae as to obtain a probability almost equal to certainty that his : 4 loss will not amount to any given sum; whereas the other, who cannot continue the play in case of loss, runs the risk of being ruined. It is thus evident, that ina multitude of cases the abstract theory of probability is not alone sufficient to give the value of an expectation, and that in dealing with We contingent events, an individual must be guided to a cer- 7 tain extent by considerations of relative advantage. 34. Various hypotheses have been imagined for the pur- : pose of reducing such relative or moral considerations to ac- curate calculation ; but that which appears the most natu- en ea nn en CR re ee ene si . See PE EER ae Se Bs ral, and applicable to the greatest number of cases, consists in supposing the relative value of any infinitely small sum to be directly proportional to its absolute value, and inversely [ as the fortune of the individual who has an expectation of receiving it. This principle was first proposed by Daniel | Bernoulli in the Petersburg Commentaries (vol. v.), and is there applied by him to the solution of a number of ques- tions of great practical interest. i Let x be the absolute value of the capital, or, as it is de- ; nominated by Laplace, the physical fortune, of an individu- te | al; then, according to the hypothesis of Bernoulli, the mo- ta | ral advantage which he derives from an infinitely small incre- a dx q ment of fortune =dz, is measured by the expression c—, ¢ £ being a constant to be determined by the nature of the ques- MATHEMATICAL AND MORAL EXPECTATION. 63 tion. Now, if we suppose the physical fortune to arise from the accumulation of the elements dx, and denote by y the relative or moral value of the fortune, of which the absolute or physical value is x, we shall have dx y=f Cae log. ”-4-constant. To determine the constant, we may suppose y=0, when x has a given value =a; this gives o=c log. a+-constant, whence y=e (log. e—log. a), or y=c log. — ; and it is to a be observed, that those values of x and y can never become negative, for as Bernoulli has remarked, it is only the per- son who is dying of hunger that can be said to possess ab- solutely nothing. In every other circumstance the mere pos- session of existence may be accounted a moral advantage, to which, however, it would be absurd to attempt to assign a nu- merical value. 35. From the above formula, it is easy to deduce a nu- merical expression for the value ofa moral expectation. Let a be the original fortune of the individual, and a, 8, y, &c. sums to be received on the occurrence of certain contin- gent events, E, F, G, &c. This being supposed, if the event E happens, the absolute fortune of the individual becomes a-+a, and its relative value, therefore, according to the for- , a-+-a : mula, is ¢ log. ear If F happens, his absolute fortune be- comes a+-8, to which the corresponding relative value is a clog. ;and soon. Now, let the probabilities of the events E, F, G, &c. be respectively p, g, 7, &c. (assuming P+9+7-+ &c.=1, so that one or other of the events will ne- cessarily happen), and let Y represent the relative fortune of | 1, - 7 ; 4 - y : $ S| 5 4 F # 4 § 5 a & 1 ny it 64 MATHEMATICAL AND MORAL EXPECTATION. the individual arising from his expectation, then, since the value of a benefit in expectation is equal to the amount of the benefit multiplied by the probability of obtaining it, we have Df. abs as ate aty ‘eet y oes - 4+q log. a3 = +r Jog. + &c. \ ee Let also X denote the absolute value of Y; then, by the X : formula, we have Y=c log. wa On comparing these two values of Y, we get X B log. = =plog, +g log. Rie +r log. aty + &c.; a a < : i . if and on passing to numbers, i X _ @+aPats a+) ke. a qet@tr+ &c. : SS eee ED LELOTLLEN LOMAS. ge ! | : therefore, since p+q+7r-+ &c. =], | K=(a+a) (aA) (aby); Ke. | In this expression X denotes the absolute value of the original fortune and of the expectation added together ; if, : therefore, we deduct a from X, the difference will be the ek value of the expectation, or the sum which, if it were to be ty | received certainly, would procure the individual the same relative advantage as his expectation. 36. If the sums a, 8, y, &c. are supposed to be very small 2 ‘ ; wine A a in comparison of a, so that quantities of the order (=) may 3 ' be neglected, the preceding equation becomes ; | KmaPtatet & 4 gptatr sort { patgB +ry+ &e. i a4. ae 1 | whence, since p+q+r+ &c.=1, a. X=atpat g8try+ &ce. MATHEMATICAL AND MORAL EXPECTATION. . 65 Deducting from this the original fortune a, the remainder pa+98t+ry-+ &c. is the value of the expectation, or the sum equivalent to the moral advantage. But the value of the ma- thematical expectation of the benefits a, B, y, &c. of which the probabilities are respectively p, q v, &c.is also pa+qB+ry -+&c. (31), therefore, when the contingent benefits are very small in comparison of the original fortune, the moral ad- vantage and the mathematical expectation are sensibly the same. 37. From the formula X=(a+a)"(a+8)"a-+y)’ &c. Ber- noulli deduces the consequence that gambling or betting is attended with a moral disadvantage, even when the chances of gain or loss, mathematically speaking, are perfectly equal. To shew this, he proposes the following question. A,whose fortune is 100 crowns, bets 50 crowns with B, on the issue of an event of which the probability is 4, on these terms: if the event happens, A is to receive from B 50 crowns ; if it fails, he is to pay B 50 crowns; what is the relative va- lue of A’s fortune, after undertaking the bet, and before the event is decided? In this case, we have a=100, a=50, B= —50, y=0; alsop=3, g=}, r=0; and the formula (35) becomes j AS X=(100+450)? x (100—50)2, whence X=,/ 150 x 50=87 ; and, consequently, the con- dition of A is worse by 13 crowns than it was before he ha- zarded the bet. The moral disadvantage is therefore equi- valent to this sum, though the terms of the play, according to the mathematical theory, are equal. 38. The conclusion arrived at in this particular case is easily shewn to be universally true. Let a be the capital of the player, p his probability of winning, g his probability saga AN aa NER TRG NONE ECARD ASLO GD claw - US Sa aie oe SE SE NE Gee eee 66 MATHEMATICAL AND MORAL EXPECTATION. of losing, and s the sum at stake. In order that he may play on terms of mathematical equality, the part of the stakes contributed by himself, or the sum which he can lose, must be ps (32), and the part contributed by his adversary, or that which he may gain, must be gs. The equation in (35) therefore becomes X=(a+qs)? x (a—ps)?, and if it can be shewn that this value of X is less than a, it will follow that his condition is rendered worse in conse- quence of having staked on the game. Now, dividing by a, and taking the logarithm of both sides of the equation, we get log. ee =ploe. (1+©) + ¢ log. i), the diffe- a a ay reniial of which (making s variable) is d log. See Ie : a nek 1__ Ps a a But the second side of this equation is evidently negative ; therefore d log. Xa is negative ; consequently the loga- rithm of X-+-a is negative, and X must be less thana. In all cases, therefore, the bet, if on even terms, produces a moral disadvantage. 39. Another consequence deduced by Bernoulli from this theory of moral expectation, is, that when property of any kind is exposed to a risk or hazard, it is more advantageous to expose it in parts to several risks independent of each other, than to expose the whole at once to a single risk, al- though the probability of loss be in both cases precisely the same. To prove this, he takes the followingexample. A mer- chant has a capital of L.4000, besides goods of the value of L.8000, which must be transported by sea. The probabi- lity of the loss of a vessel in the voyage being +, let it be MATHEMATICAL AND MORAL EXPECTATION. 67 proposed to find the value of the moral expectation of the merchant in the case of the goods being embarked ina single vessel, and also in the case of one half being embarked in one vessel and the other half in another. Supposing the mer- chandise embarked in one ship, the absolute fortune of the merchant will be increased to L.12,000 in the event of the safe arrival of the ship, and will be reduced to L.4000 in the event of its being lost. The probability of the first of these events is =°, and of the second 753 therefore his absolute fortune becomes, in virtue of his expectation, X=(12,000)1° ¢ (4000)7, whence X=10751. Deducting his other capital, L.4000, there remains L.6751 for the value of the moral expectation in respect of the venture. Let us next suppose the merchandise embarked in equal parts in two ships. In this case there are three compound events to beconsidered, Ist, Both vessels may arrive in safety ; ie Seen, 9 8] the probability of which is — ~% + = —. 2d é the probability of which is 10 * 10 = 100 2d, One may arrive in safety and the other be lost; the probability of which, as it may happen in two ways, (11) is 2x ES x To 18 =o’ 3 Both may be lost; the probability of which is ] I ] 10 * 10 = 100" capital of the merchant will become L.4000-+ L.8000= L.12,000; if the second happen it will be L.40004 L.4000 =L.8000 ; and if the third happen it will be only L.4000. With these numbers the formula becomes If the first of these events happen, the sie as is ats is X=(12,000) 100 x (8000) 16 0 x (4000) 10 0, es ye 1 i pa 5 y F Sige 68 MATHEMATICAL AND MORAL EXPECTATION. whence X=11033. Deducting his other capital, which was exposed to no risk, there remains L.7033 for the value of the moral expectation. This sum exceeds the former by L.282 ; and it is easily found by following the same process of rea- soning, that in proportion as the risk is divided among a greater number of ships, the moral expectation is increas- ed, and approaches its limit, which is the value of the ma- thematical expectation, or ;%, of L.8000=L.7200. 40. The theory of moral expectation enables us likewise to assign the circumstances in which it is advantageous or otherwise, to insure property against particular hazards. There are three principal questions to be considered in refe- rence to this subject; 1. The amount of premium the insur- ed may pay without disadvantage ; 2. The ratio of his for- tune to the value of the sum exposed to risk, in order that it may be advantageous to insure at a given premium; and 3. The capital which the insurer cr underwriter ought to pos- sess, in order that he may insure a given risk with probable advantage to himself, and safety to the insured. Let s be the value of a cargowhich a merchant embarks in a ship, p the probability of the safe arrival of the vessel, and a his capital independently of s. ‘The mathematical value of the premium for insurance is gs; for, if we denote the pre- mium by y, then yis the sum the insurer will gain if the vessel reaches its destination in safety, and s—y is the sum he will lose if it does not ; and by the theorem for the mathematical expectation py=—q(s—y) ; whence, since p+q=1, y=@s. If, therefore, the merchant insures the cargo, his absolute for- tune becomes a+-s—gqs=a-+ps; and if he does not insure, it is the value of X in the equation X=(a@+5)’a!. Hence it will be advantageous or otherwise to insure according as a-+-ps is greater or less than (a-+4s)’a’. Now the logarithm MATHEMATICAL AND MORAL EXPECTATION. 69 of the first of these expressions, or log (a +-ps), is equiva- ; pds J lent to the integral s ; and the logarithm of the se- A-- ps pds cond, or p log (as) +-¢ log a, is equivalent to r ; but a-+-s since p is a proper fraction, a+-ps is less than a+s, and therefore the first integral is greater than the second. Con- sequently a-+-ps is, in general, greater than (a+. s)’at, and the insurance is attended with advantage. Let us now as- sume a=a + ps—(a-+-s)’a’, and x will be the sum the mer- chant could afford to pay the insurer above the mathemati- cal value of the risk without moral disadvantage. Ifhe pays less than gs-+2, his relative fortune is increased by insur- ing; and if he pays more he isa loser. In practice the pre- mium may be considered as less than qs+4-a, but greater than gs; so that while the insured pays more than the ma- thematical value of the risk, he gains a moral advantage by the transaction. To solve the second question, let e be the premium de- manded for insuring the amount s; then, the other capi- tal of the merchant being a, his fortune after being insured is @-+s—e; while if he takes the risk on himself, its value becomes (a-+-s)?a%. If, therefore, the value of a be deter- mined from the equation a-++s—e=(a-+)?a’, we shall have the amount of capital he ought to possess in order that it may be morally a matter of indifference to him whether he insures or not. Asan example, let the value of the mer- chandise, or s, be L.10,000, e= L.800, and p=i2. The equation then becomes Dee) yee WE 2 Og? 0 ; a+-92U0=(a-+- 10,000) whence a is found by approximation =5043. It follows, > 8 LF SAO LT Nig TE te Me 70 MATHEMATICAL AND MORAL EXPECTATION. therefore, that unless his other capital amount to L.5043, it would be disadvantageous to neglect insuring, although the premium demanded exceed the mathematical value of the risk (which is ;'5 x L.10,000=L.500) by L.300. The third question, the amount of capital the underwriter ought to possess, is determined precisely in the same way. Let b-be his capital. After accepting the risk of the sum s for the premium e, his capital will become 6--e in the case of the vessel arriving in safety, and 6—s-+e in the case of its being lost. The formula of the moral expectation there- fore becomes X=(b-+-e)?(b—s +e)’; and in order that there may be neither advantage nor disadvantage in undertaking the risk, this value of X must be equal to his original capi- tal, d. Supposing, therefore, s, e, p, g, to have the same sig- nifications as above, the equation from which 6 is to be de- termined is b=(b+800)? 3(b—9200)?®, whence b= 14243, Unless, therefore, the capital of the insurer amounts to L.14,243, there would be a moral disadvantage in undertak- ing the risk of insuring a cargo worth L.10,000 for a pre- mium of L.800; and it is easy to see, that if a smaller pre- mium were demanded, the capital ought to be still greater. On making e=600, (which still exceeds the mathematical value of the risk), the value of b becomes L.29,878. Hence it follows, that a company possessing a large capital may not only with safety engage in speculations which might prove ruinous to another whose rescurces are more limited, but even derive from them a sure profit. 4|. The theory of moral expectation which we have now been considering had its origin in a problem proposed by 1 See the Commentarii Acad. Petropolitane, tom. v.; Laplace, Théorie des Prob. p. 482; Lacroix, Traité Elémentare, p. 132. MATHEMATICAL AND MORAL EXPECTATION. i Nicolas Bernoulli to Montmort, which, from its having been discussed at great length by Daniel Bernoulli in the Peters- burg Memoirs, has been usually called the Petersburg prob- lem. It is this; A and B play at heads and tails. A agrees to pay B 2 crowns if head turn up at the first throw ; 4 crowns if it turn up at the second, and not before ; 8 if it turn up at the third, and not before ; and, in general, 2” crowns if it turn up at the mth throw, and not before: required the value of B’s expectation? Here the proba- bility of head turning up at the first throw is 4; the proba- bility of its turning up at the second, and not at the first, is + X4=4; the probability of its not turning up either at the first or second, and of its turning up at the third, 1 xxi = and soon. Hence the probabilities of B receiving 2, 4, 8, 16......2" crowns le Eve ] l are respectively i Ghia 16777771 ge? Consequently (31) the mathematical value of B’s expectation is ] I! ] : i ee 5 — — 8 as ] eve =e 20 7 S. aaa ta et g X54 7gXx 16 + 5, X 2” crown Now, as no limit can be assigned to 7, inasmuch as it is possible that head may not turn up till after a very great, or any assignable number, of throws, this series, of which each term is unity, may go on for ever, and consequently the value of B’s expectation becomes infinite. Yet it is ob- vious that no one would pay any considerable sum for the expectation. This disagreement between the dictates of common sense and the results of the mathematical theory, appeared to Montmort to involve a great paradox ; although the question differs in this respect from no other question of chances in which the contingent benefit is very great, and 72 MATHEMATICAL AND MORAL EXPECTATION. the probability of receiving it very small. If the play could be repeated an infinite number of times, B might undertake to pay without disadvantage any sum, however large, for his expectation. A result, however, more in accordance with or- dinary notions, is obtained from the principle of Bernoulli. Let a be the amount of B’s fortune before the play begins, | a the value of his expectation, or the sum he pays A in con- sideration of the agreement, and make z=a—2z. If head t turn up at the first throw, B’s fortune becomes z-++-2; if at I the second, and not before, z4+ 2°; if at the third, and not before, z-4-2°; and so on. But the probabilities of these events being respectively ae Ee 24? & | for the moral expectation becomes (35) X=(2-4.2)2(24-2?)8 (242%) ist (2-42")2". Now the sum which B ought to pay will be determined by — oe ee peane = — os aro am or _ SS A a the formula making the value of his moral expectation, after the bet, and before the play begins, equal to his previous fortune ; we have therefore a= X, that is, t dL a8 h a= (24+2)2(z4+27)4(242°)8...... eh The general term of this series being (2-4-2")?" = Ol SS ES CS REO We Wriacinorsen aoe re ee 2 ] Tass ae the equation may be put under the form a6 fe Pea ale 2g 3 4. 2\2 2 8 am (27 +2* 42° 421°...) X (1 +5) (1+; at (1 A 5) 03 and since the logarithm of the first factor of this expression ($+5 au a 7 4 =p - ar Se. log 25 Of x 1+2(5) +3(5 | way + &c. \ og? =F) hg a= MATHEMATICAL AND MORAL EXPECTATION. 73 3 l Daihame 2 log 2, we have log a=2 log 2+ x og(1 ay, + 7G log (1 +=) + es log(1 +3] +&c. from which a value of z may be found by trial and error for any given value of a. Suppose 2=100 ; on computing the first 10 terms of the series there results a= 107°89, whence (since x=a—z) x==7°89; that is to say, if B pos- sessed only 100 crowns before beginning the play, it would be morally disadvantageous for him to risk 8 crowns for the expectation, although its mathematical value be infinitely great. If we suppose z=1000, the sum of 11 terms gives a=1011, nearly; so that if B possessed a fortune of 1011 crowns, the value of the moral expectation would, to him, be about 11 crowns. It is scarcely necessary to remark, that the results de- duced from the principle of Bernoulli are of a character widely different from those which are calculated according to the mathematical expectation. The latter gives the pre- cise value ofa contingent benefit, without any assumption or hypothesis respecting the personal circumstances of the indi- vidual who may gain or lose it ; whereas the considerations of relative advantage, of which it is the object of Bernoulli’s theory to take account, are entirely arbitrary, and by their very nature incapable of being made the subject of accurate computation. It is evidently impossible to have regard to, or appreciate, all the circumstances which may render the same sum of money a more important benefit to one man than to another; and consequently everyrule that can be given for the purpose must be liable to numerous exceptions. The principle, however, is thus far valuable, that it gives in the most common cases a plausible and judicious estimate of E 74 MATHEMATICAL AND MORAL EXPECTATION, the value of things which are not susceptible of exact ap- preciation ; and it has the advantage of being readily sub- mitted to analysis. A different principle, proposed by the celebrated naturalist Buffon, consists in making the value ‘tself of a casual benefit, instead of its infinitely small ele- ments, inversely proportional to the fortune of the expec- tant; but as this hypothesis has seldom been adopted, it is unnecessary to discuss it in this place. FUTURE EVENTS DEDUCED FROM EXPERIENCE. 75 SECTION V. OF THE PROBABILITY OF FUTURE EVENTS DEDUCED FROM EXPERIENCE. 42. In the preceding part of this article it has been as- sumed, in every case, that the number of chances favour- able and unfavourable to the occurrence of a contingent event is known @ priori, and consequently, that the proba- bility of the event, or the ratio of the number of favourable cases to the whole number of cases possible, can be abso- lutely determined. But in numerous applications of the theory of probabilities, and these, generally speaking, by far the most important, the ratio of the chances in favour of an event to those which oppose it is altogether unknown 3 and we can form no idea of the probability of the event except- ing from a comparison of the number of instances in which it has been observed to happen, with the whole. umber of instances in which it has been observed to happen and fail. In order to assign the probability of a contingent event in such cases, it is necessary to consider all the different causes or combinations of circumstances by which the event could {a possibly Be produced, and to determine its s probabilities suc-._/” Ed cessively‘on the hypotheses that each of these Causes existS/ to the exclusion of all the others. The comparative facili< y yg pif ies which these hypotheses give to the occurrence of the / event which has actually arrived, will then enable us to de- rs PERALTA DIE MTN NR ANTE 2 Oa Ds See aeaee = Lar ect Thiet eas een dente ae, 2, npn vars me Siti eSB 76 PROBABILITY OF FUTURE EVENTS termine the relative probabilities of the different hypothe- ses, and consequently their absolute probabilities, since their sum is necessarily equal to unity 5 and when the probabili- ties of the different hypotheses, and of the occurrence of the event on each hypothesis, have been determined, jthe pro- bability of the event occurring in a future trial will be found by the methods already explained. 43. Taking a simple case, let us suppose an urn to contain 4 counters, which are either white or black ; that the num- ber of each colour is unknown, but in four successive draw- ings (the counter drawn being replaced in the urn after each trial) a white counter has been drawn three times, and a black one once ; and let it be proposed to assign the pro- /° bability of drawing a counter of either colour at the next “~ f trial. den ne ere Carll L7 we In the present case three hypotheses mary be formed re- lative to the number of white and black counters in the urn. Ist, The urn may contain 3 white counters and 1 black 2d, It may contain 2 white and 2 black; 3d, It may con- tain 1 white and 3 black; for a counter of each colour hav- ing been drawn, the other two possible cases, namely, that they are all white or all black, are excluded by the observa- tion. Now, let p,, 2 Ps be the probabilities respectively of drawing awhite counter on each hypothesis, and qi, G2 4s, the probabilities of drawing a black. Supposing the first hypo- thesis to be true, or that the compound event which has been observed was produced by the cause indicated by that hypo- thesis, we have »,==3, g:=4; and the probability of the ob- served event, or that 3 white counters and | black would be drawn, (12) is 4p,> q=32- The second hypothesis gives ps=ty qo}, whence 4p,° q.=59. The third hypothesis gives Pot, qs== 3, whence 4p3 qs=% The probabilities (YT 6A Ly ey s + . & DEDUCED FROM EXPERIENCE. BD of the observed compound event, on each of the three hy- potheses, are therefore, respectively, 27, $$, ¢- 3 and the question now arises, how are the probabilities of the differ- ent hypotheses to be estimated? As we have no data, a priori, for determining this question, we must assume the probabilities of the different hypotheses to be respective- ly proportional to the probabilities they severally give of the observed compound event; in other words, we must assume the probability of any hypothesis to be greater or less according as it affords a greater or smaller number of com- binations favourable to the event which has been observed to take place. Thus, if C and C, be two independent causes from which an observed event E may be supposed to arise, and C furnishes 20 different combinations out of a given num- ber, favourable to the occurrence of E, while C, furnishes only 10 such combinations out of the same number, we na- turally infer that the probability of the cause C having ope- rated to produce E, is twice as great as the probability that the event was produced by the operation of the cause C,. Applying this principle to the present example, the probabi- lities of the three hypotheses are respectively proportional to the three fractions 27, 34, ¥, or to the numbers 27, 16, 3; and as no other hypotheses are admissible, the sum of their probabilities must be unity; therefore, making a, the probability of the first hypothesis, w, that of the second, and wz that of the third, we have 27 16 3 Pa a he lene Va bal eal Eo 44. Having found the probabilities of the different hy- potheses, that of drawing a white counter at the next trial is obtained without difficulty; for according to what was Ba ete eh si en Cn ed ents < ecaet ‘i ER ann el SR er Se ee wren oe ee Tee ta 3 nee onde ES sees I 5 [oon Pe = So he = tS comin smcnee sa MPR c0 iz, RS ai he aR 6 A pein as IIE wt Aah sige fam ~: ee ee! 78 PROBABILITY OF FUTURE EVENTS shewn in (9), the probability of this simple event must be equal to the sum of its probabilities relative to the different hypotheses, each multiplied into the probability of the hypo- thesis itself. Now it has been seen that, on the first hypo- thesis, the probability of drawing a white ball is #; on the second 2, and on the third +; and that the probabilities of the hypotheses are respectively 24, 18, 73; therefore the probability of a white counter being drawn at the next trial is 3 Ved ene dS ae 4°46 bf a6 a “46— rer In like manner, the probability of a black counter being drawn at the next trial is 1} 2 2°. 568 4 ; :. te 723 4 DEDUCED FROM EXPERIENCE. 79 event) of the existence of the different causes be respective- TY 35 Wo) Weyee0e000068 ye From the principle laid down in the preceding paragraph, namely, that the probabilities of the different causes or hypotheses are proportional to the probabilities they respectively give of the observed event, we have Megs Wessteen Wyte by th, YE geectas fhe whence, making P, +P,,+P.,...... 4+. P,, =>P,,andobserv- ing that w;+a,+7,......ta,=1 (since it is assumed that there are no other causes than those specified from which the event could arise), we have Le Pp ae Dyce: —————— Go aw =—— eeesee ed ee MSE rr SP ek cos ee Se a Ww whence it appears that the probability of each hypothesis re- specting the cause of the observed event is found by divid- ing the probability of the event on the supposition that that particular cause alone existed, by the sum of its probabili- ties in respect of all the causes. Let us now assume the probabilities of a future event E’ (which may be the same with E or different, but depending on the same causes) in respect of the several hypotheses, to be, p,, Pos P5y.++++ Dia so that if the particular cause C; be the true one, the proba- bility of E’is p ;; and let I be the probability of E’ in respect of all the causes, then by (9), 1 will be equal to the sum of the probabilities p,, p., p,.+..».P, relative to the different hypotheses, each multiplied by the probability of the hypo- thesis ; that is to say we shall have =P yD eT o+D5 M5 sores + PaTni or II=3p,a, the symbol = indicating the sum of all the dif- ferent values of p and a in respect of the different causes Y ‘ ( be Oe eaiuiers Cos ti 80 PROBABILITY OF FUTURE EVENTS 46. It may be worth while to remark that the word cause is not here used in its ordinary acceptation to denote the com- bination of circumstances, physical or moral, of which the event is a necessary consequence. In the sense we have used the term, the cause C is that which gives rise to the determinate probability P, that the event E will happen ; but so long as this probability falls short of certainty, its existence also implies that of another probability, 1—P, that the contrary event F will happen. Ifwe make P=], the existence of the cause C would necessarily involve the occurrence of E; and it is in this particular sense that the word cause is ordinarily used. In the theory of probabi- lities the causes of events are considered only in reference to the number of chances they afford for the occurrence of those events which they may possibly, but do not neces- sarily, produce. 47. The following example may serve to illustrate the method of applying the preceding formule. An urn contains 2 balls, which are known to be either white or black. Z d nua event E fs -, whence P= 3 and therefore, making 7 succes- 1 it he £3 th LAP j a ce: oe sal ; 2S bn DEDUCED FROM EXPERIENCE. 8] : ] a sively equal to 1, 2, 3,....2, we have sP=—(1 +2+3.. 6x.) Me, ; . . antl But the sum of this arithmetical series is OED therefore y 4 2 2P=4(n + 1), and consequently, x P, 2i 2 — = —__- SP, (gee) which is the probability of the assumption that the event proceeded from the cause C,, or that the urn contained 2 white balls. If we suppose ¢=” we have a,= for the n+ 1 probability that all the balls are white; and if we also sup. 7 pose x=3, this becomes 4; whence if an urn contain 3 balls which must be either black or white, and a white ball be drawn at the first trial, it is an even wager, after the trial, ~~ that all the balls are white. 48. Having found, from the observed event E, the pro- babilities of the different hypotheses, we have now to deter- mine the probability I of the event E’ (the drawing of a ‘“ white ball) at the next trial. Here two cases present them- selves; according as the ball is replaced in the urn, or is not; or in general, according as the law of the chances remains constant during the series of trials or varies. Ist, Let us suppose that the ball has been replaced in the urn. In this case the probability of the event E’, on the hy- rte et ; pothesis that the urn contains 7 white balls, iss that is te say p=—. But the probability a; of this hypothesis, as n found ab i therefe =— BAe wh ound above, is ; theretore p, 7, ; whence : : Pim n?(n-+4-1)’ 22 n(n-+-1) I a i \ ae ie re) | Pas - i [i Bit ae Bix ore Cee : Sieh esses ce staray ec erae not c — MRR 2 SS ee ee ee ear one a et . . wher teas aa int Far het pe oe pen eS == = 3 i. RE ene 2 ES EA PORES se ny 82 PROBABILITY OF FUTURE EVENTS eee the general formula (45) I= =p,w,becomes1== pas abieea n?(n+-1) wor Now 22?==7(¢-+1)—7. But by the pro- perty of the figurate numbers referred to in (23), the sum of the series of numbers obtained by giving z every value n(n-+1) Dep from 7=] to t=” in the formula is expressed by Miss ue ee ; therefore 22(2-++ pj ee us ah + 2M We have also as above Til 4243 consequently, Sita (2n-++1) 2 : and therefore 2 n(n+-1)(2n41) 2n+1 2 t= —. el n*(n+ 1) PRE es 3n 2d, Suppose the ball which has been extracted is not re- placed in the urn. In this case, on the hypothesis that the urn at. first contained z white balls, the probability of draw- i] es ing a white ball at the next trial is — 3 that is, ee ; and the probability of the hypothesis is the same as in the for- 2i7—l) . (n—1)n(m-+ 1)’ si(i—l). Now mer Case, or 7,;= ; therefore p, a= 22 n(n-+1) 2 and consequentlyli=2pa,;= CREWE TE the value of 3i(¢—1) will evidently be found by writing n—I1 for n in the above expression for 3¢(¢-++1); whence DEDUCED FROM EXPERIENCE. , and, therefore, in this case Si(i 1) =) 3 a 2 ye =D) _ 2 (n—1 )n(n+-1) 3 3 When x is a very large number, the ratio of 2n+1 to 3n, the value of If in the former case, does not sensibly differ from , and therefore in both cases 17 =%. Hence it follows, that if an event, depending on unknown causes, can happen only in one of two ways, and it has been observed to happen once, the odds are two to one in favour of its happening in the same way at the next occurrence. 49. The expression for « in (45) was determined on the supposition that previously to the experiments being made, we are entirely ignorant of the relative numbers of the two sorts of balis in the urn, and have no reason to suppose one hypothesis more probable than another. If, however, we happen to know, previously to the experiment, that the dif- ferent causes C,, Cg, C., &c. have not all the same num- ber of chances in their favour, or that the probabilities of the different hypotheses have relative values, it becomes ne- cessary to introduce those relative values, in consequence of which =, w,, &c., will receive a modification. Let us conceive a number of urns, each containing balis of two colours, black and white, to be distributed in x groups, A,, Ag, Aj.......+.A,, in such a manner that the ratio of the number of white balls to the number of black balls is the same in respect of each urn belonging to the same group, and consequently that the probability of drawing a ball of either colour is the same from whichever urn in the group it may happen to be drawn, but different in respect of the different groups ; and let the probabilities of drawing a white ball from each of the different groups be re< ee poe — —s ee ae a ee a ee ee a rae oe IEG Pie WS MA ia Jaa SEP Sap apuaO I can Bae: 84 PROBABILITY OF FUTURE EVENTS spectively P,, P,, P,,.....-P,» Now, let us suppose there are a, urns in the group A, a, in the group A,, and soon, and let s = the whole number of urns, so that s=a,+ 4, : a a +a, ; then, if we make +=),, =),, and so s s on, A, will be the a prior? probability that a ball drawn from any urn at random, will be drawn from the group Ay3 A, the probability it will be drawn from the group A, ; and, in general, \, the probability it will be drawn from the group A, This being premised, suppose a trial to be made, and that the event E is a white ball ; the probability a, of the hypothesis that the ball was drawn from the group A, is found as follows. The a priori probability of the ball be- ing drawn from the group A, is \,; and if the ball is actu- ally drawn from that group, the probability of its being white is P,; therefore the probability of both events is XP, d, P;; and consequently (45), mS —, the symbol of sum- mation = extending to all the values of ¢ from 7z=1 to i=n. 50. In the applications of the theory to physical or moral events, the different groups of urns here imagined may be regarded as so many independent causes C,, C,, C,, &c. by any one of which the event E might have been produced ; aw, is the probability that the event was produced by the par- ticular cause C;; P,; is the probability that the cause C;, if it had alone existed, would have produced the observed event E; and ), is the probability, previously to the experiment, that C, would be the efficient cause. The formula oO, r,P; SP? therefore, shews that the probability of any one of 2 the possible causes (C,) of an observed event is equa! to the product of the probability (P,) of the event taking place if Sot A. ty fif f DEDUCED FROM EXPERIENCE. R5 that cause acted alone multiplied into the probability 2, that the cause C,is the true one, and divided by the sum (2), P,) of all the similar products formed relatively to each of the causes from which the event can be supposed to arise. 51. The formule now obtained can only be used when the number of hypotheses is finite ; but in the applications of the theory it most frequently happens that an infinite number of hypotheses may be made respecting the causes of an observed event, as would be the case in the above ex- ample if the number of balls in the urn had been unknown. In such cases, in order_to find the values of a and 0, it be- comes necessary to transform the sums & into definite in- tegrals, which is accomplished by means of the theorem 2xX= /) X dx, where X is a function of 2 Suppose a ball to have been drawn a great number of times in succession from an urn (the number in which is unknown) and re- placed in the urn after each drawing, and that the result has been a white ball m times and a black ball 2 times, the probable constitution of the urn, and thence the probability of drawing a white ball at a future trial will be found as follows. Assume the hypothesis that the ratio of the num- ber of white balls to the whole number in the urn is xv: ie and let z be the probability of the hypothesis. On this hypothesis the probability of drawing a white ball in any trial is 2, and that of drawing a black ball 1—a, and consequently, the probability of drawing m white and » black in m+-x trials is Ux"(1—a)" by (12). We have there- fore for the probability of the observed compound event P=Ua”(1—x)"; whence in consequence of the above formula for transforming a sum into a definite integral EP=U f,x”(1—zx)" dx (U being independent of x) and therefore — . ween yeealanat OR DL PROBABILITY OF FUTURE EVENTS P Vam(—ay =SP 7 Fyfe xy de’ The value of the integral in the denominator of this frac- 4 a koe | tion is obtained by the usual method of integrating by parts. hed ‘ad fabelh | Since : pat ( 1 )* m -+- 1 therefore xmri(]—_9)" n I? oF ie eae m+ 1 m-+1 In like manner we get fau™t!(1—a)”—Idau AS 7 a eAAe 4 = w"(1—x)"da— amt1(1—a)"—ldx, m+ 1 farti(l-x)de, i asi eS . ies font(1—v)2de. Continuing this operation z times, or till the exponent of (1—#x) becomes »—n=0, the last integral will be gm-+n+1 mn’ therefore, collecting the several terms into one sum, we have xm+i(]_7)" — namt2(1__x)n—! Bee oe re cet 3 eae n(n— 1) n—2 ares cies 2o Las etek tet Cm+1)(m+2) 2.0.0... m+n+l When 2=0, all the terms of this series vanish, and when if xmtrdy= Be f en ba i” \ Wir | es fy pi F pif x=1 they all vanish excepting the last; therefore between the limits z=0 and x=1, the value of the integral is the Jast term of the series when x in that term =1; that is to say, of x” (1—2) pate ite eee 21 (m+ 1)(m+2)......m+n+] For the sake of brevity, let the symbol [a] be adopted to Cee oe at a DEDUCED FROM EXPERIENCE. 87 represent the continued product 1 .2.3...2 of the natural numbers from 1 to x,! whence by analogy [a+y] will repre- sent the continued product of the same series from 1 to the number denoted by #4+y. Multiplying, then, the numera- tor and denominator of the above expression by 1.2.3... .-.m=[m], we get Ng myn [ m ] _[m)[n] _ So ren i [m+n+1]’ whence the probability of the hypothesis, i in Conse Da of the equation above found, becomes pail aa og A x™(1—wx)". [m |[ 2] From this value of z we are enabled to deduce that of 1, the probability of drawing a white ball at the next trial. By (45) I= Sap. Now, since by hypothesis the number, of white balls in the urn is to the whole number of both co- lours in the ratio of # to 1, the probability of drawing a white ball is #; consequently p=a, and therefore 1=Saat he axrdx— a a hes amtl(]—a)"dx. But the value of f, x™+1(1—2x)"dx will evidently be obtained by substi- //“ tuting m-+-1 for m in the expression found for if gm (1—2x)"dx. This substitution gives fou —a)de= eee whence, observing that [m-+1] + [m]=m +1, and “atlas “Ete a a ate we have a: Ie aoe | ~ m+n+4+2° 1 This convenient notation has been adopted by Mr. De Morgan. OP GETS TEES a EIA TR nas RSE tcc SS eS SM a ne ae ee ~ en nese cormgeRinin, aE ee SLEPT Ty NSS FERNS Pare SE ar jus SSA h | } \ | 838 PROBABILITY OF FUTURE EVENTS The probability of the contrary event, or of drawing a : n+-1 black ball, is 1—O0= bin Ra As the numbers m and x m+n+-2 become larger, these two fractions approach nearer and n , whichare the apriort m+n nearer to their limits probabilities » and qg of the respective events when the ratio of the number of white balls in the urn is to that of the black balls as m to x. 52. The probability of drawing m’ white balls and 7’ black balls in m’+-7’ future trials is found in a similar man- ner, and the problem may be thus stated. E and F are two contrary events, depending on constant but unknown causes; and it has been observed, that in m+-n=/ successive in- stances the event E has occurred m times and F » times, required the probability that in m’4-n'=A’ future instances, E will occur m’ times and F 7’ times. Assume, as in the last case, the facility of the occurrence of E to that.of F to be in the ratio of x to l1—w; we have then, as before, for the probability of the hypothesis, a= liane a”(1—a)". Now on this hypothesis the probabi- [m] L7] lity of E in the next instance is x, and that of F is 1—a, whence the probability of m’ times E and 7’ times F in the - next A’ trials being denoted by p, we have (12) p=U’ 2™ be eae n’ Bay Lt ee a eee (4-2) , making Ue 3 M1 ee = ney We have therefore ap=U’ Lay Cinta ie i? Lm] [n"] (1l—a)"+' for the probability of the compound event on this hypothesis. To find its probability I on the infinite f / DEDUCED FROM EXPERIENCE. 89 number of hypotheses formed by supposing x to increase by infinitely small increments from #w=0 to v=1, we have Le Sap— of fe apdx. On substituting for ap the value just h found, we get 1=U’ ae vi amt (12) "te de, “and it is manifest that the value of this integral will be obtained by substituting m-+-m’ for m, and n+-n’ for z in the value of fC a™(1—x "dx found above. This substitution gives 0 S$ |m—m ]|[n+n’ | [A+A' +1] ? Wks an Ere (1 ot) n--n' dy = whence we conclude Sra ur eck nn’) Fh 1 [moh 41 of 2) The most probable hypothesis. will be found by making thevalue of a a maximum, or its differential coefficient equal bir Ble Meee [m]{n | Geen? Jel ila and making — =0, we get m(l—ax)=zx, whence x= dix to zero. Differentiating the equation a= 7 m -+- n ing the contents of the urn is, that the two sorts of balls are The most probable supposition, therefore, respect- in the same proportions as have been shewn by the previous drawings. We shall have further occasion for these for- mulz when we come to consider the cases in which m and m are large numbers, “et OLLIE ALOT AE GRA r= a pa ‘ . na ‘ ca = iceman ee ee ae ee en isaiatiibditihcteesnuinadiatat see Beye ee Wea rar ange een BENEFITS DEPENDING ON THE SECTION VI. OF BENEFITS DEPENDING ON THE PROBABLE DURATION OF HUMAN LIFE. 53. In applying the principles of the theory of probabi- lity to the determination of the values of benefits depend- ing on life, the fundamental element which it is necessary to determine from observation is the probability that an in- dividual at every given age within the observed limits of the duration of life, will live over a given portion of time, for instance one year; for when this has been determined for each year of age, the probability that an individual, or any number of individuals, will live over any assigned num- ber of years, is easily deduced. Thus, if the probabilities that an individual A, whose age is y, will live over 1, 2, 3...a years, be denoted respectively by p,, p,, P5-+-Pr3 and if d1> a> Y5++-Gx denote the same probabilities in respect of an individual whose age is y.1 years; r,,7,,7,...7,, the same in respectof an individual whose age is y+-2 years, and so on; then, since the probability p, which A has of living over 2 years is obviously compounded of the probability p, of his living over 1 year, and of the probability g; that, hav- ing attained the age y-+1, he will live another year, we have, by (7), Pe=P,q,- Again, the probability p, that A will live over three years, being compounded of the proba- ce ghienrrner= S SO = a = a n oS SS a a ecab tame . | PROBABLE DURATION OF HUMAN LIFR. O] bility p, that he will live over two years, and of the proba- bility *, that, having attained the age y42 years, he will survive another year, we have p;=p,7,=p,q,7,- Inlike manner Py=p; 7, 7; 8, and so on; so that the probabili- tieS Po, Ps, Py-+-Ps are successively derived from p,, Oise r,,8,, &c. which are supposed to be the data of observation. {f a large number x of individuals, all born in the same year, were selected, and if it were observed that the num- ber of them remaining alive at the end of the first year is w,, at the end of the second year x,, at the end of the third 2;, and so on, then the probabilities Pi Dos Dos Oe. would be given directly by the observation, being respec- tively equal to the quotients Sih ae ie: &c. But the most ‘an Hn accurate observations of mortality are furnished by the ex- perience of the annuity and assurance offices, where they are not made on an isolated number, diminishing, and con- sequently giving a less valuable result every year, but on a comparison of the numbers which, in a series of years, en- ter upon and survive each year of age. This observation GIVES P1, Y1, Ty, 8, &c, whence p,, P5> Pa» &c. are found, as above, for every year of life.! 54. The values of annuities on lives, and of reversionary sums to be paid on the failure of lives, are found by com- bining the probabilities p,, p,, p;, &c. with the rate of in- terest of money. Let r= the rate of interest, that is to say, the interest of L.1 for a year, and v= the present value of L.1 to be received at the end of a year, we shall then have v=1+-(1+7r). Now an annuity, payable yearly, is always understood in this sense, that the first payment becomes due * For further details on this subject, see Morrauiry, vol. xv. p. 550. ET Ta tO A a Gta a ne 0 eh ep SS Se ee 7 rote . - . : ee — ae oeeaes Se +", gah mann een A pe Cire G2 BENEFITS DEPENDING ON THE at the end of a year after the annuity is created. Suppose then the annuity to be L.1, the present value of the first payment, if it were to be received certainly, is v; but the receipt of this sum is contingent on the annuitant being alive at the end of the year, the probability of which we sup- pose to be p; ; therefore (7) the present value of L.1 sub- ject to the contingency, is up,. In like manner, the present value of L.1 to be received certainly at the end of # years is wv’; but the annuity will only be received at the end of the zth year if the annuitant be then living, the probability of which is p,; therefore the present value of that particu- lar payment is vp,. Hence if A denote the present value of the annuity, or the sum in hand which is equivalent to all the future payments, we shall have A= 3v*p,; the sum = including all values of x from #=1 to «= the number for which p=0. If the annuity be a pounds, its value is ob- aA. 55. The series denoted by =v*p, may be divided into two viously =asv*p,—= parts, Sv"p, -+-Sv*p,, where 7 is to be taken from 1 to , and zfrom 2-4-1 tothe number for which p vanishes. The first gives the value of the ¢emporary annuity on the given life for 2 years, and the second the value of the deferred annuity, that is to say, of the annuity to commence % years hence if the individual shall be then living, and to continue during the remainder of his life. Let A be the value of the annuity on the life of a person now aged y years for the whole of life, A“) the value of a temporary annuity on the same life for m years, and A") the value of an annuity deferred x years on the same life, we have then A= A) + A(@), To find A@™, let A, be the value of an annuity ona life aged y+ years. If the person now aged y years lives over n years, the value of an annuity on the remainder of his Os PROBABLE DURATION OF HUMAN LIFE. 9 life will chen be A,» The present value of this sum, if it were to be received certainly, is vA,, and the probability of receiving it is p,; therefore its value isv"p,A,. Hence A@)=v"p,A,, and A”)= A—v"p, A, 3 so that the values of temporary and deferred annuities are readily computed from tables of A and p for all the diffe- rent ages. 56. The equation A=A)4+ A(@” gives a formula by which the values of A are readily deduced from one ano- ther. Let m=1; we have then A=A)-+ep,A,. But A), the value of an annuity for one year, is merely the value of the first payment to be received in the event of the given life surviving one year. Its value is therefore vp; ; and we have consequently A=vp,-++-rp,A,, or A=up,(1+-A, ). This formula, which gives the value of an annuity at any age in terms of the next higher age, and greatly facilitates the computation of the annuity tables, is due to Euler. 57. The value of an annuity on the joint lives of any num- ber of individuals, that is, to continue only while they are ad/ living, is calculated precisely in the same manner as the an- nuityon a single life. Let there be any number of individuals, A, B, C, D, &c. and let the probabilities of each living over one year be respectively p,, 7,, 71, 5,, &c. and let P; be the probability that they will ad/ live over one year ; then Pe Pepe 7) 85 ee: Po= PoX YoX 1X Sg, &e. PSS Oy OS KC and the value of an annuity of L.1 on the joint lives is =v"P,, from #=1 to «= the number which renders any one of the probabilities p, g, 7, s, &c. nothing. 58. The value of an annuity on the survivor of any num- AR STAN EIS cit Fa GION TN EI Nl EP i DEG EET i a De eT - 94 BENEFITS DEPENDING ON THE ber of given lives, that is, to continue so long as any one of them exists, is thus found. The probability that A will be alive at the end of the xth year being p,, the probabi- lity that he will not be alive at the end of that time is 1—p,. The probability that a// the lives will be extinct at the end of the ath year is therefore (I—p, )A—g¢.)A—r, )(1—s,), &e. and the probability that they will not all be extinct, or that at least one of them will be in being, is I— (1—p, )\1—¢,,)(1++7,)(1—s,), &e which becomes by multiplication Pst Get? et Set Ke. —P Jul geeee aT gage 0000+ — TS p— BCs APs Ya Pr +Pr Yo Sx serve Ye VaiS2 + &e. —Dx Ua Tx Sy —®&C. + &c. Multiplying each of the terms byv*, and taking the sums of the respective products from w=1, and observing that 2v"H2qz2 is the value of the annuity on the joint lives of A and B, =v*p,qg27, that on the joint lives of A, B, and C, and so on, we have this rule :— The value of an annuity on the survivor of any number of lives is equal to the sum of the annuities on each of the lives, minus the sum of the annuities on each pair of joint lives, plus the sum of the annuities on the joint lives taken by threes, and so on. When there are only two lives, the value of the annuity on the life of the survivor becomes Lv? Dz F ZV" Oy —ZW" Pz Ya + 59. Let V denote the value of an assurance on the life of A, or the present worth of L.1 to be received at the end of the year in which A shall die. In respect of any year, the ath, after the present, the probability of A dying in the creme apace semen ee - _ san age a See nee NLT LIRA FO MELE RON OE ET ae a” PROBABLE DURATION OF HUMAN LIFE. 95 course of that year isp,1—pz- For let « be the probabi- lity that a life z—1 years older than A will live over one year, then 1—w is the probability of a life of that age not liv- ing over one year ; therefore p,_; being the probability of A living over e—l years, p,—;(1—w) is the chance of his liv- ing over w—1 years, and dying in the following year (7), But p, (1—u)= p,-1 —prr43 and by (53), pri U= pe 3 therefore p,—;—pz is the chance that A will survive a—l years and not survive « years. Nowv- is the value of L.1 to be received certainly at the end of the wth year ; there- fore in respect of the ath year the value of the expectation is U;(Px-1—p x )3 whence we have for the value of the as- surance V=Ev"(pri—Ppz)> from x=1 to r= the number which makes p=0. Now, if we observe that pol, and 2v*p,-1 =vzv"*—'p,_1, it will be obvious that Sv*p,-1=v(142e"p,); whence, denoting Sv? p, by A, (A being as in (54) the value of the annuity on the given life), we have V=r(14+A)—A3; or V=v—(1—+) A. 60. The values of assurances on joint lives, (that is, to be paid at the end of the year in which any one of the lives shall fail), or on the survivor of any number of joint lives, are cal- culated from the corresponding annuities by means of the same formula. Thus, let A’ be the value of an annuity of L.1 on any number of joint lives, and V’ the value of an assur- ance of L.1 on the same joint lives, then V’=v—(1—v) A’. If A” be the annuity, and V” the assurance on the life of the survivor of any number of given lives, we have still V"=v—(1—v) A”. 61. Assurances on lives are usually paid not in single pay- ments, but by equal yearly payments, the first being made Sa a - ry y PER a RPRMENINMER eRe ON - oe semenen oer 96 BENEFITS DEPENDING ON THE at the time the contract is entered into, and the succeeding ones at the end of each future year during the life of the assured. The present value of the sum which the assured contracts to pay is therefore equal to the first payment add- ed to the value of an annuity of the same amount on his life ; and if the assurance is made on terms of mathematical equa- lity, this sum must be precisely equal to the value of the assurance in a single payment. Therefore, if y denote the amount of the yearly payment, we have the equation y(1+A)=V; whence y=V+(1+ A). 62. The value of a temporary assurance for 2 years, that is, of an assurance to be paid only in the event of the indi- vidual dying before the end of 2 years is thus found. Let V be the present value of L.1, to be paid on the death of a person now aged y years, and V, the present value of L.1, to be paid on the death of a person now aged y-+-7 years. At the end of 7 years from the present time, the value of L.1 assured on the life of a person now aged y years will be V,, if he be then living. But the present value of L.1 to be received certainly at the end of 7 years is v” ; and the probability that the life will continue 7 years is p,; there- fore the present value of V,, subject to the contingency of the life continuing 7 years, is vp, V,. If, therefore, we sub- tract this from V, we shall have the value of the temporary assurance in a single payment, namely V—2"p, V,,. The equivalent annual premium is found by observing, that as the first payment is made immediately, and 7 pay- ments are to be made in all, the value of all the premiums after the first is that of a temporary annuity of the same amount for »—Il years. Denoting therefore the annual premium by w, and the value of a temporary annuity for m—1l years by A‘), the value of all the premiums is Pere pe pene S PROBABLE DURATION OF HUMAN LIFE. G7 uf uA = u(1 +)A) ; “and we have consequently u( 1 A\™))—V__o"», V_, whence V—v"p,Vi, oe TpAwy 63. The following question is of frequent occurrence. Required the present value of a sum of money to be receiv- ed at the end of the year in which A dies, provided he die while B is living. Let the sum be L.1, W= its present value, p,= the probability of A living over a years, and d= the probabi- lity of B living over x years. The chance of receiving the sum at the end of any given year, the xth, depends on two contingencies ; 1. A may die in the course of that year, and B live over it; 2. A and B may both die in that year, A dying first. The probability of A dying in the ath year has been shewn (59) to be P2-1—Pxs whence (7) the probability of the first contingency is (p,—;—p 2)92- The probability that A and B will both die in the eth year is (p s1—P « )( Yx—1— x); and forso short a period as one year, it may be considered an even chance whether A or B will die first, whatever be the difference of their ages ; therefore the probability in respect of the second contingency is 3(P2-1—P 2 )(2—1—Y 2)» Hence the whole probability of the sum being received at the end of the xth year, is (Pei1—P 2) 2+ 3(P21—Ps) (Fs1—Y2 =H(Po-1—p ») (9x41 +9.), which being developed, and multiplied by v, becomes 30° (Ps AF Pa lYa—P 2 V2—1—P 2 V2) and the sum of all the values of this expression from x= 1, gives the value of W. : It has been already shewn (59) that 2v?(p,1—pz)= v—(1—v)A, where A= the annuity on the life of A. In like : F St nn oe TEETER ea ghes Hae ae he a ee, RETRAIN ETE RRS Ne pt Arad ee eee i RS. e 3 " i 7 i 4 % it ota SS oe RS SSE nm PPT I AT OT PI —— Sener oem 98 BENEFITS DEPENDING ON THE manner, if we denote by AB the value of an annuity on the joint lives of A and B, we shall have 2v*(p2—192~1—P 7") =v—(1—v) AB, which is the value of an assurance to be paid on the death of the first dying. Assume p’ such that p'.=P’' |Po—y» then p’, is evidently the probability that an individual A’ one year younger than A, will live over # years 1 } — : (53), and Sv*p 5192=—— =U"P'.Jr = — A’B; denoting by Pa ge A’B the value of an annuity on the joint lives of A’ and B. Again, let 9’2=9' ,7—1, then q/, is the probability that B’, who is one year younger than B, will live over x years, UV 3, Sade G9 Bae mm i =v" p,g' = AB’; denoting by AB’ the Yi value of an annuity on the joint lives of A and B’. Collect- ing the different terms, we have therefore Siti es Se acer W=3fe—(1v) AB 4 AB—— AD’ }> whence W Pi qi is easily computed from tables of annuities on joint lives. If A and B are both of the same age, the two last terms destroy each other, and W is equal to 4 the value of L.1, to be paid on the failure of the joint lives, as it evidently ought to be, since there is in this case the same chance of A dy- ing before B as of B dying before A. The formula gives the value of L.1 in a single payment ; the equivalent yearly payment is W divided by 1+-AB, for the contract ceases on the failure of the joint lives by the death of either. It would be easy to extend the formula to the case of an assurance to be paid on the contingency of the failure of any number of lives during the continuance of any number of other-lives, or of an assurance to continue only during a stated time; but as it is not our purpose to give solutions x PROBABLE DURATION OF HUMAN LIFE. 99 of the various problems of this kind which may occur in practice, but merely to shew the manner in which the ge- neral principles of the theory are applied to them, we shall not pursue the subject farther, but refer the reader to the article ANNUITIES, and to the standard works of Baily! and Milne,” in which it is treated in detail. The Doctrine of Life Annuities and Assurances analytically inves- tigated and practically explained, &c. By Francis Baily. London, 1813. This work is now out of print, but a French translation of it has recently been published at Paris. * A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships, ¥c. By Joshua Milne. London, 1815. 100 APPLICATION TO THE SECTION VII. OF THE APPLICATION OF THE THEORY OF PROBABILITY TO TESTIMONY, AND TO THE DECISIONS OF JURIES AND TRIBUNALS. 64. The case of a witness making an assertion may be represented by an urn containing balls of two colours, the ratio of the number of one colour to that of the other being unknown, but presumed from the result of a number of ex- periments, which consist in drawing a ball at random, and replacing it in the urn after each trial. A true assertion being represented by a ball of one colour, and a false one by a ball of the other, it follows from the theorem in (51), that if a witness has made m-}-n assertions, of which m are true and 7 false, the probability of a future assertion being true is a ae and that of its being false Be Mem Let m+-n+2 M+-n+-2 ° 4 thé first of these fractions be represented by v, and the se- | cond by w, then v is the measure of the veracity of the in- 7 dividual, or the probability of his speaking the truth, and w the opposite probability, since v-+-w=1. In general, the existing data are insufficient to enable us to determine the numerical values of v and w in this manner; and therefore in applying the formule to particular cases, we must assign arbitrary values to these quantities, founded on previous knowledge of the moral character of the individual, or on EP TR PEPE DECISIONS OF JURIES AND TRIBUNALS. 101 some notions, more or less sanctioned by experience, of the relative number of true and false statements made by men in general, placed in similar circumstances. 65. Having assumed v and w, let us suppose a witness to testify that an event has taken place, the @ prior? probabi- lity of which is p, and let it be proposed to determine the probability of the event after the testimony. In this case the event observed (E) is the assertion of the witness, and two hypotheses only can be made respecting its cause ; Ist, that the event testified really took place; and 2d, that it did not. On the first hypothesis the witness has spoken the truth, the probability of which is v; and an event has eccurred of which the probability is 3; therefore (7) the probability (P,) of the coincidence is yp. On the second hypothesis, the witness has testified falsely, the probability of which is w; and the event attested did not happen, the probability of which is g; therefore the probability (P,) of the coincidence is wg. Hence, by the formula (47) 7,= P,--=P,)the probability (#,, of the first hypothesis becomes ees , and the probability (a,) of the second ek up + wg Up + wg The sum of these two probabilities is unit, a condition which ought evidently to be fulfilled, since no other hypo- thesis can be made, and consequently one or other of the two must be true. It is to be observed, that these values of w, and za, are the respective probabilities, after the tes- timony has been given, that the event attested took place, and that it did not. p(v—vp—wq) , 7) since w,= sa we have 7,—p = mpg > | ey | - 4 4 4 hiviocosmiga * eee) 20h = POI 09) Ot omy 1 pp—wq up—wgy Se Sa a eee BNE ee BMS sft Ry PI Ae 2 ETERS ee eae ery wei bo Aneta UPL face Reape 2a Se eae Se aw Eee 7 - SE cits Sina aired a apse aan” elena eae eae Mime aidan oma 1) Meenas! ims alia ii ei Pinata PNT wp Abt i i ana BR cS = MAREN ay ye Race and RSS OTC i dense neo Ao (SSS Se ein asia are ees Etre yh BS 4.0 Ee ae Be eee: —w BLS ANE TE AINE NTI ta a Oe yn Rag eR ae ae ee go Ss ae DOSNT IO 5 102 APPLICATION TO THE 2v—l, therefore «,—p= pores . This fraction bein positive or negative, according as 2v—l1 is greater or less oe fo) than unity, or as v is greater or less than 4, it follows that if v4, then «,>~p; that is to say, the probability of the event after the testimony is greater than its a priori proba- bility when the veracity of the witness is greater than 2. On the contrary, if the veracity of the witness is less than 4, the effect of the testimony is to render the probability of the event less than its a priori probability. 66. If the event asserted by the witness be of such a na- ture that its occurrence is a priori extremely improbable, so that p is a very small fraction, and ¢ consequently ap- proaches nearly to unity, although at the same time the ve- racity of the witness be great, and measured by a fraction approaching to unity, the value of 2, becomes nearly equal to p--w, (for on this supposition p+-wg-+v is nearly equal tow). But it is obvious, that however great the improba- bility of a witness giving false testimony may be supposed, the improbability of a physical event may be any number of times greater ; in other words, however small a value may be given to w, the value of p may still be any number of times smaller ; so that notwithstanding the veracity of the witness, the probability of the event after the testimony, namely «,=p-—-w may be less than any assignable quan- tity. On this principle mankind do not easily give credence to a witness asserting a very extraordinary or improbable event. The odds against the occurrence of the event may be so great, that the testimony of no single witness, how- ever respectable his character, would suffice to induce be- lief. 67. In the case of the character of a witness being alto- i Ce = ee DECISIONS OF JURIES AND TRIBUNALS. 103 géther unknown, we may suppose v to have all possible values within certain limits, and to find the value of w, by integrat- ing the fraction fa ,dv between those limits. Since 7,= up pvdv ———. we have fa, dv= » which on substitut- Up + w¢ up + wy ing 1—v and 1—p for w and gq respectively, becomes prdv 1—p +. (2p—1 )v v } oe Ses Oo~°.t 1L— 9n—] a ra (Qp— Tee log.(1 p+ (2p—1)e )h+e C being a constant, the value of which will be determined , the integral of which is from the assumed limits. If v be supposed to vary between the limits v=0 and v=], then 1—p, p Sa tox =P 4 1— Ip—1 eq} and if we assume p=3, we have /a,dr=}(1—4 log.3), which, since the logarithm is the Napierian logarithm, and Nap. log. 3=1.0986, becomes 3 x .4507=.676, or nearly 2. Whence we see, that on this hypittleate the probability of the event is diminished in consequence of the testimony. 68. The credit due to the testimony of a witness depends not merely on his good faith, but also on the probability that he is not himself deceived with respect to the event he as- serts. The chances of a witness being deceived through credulity or ignorance are much more numerous in general than the chances of intentional fraud; and this must be the case more particularly when the event is of such a na- ture that it may happen in various ways which may be mis- taken one for another: as for instance, in the case of a lot- tery ticket being drawn, and the witness asserting that it bears a particular number, which might with equal proba- bility be any other number on the wheel. The following PETS ae ees SENN IL lo NA Net ge RIESE S02 mT Be ase, ry 5A 2, i Be ui it bel apr i 42 ie ee j ay! .t *£, ry) 7 % 4 a i = if tua’ \ | ie Hy) if os ee ccm Sara 104 APPLICATION TO THE question will illustrate the method of applying the calculus when a distinction is made between these sources of error. An urn contains s balls, of which a, are marked A as marked A,......a@, marked A,. A ball having been drawn at random, a witness of the drawing affirms that the ball drawn is marked A,,; required the probability of the testi- mony being true. Here we have s=a,-+a,+4,......44,) (” being the number of the different indices or sorts of balls) ; so that if we make p,=4,+-5, Py=@y--S..+4.-Pm= 4,5, then py, is the @ priori probability that the ball drawn is of the class marked A,, p, the probability that it belongs to the class whose index is A,, and soon. It is evident that » diffe- rent hypotheses may be made respecting the index of: the ball which has been drawn, for it may belong to any one of the different classes A,, A,...A,. Let the probabilities of these hypotheses be respectively a, z,...a,, (that is, in re- spect of any particular index ¢, w, is the probability after the assertion that the ball drawn is marked A,); and Jet the probabilities of the assertion on each of these hypotheses be respectively P,, P,...P,,, (that is, if the ball drawn be mark- ed A, then P, is the probability the witness will assert it to be marked A,,). Lastly, let v be the veracity of the wit- ness, and w the probability that he has not been deceived. (1.) Let us first consider the hypothesis that the ball drawn is marked A,,, and consequently that the assertion is true. In order to find P,,, the probability of the assertion being made, there are four cases to be considered. Ist, we may suppose the witness is not deceived himself (w), and that he speaks the truth (v). The probability of the assertion in this case is wv. 2d, The witness knows the truth, but in- tends to deceive, or testifies falsely. In this case the proba- DECISIONS OF JURIES AND TRIBUNALS. 105 bility of the assertion being made, on the hypothesis under consideration, is 0. 3d, The witness has been deceived him- self, but intends tospeak thetruth. Inthiscase also the proba- bility of the assertion being made is 0. 4th, The witness has been deceived himself, and intends to deceive. In this case the assertion might be made; and to find the probability of its being made we have to consider, that since the witness has been deceived, he must have supposed some other index than A,, to have been drawn; and since he intends to de- ceive, he must assert some other index to be drawn than that which he supposes to be drawn. Setting aside, there- fore, the index which he supposes to have been drawn, there remain 7—1 others, any one of which he is as likely to name as any other. The probability, therefore, of his naming A,, when he intends to deceive is 1+(n—1). Hence the pro- bability of the assertion in this case is compounded of the probabilities of three simple events, as follows: 1. Probabi- lity the witness is deceived =(1—z) ; 2. Probability he in- tends to deceive =(1—v); 3. Probability he names A,, =1-: (n—1). Theprobability of the assertion is therefore in this case =(1—u)(1—v)-+(n—1). Adding this to the pro- bability found in the first case, we have P,,,, the whole pro- bability of the assertion being made on the hypothesis that the index of the ball drawn was A,,, namely —u)(1 0) = yet 1 P,,=uv+ ( (2.) Let us now consider one of the remaining hypotheses, and suppose that the ball actually drawn was marked A, and not A,,, as attested by the witness. As before, there are four possible cases for consideration. Ist, The witness knows the fact and speaks the truth. In this case the as- sertion could not be made, or its probability is 0, 2d, The a ee tet ma game wing PT re aT pk Se Savane oan ge it oh z ss Oe pe ne NO Ro yeep ern Ro San eae a 106 APPLICATION TO THE witness knows the fact, and intends to deceive. In this case the probability of his asserting A,, to be drawn is compound- ed of the probability that he is not deceived (w), the proba- bility that he testifies falsely (1—v), and the probability that, knowing the index A, to be drawn, he selects A,, from among the x—1 which remain after rejecting A, (1+(n—1)). The probability of the assertion being made in this case is there- fore u (1—v)+(2—1). 3d, The witness is deceived, and intends to speak the truth. By reasoning as in the last case, it is easy to see that the probability of the assertion being made in this case is v (1—u)+(n—1). 4th, The wit- ness is deceived, and intends to deceive. The prebability of the assertion being made in this case will be found by con- sidering, that as the witness is himself deceived, he must suppose some particular index to be drawn different from A,, (which is drawn by hypothesis), for instance A, the proba- bility of which is ]~-(#—1); and intending to deceive, he must fix on some index different from A, which he sup- poses to be drawn ; and he announces A,,, the probability of which selection is also 1+(mn—1). The probability, therefore, that the witness supposes A, to be drawn, and annnounces A,,, is 1+(u—1l)*. But it is evident, that whatever can be affirmed with respect to the particular in- dex A,, may be affirmed with equal truth of every one of the other indexes, excepting A, which is actually drawn, (since by hypothesis the witness is deceived), and A,,, which he announces, (since by hypothesis he lies). There are therefore n—2 different ways in which he may at the same time be deceived, and intend to deceive, and announce A,,3 consequently the probability of this announcement in any of these ways is (n—2)-(n—1)?. Multiplying this into the probability of his being deceived (1—z), and the probability DECISIONS OF JURIES AND TRIBUNALS. 107 of his giving false testimony (1—v), the probability of the (1—w) (1—v) (n—2) (n—1)? the whole probability of the assertion, in all the cases in- cluded in the hypothesis that the ball actually drawn was assertion inthis case becomes . Hence marked A,, is _ u(i—v) , d—)v (1—w) (1—v) (n—2) +1 n—l GT) ‘a As this expression will evidently be the probability of the assertion on any other of the »—1 hypotheses that the ball actually drawn was marked with an index different from A,,,, the sum of the probabilities of the assertion on all these hy- potheses is 3P,, where 2 is successively each of the num- bers 1, 2, 3,-..2, excepting m. We have now to find z,,, the probability of the first hypo- thesis. Since the hypotheses, in the present question, are not all equally probable @ prior?, we must have recourse to the formula (49)a,;=),P,+ >,P,, and consequently in the present case we have ts Nek re Tek ES) A as the sign of summation = including every value of 7 from @ 0 to n, excepting i=m. Now the value of P, being the same in respect of each of the hypotheses which suppose the assertion untrue, =\,P,;=P,2);3 and the sum ofall the values of \, from i=0 to z=” being 1, on excluding A,2=An--s, we have LA=(s—A,) +S: Substituting this, together with the values of P,,, and P,, as above found, and making w’= 1—w, v’=1—», the formula becomes, after the proper reduction, Ay, (N—] Juv+ u'r’} dg (Len 10°F (Stl) fee pee’ HFS wes — A im— Se et IRI Abide eNO A AER LEN Si BE ne hte BaD Ne I ei TR eS Ot SAS Bi aS Sn ee ee ao Ee RE NR SIN EI NC NO RENEE AIM EE meen Tears : Se ; Se SRE. 108 APPLICATION TO THE which is the probability of the hypothesis that a ball mark- ed A,, was drawn, or that the testimony is true. When there are no two balls in the urn having the same index, the numbers a,, @,, @;, &c. become each =1, and s=n. In this case the formula gives (n—1 Juv +-u’v’ hea (n—1 )uv-u'v' + (n—1) (w'v+-ue’) + (n—2)u/v” which, on observing that wv-+-w’v-+-wv'+4+wu/v’=1, becomes by reduction 1—u)( 1—v Wu + foee Me This is the probability of the truth of the testimony of a witness, who affirms that the number m is drawn from an urn which contains 2 balls, numbered 1, 2, 3...%. It is ob- vious, that when w and vare fractions approaching to unity, and 7 is a considerable number, the second term becomes very small, and may be neglected. The probability then becomes simply a,,=wv. 69. We now proceed to consider the probability of anevent attested by several witnesses; and first let us suppose the wit- nesses to agree in their testimony. The measures of the ve- racity of the several witnesses being respectively 7, v2, v5, &c., and the a priorz probability of the event being p, we have by (58) for its probability after the testimony of the first witness, a — pide EST hie en ‘ »p+(1—v, )(l—p) In order to find the probability of the event after the se- cond witness gives his testimony, we may suppose the a priori probability to be changed from p to a, by the testi- mony of the first witness, and the same formula gives DECISIONS OF JURIES AND TRIBUNALS, 109 paella iad honeliareege wi ox 0.7, +(1-v,)( l-w) te vp + (1-v,) (1-2, )(1-p) Let a third witness now come forward, and give testi- mony in favour of the same event. Its probability after his testimony will become in like manner a U3A%o nia V,V.U3P * v,a7,+(1l—v, (l—a,) v,v,v,pt(1-v, )(1-v,)(1-v, )(1-p) In general, let z, be the probability of an event after it has been attested by x witnesses, and let v, be the veracity of the last witness, then w,_; being the probability of the event after e—l eyewitnesses have each testified in its fa- vour, we have UDE ise. Og P UVo+--Urp + (A—v,)(1—v). ..(1—v,)(1—p)y If we suppose the witnesses all equally credible, or that ie U,;=V,7=0;...=v2, this becomes a vp mi 1 vp -+(1—v)?(1—p) a 14 ear 1p ° v Pp Now, if v4, then (—v)+v=1, and a ,==p; whence it appears that the probability of an event is not increased @ x by the testimony of any number of witnesses, when the vera- city of each is only $3 but when v is greater than 4, the event becomes more probable as the number of witnesses is greater, and when v is a considerable fraction, its probability in- creases very rapidly with the number of witnesses. 70. When the values of v and p are given, that of x in the last formula may be found so as to render wz, of any given value. Hence we may find the number of witnesses required to make it an even wager, whether an event ex- _ ceedingly improbable, and in favour of which they give una sect mae ae CR ae =a See ie a NS aN a Ba “ine ei eh é r i 7 st POS —_ 2 —— > RE TS eT te pe OR er are = nt rae CA Sel SSA OMAR ee ’ Te ee ead aol * : 7 se ee | Sg ES EE RTS a ae eel ot Fyn cat < - == = SSS ¥ See nie Da ae Sh Stile AT an ane ile ae somone Pt Se os oS I ee eS ee _ ak ae ee Sa = peeping re antisite Sa 110 APPLICATION TO THE nimous testimony, has happened or not. For example, let the odds against the event be a million million to one, ] ] Laer OP Pe PSY Ore Lae, Gk t 1 1,000,000,000,001 = Tompy 2nd let & the that is, let p= ; 9 veracity of each witness be i0° In order that wz may equal $, 194 1 eh we must have ( 3) fom : ef a x ae —10!", therefore (5) x 10!=1; whence x log 12 ee EG 95424 ] log jon % # log 9=12 and therefore # = nearly, so that 13 independent witnesses would suffice to render it more probable that the event really took place than that it did not. This example is given by Mr. Babbage, (Ninth Bridge- water Treatise, Note E), with a view to shew the fallacy of Hume’s celebrated argument respecting miracles. What the example proves is simply this, that if we suppose an urn to contain a million million of white balls, and only one black ball, and that on a ball being drawn at random from the urn, thirteen eyewitnesses of the drawing, each of whom makes only one false statement in ten, without collusion, and independently of each other, affirm to A, who was not present at the drawing, that the ball drawn was black, then A would have rather a stronger reason for believing than for disbelieving the testimony. But it is sufficiently obvious, that the event attested in this case, though exceedingly im- probable a prior?, cannot be regarded as in any way mira- culous. Onthe contrary, the black ball might be drawn with the same facility, and was a priori as likely to be drawn, DECISIONS OF JURIES AND TRIBUNALS. lil as any other specified ball inthe urn. Let it be granted that an event is within the range of fortuitous occurrence, and that there exists a single chance in its favour out of any number of millions of chances, it may then happen in any one trial; nay, a number of trials may be assigned, such that its non-occurrence would be many times more im- probable than the contrary. 71. Let us next consider the case of a number of wit- nesses contradicting each other. If the first witness an- nounces an event of which the probability is p, then the probability, after the testimony, of its having happened is @,, and the probability that it has not happened 1—za,. Sup- pose a second witness now to appear, and testify that the event has not happened, and let the probability of the truth of his testimony be denoted by aw’, ; then 1—za, being the probability before his testimony was given that what he as- serts is true, and v, being the measure of his veracity, we v,(1—=,) %.(l—a,)+ (1l—v,)a,, have, as in (69), #’.= whence, since = Vip wr vp + (1—e)(1—p) (1—v,)v,(1—p) (l—v)v,(l—p) + v(1—v, )p for the measure of the probability that the event has not , there results , — Fier 2 happened. The probability that it has happened is there- fore 1—zw’,, and accordingly if #’, be less than $, there is a stronger reason for believing that the event happened than that it did not. The method of forming the expression for the probability of the event, after it has been attested or denied by a third witness, or any number of successive wit- nesses, is obvious. ee Se ee ee Pe ee Se ee Ie Bo Se — praptase Reames a * es a ae eee no eee 2 eee LO I LOLI LOI OGIO AGI OLED YAP SE I AD te SEARLE IE ONS RE ot wo attaertr SS eS aie a=) 2s AT ete ae eta eae 172 APPLICATION TO THE If we suppose the values of v, and v, to be equal, the ex- pression becomes #’ ,=1—~p, which is the a prior? probabi- lity that the event did not happen. It is obvious that this must be the case, inasmuch as two contradictory testimonies of equal weight neutralize each other. In general, the pro- bability of an event which is affirmed by m witnesses, and denied by witnesses, all equally credible, is the same as that of an event which is affirmed by m—wn witnesses who agree in their testimony. 72. When a relation has been transmitted through a se~ ries of narrators, of whom the first only has a direct know- ledge of the event, and each of the others derives his know- ledge from the relation of the preceding, the probability of the event is diminished by every succeeding relation In order to obtain a general expression for the probability of traditionary testimony, we may take the event considered in (68), namely the extraction of a ball marked A,, from an urn containing s balls, of which a, are marked A,, a, mark- ed A,,...@, marked A,, there being in all different in- dices. Now suppose the relation to have passed through a chain of narrators, T, T,, T,,...T,, in number 2-41, of whom the first only was an eyewitness of the event, each of the others receiving his knowledge of it from the one pre- ceding him, and communicating it in his turn to the suc- ceeding, the question is to determine the probability that a ball marked A,, was drawn, after this event has been nar- rated by T,, the last witness of the series. 73. In order to apply the general formula of (68) to this case, it is necessary to remark that the event observed is the attestation of T, of his having been informed by T,_1. that the ball drawn from the urn, the drawing of which was seen by T, was marked A,,. There are 2 different hypotheses DECISIONS OF JURIES AND TRIBUNALS. 113 ~ respecting the index of the ball actually drawn, but it is only necessary to consider two of them, namely, the hypo- thesis that the ball actually drawn was marked A,,, and any one of the other hypotheses which consist in supposing that a ball with a different index from A,, was drawn, for ex- ample A, Let the probability of the attestation, on the hy- pothesis that the index of the ball drawn was A,,, be denot- ed by yz, and its probability on the hypothesis that the in- dex was A, by y’2, (7, and y’, corresponding to P,, and P, in (68), which express the same probabilities in respect of the eyewitness T), then by (68), the probability of the hypothesis that A, was drawn is AmYx OOF Nene PERS a But since y’, is the same for all the hypotheses that the in- dex drawn was different from A,,, DA,y’,=y’,DA,; 3 and by (68) DAj=(s—a,,) +S, and d,,=A,+S; therefore anY x em ‘ he AmY x a (s—a,, Y # We have now to find y, and y’, interms of w Let », v, v,, &c. be the respective probabilities of “T, Toye, ee speaking the truth, then the probability of T, speaking the truth is v,, and the probability that he does not 1—1v,, whether because he is dishonest, and intends to deceive, or because he has mistaken the statement of the preceding witness. Now there are two ways inwhich it may happen that A,, is announ- ced by T,. First, if he speaks the truth, and has been inform- ed by the preceding narrator T,_; that A», was the index drawn; secondly, if he lies, and has been informed by T ,_1 by) >] : that a different index from A,, was drawn, Assuming ¥;—1 to have the same signification with respect to T,-; that y, has been assumed to have with respect to T,, (that is to say, 2 PP ad emi mitt til eA ek TB es se eee ie . a. 114 APPLICATION TO THE the probability of the assertion being made by T,_; on the hypothesis that the ball actually drawn was A,,,), the proba- bility of the first of these combinations is v, 7,1. With re- spect to the second case, it is to be observed, that if T, an- nounces a different index from that which has been announ- ced to him by T,-1, the chance of his announcing A, out of n—1 indexes different from that announced by T,-1 is 1+(n—1); and on multiplying this by the probability 1—v, that the testimony of T, is false, and by the probability 1—y,—; that T,_; has announced a different index from Ap», we have, for the probability of the second combination (1-v,)(1—y2-1) + (n—1). The whole probability of T, tes- tifying that A,, was drawn, is therefore, on the first hypo thesis, given by the equation, Ye=Velfra+ (1—2,)(1—ya) (21) This is an equation of finite differences of the first order, the complete integral of which is 1 C(av,—1)(nv,—1)...(nv,1—1) (0, 1} hie (n—1)* In order to determine the arbitrary constant C, it is to be ob- served, that, since y1, Y/g++-Yo a8 Well as Up Vy...Vz apply to the narrators T,, T,...T, respectively, if we suppose w=o the resulting value of the integral will be the probability that A,, was announced by the eyewitness T, on the hy- pothesis that A,, was actually drawn. Let this probability be P,, 3 then the equation becomes P,,=C- 1-- whence C=(2P,,—1)+2. If, therefore, we make. x — 1 — D1). (M1 =! )(nv,—1) (7—1)” 1 This is easily verified; for on changing x into x—l1 in the in- tegral, and forming the expression vz yx—) + (l—vz) (lL—ys-)+ (n—1), there results an identical equation. DECISIONS OF JURIES AND TRIBUNALS. we obtain, on the first hypothesis, for any value of 2, ¥,= { + (aP,,—1) xX} 7 In the same manner we find the probability y’, of the testimony given by T,, on the hypothesis that the ball ac- tually drawn was A; The probability of the event, after being testified by the eyewitness, being on this hypothesis P,, we haye, ; ae +(nP—1)X} +n. Substituting these values of y, and y/’, in the expression above found for z,,, we obtain for the probability of the event observed by T, and narrated by T,, the ndrration having passed from one to another in the manner supposed, id Tm {1 +n(P,,—l )x} on a., ; i+(nP,,—1 )X} -+- (S—Qn) yi +-(nxP—i )Xt ; : NV, —I| | ee p= 3 ; 74. Since Stas agg ee 2 and since v, is always less n— n— than unity, and 2 always greater than unity, each of the terms of the series represented by X, whether positive or negative, is a proper fraction, whence the value of X be- comes smaller and smaller as z increases. Suppose # in- finite, then X=9, and a,,=a,,-+-5, which is the a prioré pro- bability of the event. Hence we see that the probability of an event transmitted through a series of traditionary evi- dence becomes weaker at every step, and ultimately equal to the simple probability of the event, independent of any testimony. 75. When the urn is supposed to contain only 2 balls, each having a different index, the expression for w,, is great- ly simplified ; for, in this case, a,,==1,s=7 5 therefore, (since P,,+(z—1)P,=1) the denominator becomes», and we have consequently z,,= fl +(nP,,—l )x} +n, which coincides 2 3 sam Teg : Fae nc a ghapeanan sas A aco NS inne hn a aa mt myth a ae agrees "i nOvde ie set ~ yaaa? PPAR ent me e # papier oe) SSE, So See ee tase Saat Z e ” : - ‘ = = ter. as " ” So Gas ne Saw ES Te - Se ~ Pig STL TY ee adie ior Pe Saat fe hind 5 yale nal > Nea ee Ee Oe SER Sees ‘ —— Sa “ ae = Sate a BP i : . . rcv wey, ae ROR ep Emm ee = 116 APPLICATION TO THE 3 with the value of y, found above, that is to say, with the probability of the event being testified by T, on the hypo- thesis that it actually happened. Laplace, in solving this particular case of the problem, (p. 456) assumes that the probabilities here denoted by y, and a,,are identical. They are, however, as is evident from the above analysis, quite distinct in their nature, and their values are only equal in the particular case in which s—a,, is to a,, in the ratio of m—1l tol. (Poisson, p. 112.) 76. The question of determining the probability that the verdict of a jury is correct, is precisely analogous to that of finding the probability of an event attested by one or more witnesses. Let us first take the case ofa single juror, and assume u= the probability that the juror gives a cor- rect verdict, (that is, correct in respect of the facts), and p= the probability that the accused is guilty before being put on his trial. Suppose the verdict guilty to be return- ed; two hypotheses may be made respecting the cause of the verdict, first, that the accused is guilty ; secondly, that he is innocent. On the first hypothesis, the accused will be condemned if the juror gives a right verdict, the probabi- lity of which is u. On the second hypothesis, the accused will be condemned if the juror gives a wrong verdict, the probability of which is 1—w. But the @ priori probabili- ties of these causes (the guilt or innocence of the accused) being respectively p and 1—p, we have by (49) u, Te serrargemeryy (Ta) a, being the probability of the first hypothesis, or the pro- bability that the accused is guilty after the verdict has been given, and a, the probability resulting from the verdict that the accused is innocent. Ne ae DECISIONS OF JURIES AND TRIBUNALS. Oy 77. Suppose the verdict not guilty to be given, and let a’, and a’, be the probabilities after the verdict of the two hypotheses. On the first hypothesis, namely, that the ac- cused is guilty, this verdict will be given if the juror gives a wrong verdict, of which the probability is 1—w; and on the second hypothesis, the verdict will be given if the juror gives aright verdict, of which the probability is v ; and the probabilities of these hypotheses before the verdict being respectively p and 1—p as before, we have (l—u)p u(1—p) (1—u)p +-ul—p) ~ (2) p+ up) From the above value of a, we obtain 7, —p = p—p)(2u—1) up +-(1—w)(1—p) according as w is greater or less than $. Hence it appears i ; a fraction which is positive or negative that the guilt of the accused is only rendered more probable by the verdict guilty being pronounced, when the probabi- lity that the juror gives a correct verdict is greater than 5. In like manner it is shewn that 2’; (the presumption of the guilt of the accused after a verdict of acquittal), is greater than p when »w is less than $ 78. The a priori probability of the condemnation of the accused before he is put on his trial is wp-+-(l—u)(1—p) ; for there are two ways in which this condemnation may take place ; first, if the accused be guilty, and the juror give a correct verdict, the probability of which concurrence is up ; and, secondly, if the accused be innocent, and the ju- ror give a wrong verdict, the probability of which is (1—v) (1—p). Therefore, making c = the probability of a ver- dict of condemnation, we have cmup-+- (1—w)(1—p) ; and for a verdict of acquittal, 1—c=(1—u)p +u(1—?p). 79. Let us next suppose that after the verdict of the first ate ee ene i er 2S ae oe om PS eee ee 5 ini 3s sites j . > 4 | } } 2 i | | . = {hh fj ca | 7 ; { | We } ii y } ih - | I D | « | ; | | | ’ Wg | i AN at, | | PO SS EES — 118 APPLICATION TO THE juror has been pronounced, the accused is put onhis trial be- fore a second juror, and let uw, be the probability the second juror gives a correct verdict, and. c, be the probability the ac- cused will be pronounced guilty by him. After the verdict guilty has been pronounced by the first juror, the probabi- lity of the guilt of the accused is w,, and it is evident that ec, will be found by substituting w, for u, and aw, for p in the above value of ec, whence ¢,=u,a, +-(1—u,)(1—=a, ). The probability of a verdict of condemnation by both ju- rors is ce, 5 therefore, (observing that #,=up--c), we have for this probability CC.—=UU op -+- (1—u)(1—u, )(1—p). The probability of the guilt of the accused after a verdict of acquittal has been pronounced by the first juror being a’ ;, the probability of a verdict of acquittal being given by the second juror is 1—e,=(1—u,)a’ , 4-u,(1—z’,); therefore, (observing that w’ , =(1—u)p~(1—e), we have for the pro- bability of a verdict of acquittal by both jurors (1—c)1—e, )=(1—v) (1—u, )p -- uu. .(1—p). Adding the probability of a verdict of condemnation by both jurors, to that of acquittal by both, we have wu, -+-(1—w) (1—u,) for the probability of both giving the same verdict. Thisresult is independent of p, and is evidently true a priori, inasmuch as there are two ways in which the same verdict may be given, namely, when both jurors are right, and when both are wrong. The probability of acquittal by the second juror, after a verdict of guilty by the first, is 1—e,=(1—wu,)a, 4+ u (1—w,); multiplying by ec, and substituting for ¢ and zw, their values, we have for the probability of a verdict of cuilty by the first, and not guilty by the second, c(l—e, )=u(1—w. )p-+(1—w)u.(1—p). DECISIONS OF JURIES AND TRIBUNALS. 119 In like manner, if the-accused has been acquitted by the first juror, the presumption of his guilt becomes #’,, and the probability of a verdict guilty by the second is 6,62; +(1—u,)(1—w’,); therefore the probability of a verdict of not guilty by the first, and of guilty by the second is (1—c)e, =(1—u) up + u(1—u, )(1—p). The sum of these two expressions gives for the probability of a discordant verdict, w(1—u,)-+- (1A—w)uy. 80. If we now suppose u=u,, and make 1—u=w, the probability that the two jurors will agree in their verdict, whether they are both right or both wrong, is u?-+-w" ; and the probability of a discordant verdict wv--uw=2ue. The sum of the two expressions is wv? 2uw-+-w?=(u-4w)? ; and therefore the probabilities of the different cases are respec- tively given by the developement of the binomial (w+-w)’. By pursuing this reasoning, it is easy to see that if there be any number / whatever of jurors, or voters on any ques- tion which admits only of simple affirmation or negation, all being supposed to possess the same integrity and know- ledge, so that there is the same probability u of a correct decision in respect of each, the probablities of the different cases are found by the development of the binomial (w-+-w)". The probability of a correct verdict being pronounced una- nimously is w*; of an erroneous one being pronounced una- nimously is w*; and the probability that a correct verdict will be given by m of the jurors, and an erroneous one by be 2 Gers alt . Mee SU eae coda 81. The probability that the accused will be pronounced n, is Uu"w", where U = guilty by m jurors, and acquitted by 7, on the supposition that the value of w is the same for each juror, is thus found. There are two ways in which this event may take place; aT? Pe ee ee ee ee ee ee DRIER at Die AR TALI MEE tee St Bite eK. aS eT ee. 120 : APPLICATION TO THE Ist, if the accused be guilty (the probability of which is p), and m jurors decide correctly, and 2 wrongly (the probabi- lity of which is Uw"w") ; the probability of the condemna- tion taking place in this way is therefore Uuw"w"p. 2d, If the accused be innocent (the probability of which is g) and n jurors decide rightly, and m wrongly (the probability of which is Uu"w™); the probability of the event taking place in this way is therefore Uu"w"g. Let G therefore denote the whole probability of the verdict, and we have G=U(u"w"p + u"w""q). Hence the probability that the accused will be condemned unanimously by a jury consisting of / jurors is u’p+w'g ; and the probability that he will be unanimously acquitted u'g+tu"p. 82. Suppose the accused to have been pronounced guilty by m jurors, and not guilty by jurors, the probability of the verdict of the majority being correct is found from the formula in (49). Two hypotheses may be made: Ist, the accused is guilty ; 2d, he isinnocent. The probability P, of the observed event (the condemnation by m, and acquit- tal by 2 jurors) on the first hypothesis is Uw"w"; and the a priori probabilities of the two hypotheses (or the proba- bilities denoted by A, and A, in (49), being p and gq ; there- fore if a, denote the probability of the verdict being cor- rect, that is, the probability of the first hypothesis after the verdict has been pronounced, and 2a, the probability of its being wrong, we shall have (49) ” i | ae ees ‘, 2 4 . q is { f | ia ’ i oS Be | i Hh | ae, | +e q - } +e i! ¢ : ; ) \ | i : { a i 4a f : i! { } 4 is Ry {i ie | dae Gy Se | ab st j iat || } 48 im | | Sey me i | i fi =i. | ie Sade | * wie i ie Te aa { { ae | ; 7 ; a 23% 4 | wit. A giate he | pay I *' P| iy | ig ay eee Sip | 3 | } \ # ; i bh Ni) | i | H a | i 1 I f f aa ep y 2 ey. | | K, te i { { a pr = Bas Gales eee a ent eh so oe Tn ere pr TN ee ee sow a Bere u™w"p uw g a= we oe 1 u™w"p+urwng nd ump + urwg If the verdict-has been pronounced unanimously, then mah and n=0, and the formule become rien eres peels: Sr Se bengseoe heed co ery meee ME SSR 5S sane ne TG Sect *) aan germane prem me nein + = A he DECISIONS OF JURIES AND TRIBUNALS. 134 up wg @,=——_—; @.=————_ . 1 u'p-+-wg g up -+w'g If p=q=3, and m—n=z, we have then untipy nr ui oa = baiestt Noni — gin gia coe Urting” Lurunrtt ula w But this is the probability of a verdict being correct which has been pronounced unanimously by z jurors ; whence it follows that the probability of a decision rendered by a given majority being correct, is the same as that of a decision ren- dered unanimously by ajury equal in number to the differ- ence between the majority and minority, ‘and is therefore independent of the total number of jurors. This, however, is only true on the supposition that the value of wu is known a prior? ; for if w be not absolutely known, the weight of the i ae ae verdict depends onthe ratio of the majority to the whole num- cid as ber of jurors. This is in accordance with common notions, for it will readily be admitted that a verdict given unani- { Y i i i mously by a jury of 10 will be entitled to much more weight than one pronounced by a jury consisting of a large num- ber, as 100, in which 55 are of one opinion, and 45 of the Tita Bs ai ie opposite. In this case, the-opinion of the minority throws great doubt on the correctness of the verdict. It is to be observed, however, that the probability of a verdict being ee ee ee ee SEPA ND REELING tS eet 2 given by a small majority becomes less and less as the num- ber of jurors is increased. 83. When the number who dissent from the opinion of the majority is unknown, and we merely know that the ma- jority exceeds the minority by aé least 7 jurors, the proba- bility of the verdict being correct is found as follows. Sup- pose the verdict to be guilty. On the hypothesis that it is correct, the probability of the accused being found guilty by A—=a, and not guilty by 2 jurors, is by the formula in (80), G = oe en ; SARL AE EVIE om Ho eS ae RR A as sey ee a FSET TON sadn : a t 132 APPLICATION TO THE Uu'—w*. Now, if we give x successively all the values 0, 1, 2,...%, where x=43(h—~), and assume U, to denote the value of U when 20, U, its value when x1, and so on ; and also make W=sthe probability of the accused being pronounced guilty by —n at least, we shall have WU + U uO w+ U uw? fe Ue. In like manner, if W’ denote the probability of a verdict guilty by h—n jurors at least, on the hypothesis that the accused is not guilty, we shall have, WU w+ U ww 4 U o,f Ue 5 whence, p and q being as above the a priori probabilities of the two hypotheses, the probability that the verdict guilty is correct, when pronounced by h— jurors at least, becomes Wp-—(Wp-+ W’g.) 84. It is evident that no application can be made of these formule without assigning arbitrary values to u and p, un- less, indeed, we have data for determining their mean values from experience. With respect to p, we may assume, for the sake of shewing the general consequences of the formu- lee, its value to be 4 ; for it cannot well be supposed less than 4, or that a person brought before a jury is more likely to be innocent than guilty; and if it much exceeds $ and ap- proaches to unity, a verdict of guilty may be expected from any jury, however constituted. When a mean value of u cannot be determined from experience, the only way of ob- _ taining numerical results, is to suppose « to have all possible values within given limits, and to integrate the equations between those limits. As it seems unreasonable to suppose that a juror is more likely to give a wrong verdict than a right one, we may assume that wu cannot be less than 4. Suppose, then, that «increases by infinitely small increments SETS ———— ET = PEs —— g an DECISIONS OF JURIES AND TRIBUNALS. 123 from w==4 to w=1, and let it be proposed to determine the probability that a decision is correct when the accused has been pronounced guilty by m jurors, and not guilty by x. Here an infinite number of hypotheses may be made re- specting the value of w, and we must therefore have recourse to the formule in (51.) Let w=a be one of those hypo- theses, P,= the probability on that hypothesis of the event observed (that is, of the accused being pronounced guilty by m, and not guilty by 2 jurors,) a= the probability of the assumed hypothesis, and I1 = the mean probability of the correctness of the verdict from all the hypotheses. By the formulze in (81) we have P,=U fa"(1—a)"p-+ar(1—2)" (1p), and as all the hypotheses are supposed equally probable, we have (45) w,=P,-+3P, But between the proposed limits P,=p f2"(1 —x)"dx + (1—p) fa" (1—2x)"da if, therefore, we make p=4, we shall have by reason of f ta"(1—_a)"de= fem and PCE, bop qe 1ym(1—ax)dz, and therefore et eo (1—2v)" + a" ead ite ‘ Vy, ao ( 1—a)"dx for the probability of the hypothesis. But (82) the proba- a v 1 ‘ i t {i vi 1 bi ' qe 1 | f bi} bility on this hypothesis of the accused being guilty, is a” (1—ax)” ity of the hypothesis, z,, we obt for the probability of the verdict being correct 2"(1—a)"=+ fla™(1—a) "dx ; and, therefore, for the probability of the verdict being cor- ; multiplying this by the probabil- ERE FEDS OFERTAS SoBe rect on all the hypotheses from #=4 to «=I, fyu™(—ax)dx fam 1—a)y dz ge te it mare = il ammaidace Sa ak oar agenesis sea, = te REE AIRS ene oe a ET RPL ACL REDO GILT A AR OLE PERALTA PRED tora! i pete nor, sete rm gees _ —— ieee 124 APPLICATION TO THE Hence the probability that a verdict given by a majority m out of m-+-n—h jurors is wrong, is fiam(—a)"dx —T=: aa (l—a)"dx which, on effecting the integrations by the formula in (51) ] becomes after reduction 1 {1 fips Mech Ue ise ae Qh+l 1 Dae Lag (h4 1)hA—1)..---- (h—n 4-2) \ i V+) Doe 3 sive vvoneevest es n Assuming / (the number of jurors) =12, and making 2 successively 0, 1, 2, 3, 4, 5, the series gives 1 14 92 378 1093 2380 Sigz sige’ 8192’ 8192 8192’ 8192’ for the respective probabilities of the error of a verdict when pronounced unanimously by 12 jurors, by a majority of 11 to 1, of 10 to 2, of 9 to 3, of 8 to 4, and of 7 to 5. Inthe last case the probability of the error is nearly =¥. 85, From these results it appears that the chance of a verdict being wrong which has been pronounced unanimous- ly by twelve jurors is very small; but it is to be remarked, that they have been deduced on the supposition that the unanimity proceeds from agreement in the same opinion, and that the jurors are unbiassed by each other. In this country, where unanimity is compelled by law, the mean probability ofa correct verdict can scarcely be considered as greater than that of a verdict pronounced by a simple majority; for, though in most cases the verdict may be supposed to repre- sent the opinion of a larger majority than seven, it may happen, not unfrequently, that a smaller number than five, possessing greater energy or perseverance, may persuade the DECISIONS OF JURIES AND TRIBUNALS. 125 others intoa surrender of their judgment. In fact, unless the presumption of the guilt of the accused be very great, it would scarcely be possible, without concert, to procure an unanimous verdict in any case. It is also to be observed, that the assumption of all values of w from § to 1 being equal- ly probable, may lead to results widely different from the truth. The mean value of u, which depends on the general intelligence of the class of persons from amongst whom the lists of jurors are made up, can only be rightly determined from data furnished by experience. One of the elements, however, which require to be known for this purpose, is the number of jurors who concur in, and dissent from, the verdict. The forced unanimity of the law renders it impossible to obtain this element from the records of the English courts; but in France and Belgium, where the majority and minority are known and recorded, the same obstacle does. not exist, and the “ Comptes Généraux de [Administration de la Justice Criminelle,” published by the French Government, have enabled Poisson to deduce mean values of w and p for that country, and consequently to obtain the necessary data for one of the most interesting applications of the theory of Probabilities. The general results were as follows: During the six years from 1825 to 1830 inclusive, the system of cri- 4 $ 4 + a minal legislation in France underwent no change ; the jury consisted of 12, and a simple majority was only required to concur, though when it happened that the majority was the least possible, the Court had power to overrule the verdict. On comparing, according to the rules of the theory, the ver- dicts given in the cases tried before the criminal courts ERIN AU TA ila POSE BR during those six years, it was found that for the whole of France, the probability (w) of a juror giving a correct verdict was a little greater than 3 with respect to crimes 126 APPLICATION TO THE against the person, and nearly equal to 14 with respect to crimes against property ; without distinction of the species of crime, it was found to be a very little below 2. The other element, the probability (p) of the guilt of the accused be- fore the trial, was found not much to exceed 4 (being be- tween 0.53 and 0.54) with respect to crimes against the person, while it a little exceeded 2 in respect of crimes against property. Without distinction of crime, its value was very nearly 0.64. 86. On substituting these values of w and p (namely u== 2, p=.64, whence w=, g=.36) in the formula in (81), and making m=7, n=5, and consequently TEST HAD Eas = 7% we have Ga (37% .64-+35 x .36)=.07 nearly. Hence it may be ex- pected, that in a hundred trials it will happen only seven te times that the accused will be pronounced guilty by the smallest possible majority. If m—=12, and n=0, we shall have u*p--w'q=.02027=;, nearly, for the probability of an unanimous verdict of guilty, and uv*¢--w*p=.0114 for the probability of an unanimous verdict of not guilty. Making the same substitutions in the formula in (82), we have for the probability of a verdict guilty being correct, from which 5 jurors out of 12 dissent, a;=+$ ; anda ,= +4 for the probability of its being wrong. Substituting the same values in the series represented by W and W’ in (83), and supposing ” to have all values from n=0 to n=5, there results W= es x 7254, W’= 1 W. 126915984 118 4B xX 239122, whence Woa.Wa = 727992033 = 119 nearly. ‘This is the probability that a verdict guilty, pro- DECISIONS OF JURIES AND TRIBUNALS. 127 nounced by amajority of seven against five atleast, is correct. The probability of the same verdict being wrong, is there- fore zt, ; so that out of 119 verdicts, respecting which we know nothing else, than that seven at least of the jury con- curred in finding the accused guilty, we may expect one to be wrong, or that one person out of 119 so condemned will be innocent. 128 SOLUTION OF QUESTIONS amas SECTION VIII. OF THE SOLUTION OF QUESTIONS INVOLVING LARGE NUMBERS. 87. The probabilities of the different compound events which can result from the combination of any number of ee pee the nett etme eee sencnct ua % = N simple events, E,, E,, E,, &c. being (13) measured respec- tively by the several terms of the developement of the mul- tinomial (p+q+7-+ &c.)*, the most probable of those com- pound events will be that which corresponds to the term SAR aa gE RE having the greatest numerical value. Let us consider the oer griceeen eniere wee case of two simple contrary events E and F, the proba- bilities of which are respectively p and q, and suppose the is in to bay Hh a } % 1% {/ number of occurrences to be 4. Neglecting the order of — te a ST Ra Se ee occurrence, the different combinations, with their respec- tive probabilities, are the following : EEEE, EEEF, EEFF, EFFF, FFFF, p*, 4p?q, Op*g®, 4pq®, — qt Now it is evident that the numerical values of these proba- chicory parer tty ayo teenie eae Seance ie Paks SI Sea Se bilities depend on the ratio of p to q, as well as on the co- efficient by which they are multiplied, and that values may be given to p and gq, such that any one of the terms may be made the greatest or the least in the series. If we suppose p= and consequently p=3, g=4, (since p-+-g=1) the probabilities of the different cases become respectively 24 eV 4 4 nee Chee Maer kyr B47) Dt ee? Ore ee an eee ee . Se a ae oe i aaa a ST NS SS — ‘ s ? ee SES = we ny ete teen cme miteninag SEN, tiapott ten: eee ee ge ae ee ae ee potion - $ Bry A Ba dO iis eee ae ee Sa Se ee >on : pe dns Sata Ze ee ee * See J SE TNR een re ne cic Ay he ne ps Se INVOLVING LARGE NUMBERS. 129 whence it appears, that the most probable combination is that which corresponds to 6p?g?, or in which each of the simple events occurs twice, the probability of this combina- tion being 58;, while that of either of the simple events oc- curring four times in succession is only 7. When the number of trials is 5, the probabilities of the several cases are respectively Pp’, Sp*q, 10p?g*, 10p%9*, Spq*, 9°, which, when p=q, become 39 35 42, 52) 3m 399 so that there are two different combinations equally pro- bable, namely, that in which E occurs three times and F twice, and that in which E occurs twice and F three times ; and of the six possible combinations these two are the most probable, having in their favour a number of chances twice as great as the two cases in which one of the events occurs only once, and the other four times, and ten times greater than the two cases in which either of the simple events oc- curs in each of the five trials. From these two instances it may be inferred in general, “<’ that when / is an even number,, the most probable com- pound event is that of which the probability is represented by the middle term of the developement of (p-+-q)"; and that when / is an odd number, there are two compound events equally probable, and more probable than any other, namely, those corresponding to the terms which occupy the middle of the series, supposing in both cases p=q. This supposition gives (p+q)’=(1+1)"(3)’; therefore in the case in which / is an even number, the general expression for the greatest term is h(h—1)(h—2)......(h—3h + 1) (ay; |p GRO See Lh ae 1 a 4 t : i ; CDF ei AGS PEAR OT OT I EN ELLIO IT po TET RE EIT RA I a EP TES ON ae A i ae ee TE "UREFRY IN ged im mana tas BY I natn tang fs foe ah Fos SERS we i= eee iaas aS Se — 4 RR omenerpee t= i Scie Bee ee pee ty SPM aac ES as — writ 3 to —— ad ae TEE "a Taek GE | 4 ie), 3 mre | pi E i H , é 3, ES ae 130 SOLUTION OF QUESTIONS and when h is odd, the general expression for either of the two equal terms, which are greater than any of the other terms, is h(h—1)(h—2)......{h—4(A+ 1) 41} ay 1 fO9S CP Eee Pana) a 88. When p and g are unequal, the greatest term of the expansion of (p+q)" will not occupy the middle of the series, but its place may be found by comparing two con- secutive terms. Leth=m+zn. The general term of the series then becomes U2 Ponte eh 7 | oa [20S Teese CONS.Sh en ee and the term immediately preceding is 182 SSR ee - Ps, 1.2.3......(m-1)X1.2.3......(a—l) Dividing the first ‘of these by the second, we get for the wk quotient (m-+-1)q--np, which, therefore, is the ratio of two consecutive terms taken at any part of the series. If this ratio be greater than 1, the term which has been taken as the dividend is greater than the preceding one which has been taken as the divisor ; and it is evident that the terms must go on increasing, from the beginning of the se- ries, so long as the ratio in question is greater than 1. But if the ratio be less than 1, the preceding term is greater than the succeeding, and the terms will become less and less as they are nearer the end of the series. Let (m-+-1)q+-p=1; then, since p-+-q=1, and m+-n=A, we have n=(h-+1)q, and consequently the ratio of any term to the next preced- ing is greater or less than 1 according as 7 is less or greater than (A+-1)g. Now » is necessarily a whole num- ber; therefore if (4+ 1)gbeawholenumber, taken=(h-+ 1)q, and the two terms of the series given by the expansion 4A P FPL Ft & 7 INVOLVING LARGE NUMBERS. 131 of (p-+-q)’, in which the exponents of q are x—1 and 2, will be equal to each other, and each greater than any other term of the series. But if (2-+-1)q be not a whole number, let (h4-1)qg—w be the nearest whole number /ess than (h4+-1)g, and make n=(h +1)q—z«; then the greatest term of the developement will be that in which the exponent of q is 2. Sincen=(h-+ 1)q-7,we have g=(n-+ 2) +(A-+1), whence n+x2 m+)1—x fete: N+ x _—1|— ——- = — her ga 3h Tn | hl Pion and therefore Aaa Now x is by hypothesis less than 1, therefore if m and x are large numbers, we have, very nearly, g:p=m:m; or, since m+n=h, m=hp, n=hq. It follows therefore, that the greatest term of the developement of the binomial (p-+q)’ is that in which the exponents of p and q are to each other in the ratio of p to q, or more nearly in that ratio than are any other two numbers whose sum is 4. In other words, the most probable combination of two simple events, E and F, in any number of trials, is that in which the num- ber of occurrences of E is to the number of occurrences of “7 F in the ratio of their respective probabilities. 89. In the same manner it may be shewn, that when there are more than two simple events, of which one must occur in every trial, the most probable result of any number ,/; of trials is that combination in which the number of repe- titions of each simple event is in proportion to its probabi- lity in a single trial. Thus, the probabilities of the simple events being respectively p, q, 7, &c. the most probable com- pound event is that whose probability is expressed by that term of the expansion of (ptqatr+ ke.) which has for its argument p"”, gq”, 7", &c. 90. Having determined the form of the greatest term of ee — eee CMTS, RRM ee ret Rein ob wa a me ? e SS ee UE ae LT RAS Bees | —— iil on cain = 132 SOLUTION OF QUESTIONS. the series, we have next to find a method of approximating to its numerical value ; for its coefficient containing the pro- duct of the natural numbers from 1 to / inclusive, its di- rect calculation becomes impracticable even when h is only a moderately large number. The theorem which gives the approximate value of this product is known by the name of Stirling's Theorem, having been discovered by that mathe- matician.- As its investigation is a matter of pure analysis, we shall not stop to give it here, but refer the reader to the Treatise on Differences and Series, by Sir John Herschel, in the translation of Lacroix’s Elementary Treatise on the Differential and Integral Calculus, p-568.! The theorem is as follows: Let # be any number, then ] 1 : 1.2.3......emate*/2ra(l+ 75- + sgn t &e-) where e is the number of which the Napierean logarithm is unit, or. the number 2°71828, and za the ratio of the cir- cumference of a circle to the diameter, or 3:14159. When « is a large number, the term divided by 122 be- comes very small, and the series within the brackets may be considered as equal to unity. In this case, then, the for- mula becomes 1.2. 3......0= ae", / Irae, which gives a sufficient approximation in most cases. If, for example, z=1000, the result will be within a 12000th part of the truth. ane Now, let E and F be two events of such a nature that the _ one or the other must happen in every trial ; let p and qg be their respective probabilities, and P the probability that in a * Stirling’s investigation of the theorem, or rather of its equivalent, to find the sum of the logarithms of a series of numbers in arithme- , tical progression, is given in his Methodus Differentials, p. 135. INVOLVING LARGE NUMBERS. 133 m-+-n=h trials, E will happen m times and F 2 times ; then by (12) we have /Anléja=o ) ESE Site teh tot TAPES ees 2 aod pes Lbs When m, m, and / are large numbers, the value of this ae coefficient may be computed from the above formula, which gives . 1S OP (coed rer Vy ey 1.2.8......m=m™e—™s/ Inm, rt 1.2.3......m=n"e—"/ Inn, nt k whence d \ a ee i | a. Ate h std png (es, (-2) of h ve rt 2) tn § mnre—(™+n) / (2armn) m n 2rmn ° Me ae This expression represents any term of the series ( P+9). ee &> i The greatest term, which corresponds to the most probable | result, is (88) that in which m and 7 are to each other in i the ratio of p to g, or when m=/p, and n=hq. Let the : greatest term therefore be denoted by P,, that is to say, let P, be the chance of the most probable result of / trials, and we shall have : ( P.=V (h-+-2rmn), or P= (1-+2rhpq). : This last formula shews that the absolute probability of that combination which has the greatest number of chances in its favour becomes less and less as the number of trials is i | a. en Ags a MLE GAT aA ODE ERAS LARS OSB increased ; for the fraction 1+, to the square root of which the probability is proportional, diminishes as / is ir- --creased. a ~ 91. As an example, suppose a shilling to be tossed 100 times in succession. In this case p=q=3, Ap=50, hg=50, and the most probable result of the trials is 50 times head and 50 times tail. .We have then A=100, m=650, and VW (h--2x2mn) =1~-4/(507) for the measure of the probabi- - att Les Se aa se = SRI Se a a RG CRT EEO ie ee ne ee eee Aen Eh te Pg S en EE IO OS ee ee we oe, ores nomen neiennitee 134 SOLUTION OF QUESTIONS lity that the event will happen in this way exactly. On cal- culation, this is found =.07979 ; whence it appears, that although 50 heads and 50 tails is a more probable result of 100 trials than any other combination which can be named, its absolute probability is measured by a very small fraction. The probability of the contrary event, or that there will not be thrown 50 heads and 50 tails exactly, is1—.07979=.92021, so that the odds against the event are about 92 to 8, or 23 to 2. Had the number of trials been 1000, the probability of 500 times head and 500 times tail exactly, though more likely to occur than any other combination, would have been found 1~—-/ (5002); that is to say, “10 times, or rather more than 3 times less than in the former case. In general, when the chances in favour of the simple events are equal, the pro- bability of the combination which is more likely to happen than any other, is inversely proportional to the square root of the number of trials. 92. The formule in (90) enable us also to determine the ratio of the greatest term of the developement of (p+q)" to any other term of the series, and consequently the rela- tion of the probabilities of the different compound events. Let m:n:: p:q, whence m=hp and n=/q, and let P, de- note the probability that in / trials the event E will occur (m—zs) times, and the event F (n-+-2) times, the probabi- lities of the simple events E and F being respectively p and g: By (13) we have P 1 PA2 shan cot en ss h nae a |RSS aa (m—#) X1.2.3..1...(n4a)! 42 which by (90) becomes pe hhe—hay (27h) * (m—a)ym—te—(m—z) 4/ Qa(m—a)x(n+ar)"+%e—(2+2),/ 2a(n+2) x p™—*g" t+; whence, substituting m+h for p, and x+h INVOLVING LARGE NUMBERS. 135 for g, and leaving out the factors common to the numera- tor and denominator, we find, ae v4 (=) (m—a)—™t7-H(n 4-2)" pene”. Now log (m—x)—"t?-t= (—_m 4+«—4) log (m—w) ; and log (m—x) = log m— — — —, —Xe. therefore log (m—x)—™t? $= (—m + 2—4) log m—(—m-++-«x—4) (= + x oe sc.) whence, neglecting terms divided by m?, m>, &c., m being osed to be a large number in mes hb with 2, supp log (m—x)—™+2—-4=(—m 4- x —4) log m + x— g ( ) ( ++ :) 5 + aa +> = 3 therefore, on passing to numbers, x2 a Cr—eye — m—mt2—4 x Ee — Im Ke I 5 z 2 ; om x ¥ or, since e~ =1+ — &e. ? + 2m a) 2.4m? a r m—x)—~" te =m" +2 —4 e Nera ae Belporege \. ( ) 2m } In like manner, by changing m into n, and x into —2, we get x2 (NES eae Oma od ase eae i n and the second by 2”+?, we have x2 ais oes veg. Di; xv 4 a \—m-- t— 97 Nk moe —s od (m—x) x m"—* =m-*te (1 he ‘i r2 (n4a)-" te K n+? =n-} aes = aa . ), n ~ Multiplying the first of these two expressions by m~", 136 SOLUTION OF QUESTIONS whence, substituting these values in that of P,, and ne- glecting the quantity divided by mz, h pa LAL h By ee r= (= ) Pen eerie (ea (amy ag The term of the series (p+q)" which corresponds to this _ | | value of P, is that which is 2 places to the right ,of the (4 KRY greatest term; and it has been shewn, (90), that the great- est term has for its expression W(h+-2nmm) ; therefore the greatest term being denoted, as before, by P,, and the term which comes after it # places by P,, we have a | ier —Pp Oe? <2mn, ¢ at | ti i that is to say, the probability the event E will hone m | iF if times and fail times in m + x trials, is to the probability heay Welt ih ; . : ate : : 1 t ' 7 of its happening (m—zx) times and failing (n+4-x) times in i ' the ratio of 1 to e—h#? +2mn, c Since the numbers m and 7 enter symmetrically into the i expression ¢—?°+2m, it is evident that the result would 2 1 1 have been the same if, instead of seeking the ratio of the ors emt aa re ee ee nc eta greatest term to that which succeeds it by x places, we had sought the ratio of the greatest term to that which precedes i a oe alee ee ee RO RE a ES it by x places. Hence if the most probable result of m--n trials be that E will happen m times and fail » times, the’ probability that it will happen m—za times and fail N+-xX times is the same as the probability that it will happen m +2 times and fail n—a times. ae ee Se ee Se Pr ae epee esr . noe : = on nani ie eneeiehonaereattdacemmnetonbenn: "eraser pata eat See SES RIOTS ee ~ The following example will suffice to shew the applica- tion of the formula: A die is thrown 6000 times, required the probability that the number of aces turned up will be exactly 960? Here p, the chance of throwing ace, is > 9=2, and h=6000; whence m=hp=1000, and n=hq=5000. We have first to find P,, the chance of the most probable result, week aro + — A ae ae a Jeep peniine a t — Sn elses RN eT ERISA EE A Se SEL SE . oe eee es * INVOLVING LARGE NUMBERS. 137 ee ee or of 1000 aces. By (90), P,=W(A+2mmn); whence, substituting the above values, Pp=3 + (5000 X 3.14159). On performing the operation indicated , by the logarithmic tables, we get log P,==8.14050, whence P0138: The calculation of e—***2™ is as follows: Assumé,’ © ui —hxe?+~2mn. We have r=1000—960—40. WT? log 40=1.60206 2 a nt er ee rn ~— enemy a 3.20412 log A= log 6600= 3.77815 log ha? = 6.98227 log 2mn= log 10,000,000=7 log #?==9.98227 log e==.43429, log 43429=9.63778 log (@X 43429 )=9.62005 t? x 434292? log e= .41692 _#? log e=9.58308 add log P,=8.14050 log P,=7.72358 therefore P,=.0053, which is the chance of 960 aces ex- actly. The odds against this event are therefore 9947 to 53, or nearly 188 to l. 93. When h, m, and x are large numbers, and 2# is small, the exponential e—"*** 2 differs little from unity, and it de- creases slowly as x increases, so long as x is small in compa- rison of m and n. Suppose m=n and r=4/ m, it becomes ». 1 1 on “e 27182818 10th term before or after the greatest would still exceed the = so that if we assume m=100, the’ *"* si af 3d part of the greatest. But when a becomes greater than SNORT NEE pe SO CR SEL mY oe A Se saa ne nanan ~ See ee a rine : : wpe eee OPS aimee CRT ee —— mtn pare rr SEE eee to ERO NT negreneeeerr ei ot Oe meetin cei EE peered se ong OS eee a ~o— i , | it : 138 SOLUTION OF QUESTIONS /m or 4/n the exponential, and consequently also the terms which are multiplied by it, begin to diminish with great rapidity, and the diminution is more rapid as x increases. Cm Ifm=n= 100, and x=50, then the exponent. hz? —2mn=25, 6¢ so that e—%*?+2mn, —] 5, a quantity witole is altogether insensible, We may therefore conclude generally that when his a large number, the principal terms of the developement of (p-+-q)" are those which are near the greatest term, and that h may be taken so large that the terms towards the beginning or end of the series may at length become smaller than any assignable quantity. 94. From the proposition which has now been demon- strated it follows, that although the probability of that par- ticular compound event which has the greatest number of chances in its favour is very small when the number of trials is great, yet on account of the rapid diminution of the terms towards the beginning‘and end of the series, the sum of acom- paratively small number of terms taken on both sides of the greatest, may be very much greater than all the remaining terms of the series; and, consequently, there will bea very great probability that the compound event will be repre- sented by one or other of those terms. This consideration leads us to one of the most important questions in the theory, namely, to determine the probability that in a large number of trials, h, an event E, which must either happen or fail in each trial, and of which the chance of happening in any trial isp, will happen not less than 4p—Zd times, and not oftener than hp+Z times; or, making hp=m, hg=n, to de- termine the probability that the number of occurrences of E will be included between the limits m=/. Let x be any number between 0 and 7. Then (92) the probability that E will occur (m—z) times and fail (n-+- x) INVOLVING LARGE NUMBERS. 139 times is P, =P,e—**= 2""(where Pp=W(h--27mn). Nowif in this expression we make x successively equal to each of the numbers0, 1, 2,...d, we shall have the respective probabilities of E happening m, m—1, m—2,......m—Z times in / trials ; and the sum of these probabilities will be the probability that — E happens not oftener than m times, and not seldomer than m—TI times. The same suppositions with respect to 2 will give the probabilitiesof E happening m,m-+- 1, m-+-2,. ml times, the sum of which will be the probability that E happens not seldomer than m times, and noé¢ oftener than m--/ times. Adding, therefore, those two sums, and deducting P, the pro- bility which corresponds to z=0, on account of its being in- cluded in each sum, and therefore having been counted twice, the result will be the sum of the terms of the binomial (p+q)" comprised between, and including, the two terms of which the first has for a factor p"+’, and the last p™—’, and will therefore express the probability that the number of occurrences of E will fall within the limits m=k/. Let this probability be denoted by R, and let SP, represent the sum of all the values of P, obtained by substituting suc- cessively 0, 1, 2, 3,...2 for w, we then have R=2SP,—P,, whence, writing for P, and P,, their values, R=28,/ (== Je pe ee 95. In order to find an approximate value of this expres- Qrmn Qrmn sion we must have recourse to a formula first given by Euler for converting sums of the kind denoted by S into definite integrals (for which see Lacroix, Traité du Caleul Différen- tiel et Inté gral, _tom. ii. p. 136, or Herschel’s Treatise on Differ inch p- "613). Assuming wu to denote a function of x, the formula is as follows: EGP ian SEITE BND ic LEASE A SE REE TR BBS RE Pim ae te Karier yy Sa I EE ea - . > . Pin mate Selene es ee een ee Sg epee oer LR = : ’ rae go 2mn ; therefore, if we suppose x to be not greater than 4/m or 4/n, this differential coefficient is of the order 1+-h, (as may be easily shewn by substituting hp for m, and hq for n), and may be rejected, since / is supposed to be a very large number. The above equation therefore becomes SP,=P, fe? emnda +. dP oe—ha2= 2mn 41 constant 5 and on supposing x=0 this gives O—=—4P,+ constant, therefore the constant is equal to $P,, and we have SP, =P fom "itd: 1p AEE 2P,. Assume eR Hh. 200), whence dt=-dny gee substitute these in the above equation, and it becomes by reason of Pp=/(h-+-27mn), 1 9g ri 9 SP.= 7 edt 4-4P,e-? +P whence, from the equation R=2SP,—P,, we obtain | R= = edt +- P,e-®. The integral in this expression must be taken between the limits a= and xl. , arr ed aicissiceaeatttae ie mae Fe ET ge ee Fe ae Se EE aaoee TO * cd menenenn pen si OT NEE oe iste ETL sore 44 SOLUTION OF QUESTIONS the square root of h, and consequently the greater the number of trials, the smaller w ill /be in proportion to that number. Thus, if the number of trials be 1000, and we have a given probability R that the number of occurrences > of E will not differ more than 10 from the number which is the most, probable of all (that is, from 1000 p), then if we take 100,000 trials, we shall have the same probability R that the number of occurrences of E will not differ more than 10 X 4/100=100 from the most probable number. But a difference of 10 in 1000 is 1-100th of the whole, whilst a dif- ference of 100 in 100,000 is 1-1000th of the whole, and thus the ratio of J to A becomes smaller and smaller, or the ratio of the occurrences of E to the whole number of trials ap- proaches nearer and nearer to p, as the number of trials is increased ; and the experiments may be repeated until the difference between p and p=t/—h, in respect of a given probability R which may be as great as we please, shall be less than any assignable quantity. If, on the other hand, we suppose /--/ to be constant, then 7 is proportional to the square root of the number of trials. But as 7 increases, ©, and consequently R, approaches nearer and nearer to unity, (and it may be seen, by referring to the table, that it is only necessary to have r=3 in order to have ©=:9999779) ; whence the number of trials 4 may always be increased until we obtain a probability approach- ing as nearly to certainty as we please, that the number of occurrences of E will be comprised within the given limits (hp==l); or, which is the same thing, that the ratio of the number of occurrences of E to the whole number of trials, shall not differ from p, the probability of E in a single trial, more than a given quantity. /+A which may be less than any assigned fraction.” This is the celebrated theorem INVOLVING LARGE NUMBERS. 145 which was demonstrated by James Bernoulli in the Ars Conjectandi. 98. ‘The application of the preceding results to numeri- cal examples, is rendered extremely easy by means of the table of the values of ©. From the formula in (95) we have the probability =e+P.¢-72 | that the occurrences of E in A trials will fall within the | limits Ap=e/, the relation between / and r being given by the equation (=74/(2hpq). If, therefore, we suppose J to be given, r becomes known, and the corresponding value a of © is found from the table; and, conversely, if © be as- sumed, 7 is given by the table, whence the corresponding 4 limits 7 are deduced. With respect to the quantity P,e-’, wT e ' ve may observe that it denotes the probability that the 2 oy number of occurrences of the event E will be hp + Yor hp—t u// L, “SHA precisely (92), and is therefore always a very small fraction | when / is a large number (90). It may be regarded as a ie Hl correction of ©, which in most cases might be omitted with- out sensibly affecting the result; but when & is not very, , 7~ f } aie Ain ea oe Lae. large, or J is a small number, it becomes necessary to take it into account. In such cases its value may be computed~ : directly as in the example in (92); but this labour may be . avoided by increasing 7, so as to include it within the | limits of the integral @. Thus, let R be the probability | that the number of arrivals of E will be included within | | the limits 4px, and R’ the probability of the limits be- ing Ap==(v+1), and let @ and ©’ be respectively the correspondiug values of the integral. We have then, giving P, the same signification as in (92), the two equa- - tions GRAS eS R=0+P., R’= , =@ + Prt oR Re IEE ae Rt i. zs . types areal neetemren she ee soci aaa tyne Rw a a i an vei | = ee" & ‘ 2 . Se wh > y? : ; , = ye : ‘é fe fe ts g P* ibe fF ep he Pelee ' + , ‘ oe Sg. ‘a oe it 7 ni i) 146 SOLUTION OF QUESTIONS Yo, = / 4 7 and the difference R’—R of these two probabilities is ob- viously the double of the probability that the result of the trials will be either (Ap+#4-1) times E, or (Ap—ax—1) times E, exactly. But the chance of either of these events being P,4;, we have therefore R’—R=2P,4;. Now, when h is large, P, and P,4, are very small, and very nearly equal to each other, (their difference is in fact of the order of quan- tities omitted); hence R’—R=0©’—®, and also 2P,4;=2P,, and consequently ©’ —@=2P,, or P, =3(0’—@). Substi- tuting this valueof P, inthe equation R=e + P,, weget R= (0’+ 0); so that if we take from the table the values of 0’ and © corresponding to 7 and /+-1, half their sumwill give R. But as the interval between ©’ and © in the table is always small, half their sum will not differ sensibly from the value / of © corresponding to 7+4, whence this value of © is. equal to R, and we have the following rule for determining the limits corresponding to a given probability, or vice VETSA :— ae mre, f gees When the limits are assumed, find + from the equation 14+1=7/(2hpq); then the value of © in the table, corre- 7" sponding to r is the probability that in / trials the number of occurrences of the event E, the chance of which in a sin- gle trial is p, will lie within the limits Ap=/ both inclusive. Conversely, when © is assumed, find the corresponding va= * lue of 7 in the table, by means of which the limit Z will be ~ given by the equation /4+-4=7V/(2hpq). It is obvious, that ™/~ if the limit Z and the probability © be both assumed, then -h may be determined from the same equation, 99. We will now give some examples of the application of the preceding formule. Suppose p=q=}, and h=200, and let it be proposed to assign the limits within which there is a probability =} / | INVOLVING LARGE NUMBERS. 147 that the number of occurrences of E will fall. In this case the equation 1+ 3=r4/(2hpq) becomes /4 5==r/ 100= 10r. Now, it is easily found from the table that for e==4 we have r=:4769, whence J+ }==4°769, and l=4:269. ~On tossing a shilling 200 times, it is therefore ‘more thar an even wager that head will turn up not seldomer than 935 times, and not eftener than 105 times. : | Po ‘Suppose p=g=4, /=3600, and letit be proposed toassign the probability that the number of occurrences,of E will not exceed the limits’ 1800-30. In this cA8evthe equation l4+45=71,/(2hpq) becomes 30°5=1o/ (2 X 900) =30r/ 2, whence r= 30°5-~ 304/2="7189;. and. the table gives e=-6907=2% nearly. Hence in tossing a shilling 3600 times, the odds are 28 to 13 that head will not turn up oftener than 1800-4 30=1830 times, nor seldomer than 1806 __30=1770 times. Neglecting the second term of R (95) and taking simply /= bd;, the table gives 6='6827, which . is the solution given by Demoivre, p. 245. Suppose p=}, q=&, and let it be proposed to determine how many trials must be made in order that it may be one to one that the number of occurrences of E. will not differ more than 10 from the most probable number. For e=} we have r= 4/69; therefore the equation J-4-1=7V/ (2hpq) becomes 10:5 —="4769V/(1 04-36), whence h=3-6(10-5 +-4769)?. Oncomputing this formula A isfound 17452. Say 1746, } of which is 291; and at follows that ifa die be thrown 1746 times it is an even wager that the “2” number of aces will fall between 291-10, that is,” be- tween 281 and 301, or be equal to one of those numbers. In (92) we found the probability to be :0053, that in / 6000 throws of a die the number of aces will be exactly y ed pan aeration area nea aT a TA Ato : . tenet oe Le, stitial ae neat =< nip. tienen conte ees ee es ——~ woven taeee Sarre er Ten eating vee rnce Ta SS ee ee Peete ae 148 SOLUTION OF QUESTIONS 960. Let it now be proposed to assign the probability ©, that in 6000 throws the number of aces will lie between 960 and 1040, that is, between 10004=40. Here h=6000, = 9= 3, and 40; the equation of the limits therefore becomes 40-5==1,4/(10000+6), whence ™='405 4/6992, corresponding to which the table gives @=:8394. The following question is discussed by Nicolas Ber- noulli in the Appendix to Montmort’s Analyse des Jeux de Hazard, and is noticed by Demoivre and Laplace. From the observations of the births of both sexes in London dur- ing 82 years (from 1629 to 1711) it was found that the aver- age number of children annually born in London, was about 14,000, and the ratio of the number of males to that of fe- males, was nearly as 18 to 17, the average number of male births being 7200, and of female births 6800. In the year in which the greatest difference from this ratio took place, the actual numbers were 7037 males and 6963 females, so that the difference from the average amounted to 163. Assuming, then, the comparative facility of male and female births to be as 18 to 17, required the probability that out of 14000 children born, the number of malesshall not be greater than 7363, nor less than 7037: This question is evidently equivalent to the following :— Let 14000 dice, each having 35 faces, 18 white and 17 black, be thrown ; what is the probability that the number of white faces turned up, will be comprised between the limits 7200-163. We have therefore h=1400, p=18 G=3%: =163, and the formula /-+-1—7,/(2hpq) becomes 163.5 =74/(2 x 14000 x 18 X 17)-+35, whence r= 1-955. The corresponding value of @ is found from the table—=9943, which is the probability that the number of white faces shall Sees Ss Se INVOLVING LARGE NUMBERS. 149 not be greater than 7363, nor less than 7037. _The odds in favour of the event are therefore 9943 to 57, or about 175 to l. > 100. We now proceed to consider the case in which the probabilities of the simple events are not known, @ priort, but inferred from the results of experience. It was shewn ~ in (52) that the probability 1 of an event happening 2’ times, and failing ’ times in A’ trials, (A'=m’ +n’), when it has been observed to happen m times, and fail 2 times in h previous trials, is expressed by this equation [m+m' lin+-n'][h+1] [min j[h+A’+1] ; Now, when m, n; m’, n’, are large numbers, an approxi- l= mate value of I, more accurate in proportion as those numbers become larger, is obtained from Stirling’s theorem (90), which for any number 2 gives Ce ]—are—*)/ (27x). Applying the theorem therefore to the expressions within the brackets in the above equation, and assuming rs h+1 (m+m’)(n+n)(A+1) K=-—_ ni haemuonen rd aa ig oa h+h'+1 mn(h +h’ +-1) we obtain, in consequence of m-+--n=A, gm 4-m/ yn (n +n’ rth +1)" Kat.” mn” (hh! 4-1 )r+ Let m’=6m, n/=6n, and consequently 4’ =6h; then taking hh fee (h+-1)" (h4h’)rtr (hh! + 1)err out sensible error, since / is by supposition a large num- ? I= (which may be done with- ber,) the equation becomes yd 4 Oymtmn(1-d)rrr hi | ae mn” ( 1+6)rt" s! or, since m+n2=A, m’ +m =", © ete aan nl cnt aR cht = 5 ann trerton seen aes me toe Lat AE eR Se ee ers CE EY em PR ee pee A ees ar ee re nes re DENSE an a SS 2S 150 SOLUTION OF QUESTIONS wi m™* nn (J oh pr -=UK(7)"( n K AY NRPS Making the same substitutiens in the expression denoted by K, we get, after reduction, K=] + /(1+6); whence, il VU’ m\m n\n ray: rene &. ; The value of 1 now found, is the probability that in a hh ( ] = Gyre future series of trials the ratio of the occurrences of E to those of F will be the same as in the preceding trials, which are supposed to have been very numerous. If the chances of E and F had been given a priori equal to mh and nh respectively, the probability of m’ times E, and x’ times F in m’+7’ future trials would have been P— u(z)" (=) by (12) ; hence (since m’+-h’=m—~h and n'—h'=n--h), the relation between the probability P, of that combination of simple events which has the greatest number of chances in its favour, when the chances of the simple events are known a priori, and the probability of the same combination when the chances of the simple events are only presumed from previous trials, is expressed by this equation, =P? +-4/(1 +0). 101. When A’ is very small in comparison of h, 6 be- comes a very small fraction, and may be neglected, and we have then N=P,. But when /’ is a number comparable with /, 11 is less than P,; and it diminishes rapidly when 6 exceeds 1. The reason of this is obvious. H the con- tents of the urn are not known a.priori, however numerous the trials may have been there is only a presumption that the chance of drawing a white ball in a single trial is measured by m+/; whereas, in the ease of the ratio of the balls being INVOLVING LARGE NUMBERS. 151 previously known, the measure of the probability is cer- tain. As an instance of the manner in which the proba- pility of an assigned series of future events diminishes, when the probabilities of the simple events are inferred from experience, let us suppose h’=h, whence 6=1, and consequently 1=P,+/2=7071 x P, Nowit was shewn in (91) that if a ball be drawn at random 100 times from an urn which contains an equal number of black and white balls, the probability P,, that the result will be 50 white balls, and 50 black, precisely, is -07979. It follows there- fore, that if the contents of the urn be unknown, and we can only judge of the relative numbers of the two sorts of balls it contains from having observed that in 100 trials there have been drawn 50 white balls and 50 black, the probability 1 of that combination in 100 future trials, be- comes 07979 X°7071=3 05642. %*: $ ied » pr 102. The result obtained in (100) enables us to de- termine the probability that the number of occurrences of E in /’ future trials, will not differ in excess or defect from the most probable number, by more than a certain given number J. It has been shewn (95) that in the case of the probabilities p and q of the simple events being given @ priori, if we determine 7 from the equation /=rv (2hp7),, the formula R=04V(1 -Inhpq)je—* gives the probability R that m will be comprised within the limits hp=terV/(2hpq)s or, dividing by h, the probability that the ratio of m to / will be comprised within the limits poerv (2pq--h)- Conversely !, whenpandqare not known, 1 This inference, though admitted by both Laplace and Poisson, is not strictly correct. In a paper published in the Transactions of the Cambridge Philosophical Society, (vol. vi- part iii.) Mr. De Mor- pre AT = er onastenarah arena saiteatnn emmandnditttis ‘that have been rejected in the approximations, 152 SOLUTION OF QUESTIONS but the event E has been observed to happen m times ink trials, then R=0-+-(h--2mn)e-™ fi? gives the probability R that Pp is comprised within the limits m ve 2mn ps h h h These limits approach more nearly to each other as h in- creases; and when his a large number, the ratios M321 ° . vA 2 -h maybe assumed, without sensible error, as the chances Ug 3 of I and F in computing the probable result of a future series of h’ trials, provided, however, that h’ (though abso- lutely a large number) be small relatively tof. When this_ A pete \i « j i, condition is not fulfilled, the assumption of m-+-h andn——-h as the a priori chances of E and F, might lead to consi- derable error; but an approximation to the limits corre- sponding to a given value.of R may be obtained from the following considerations :— Suppose a large number / of events to have been observ- ‘ed, and that the result of the observation gave m times E and m times F. Let a new series of j’ trials be made, and suppose that in this new series p is the real chance of E gan has shewn by a direet analysis that in the case of p and gq not being known a priori, but made equal to the observed ratios mh, n--h, the presumption of the true value of p lying within the limits : ; : I —r? stated in the text is not, as there inferred, O + ct ——- @ V (2ahpq) ‘ 13p?—13 | an : ; but aol Eg niet a hee last correetion to © is smaller ChpqVv « than the former, and being divided by h, is of the order of quantities It is right to state that the method of simplifying the calculation of R in the direct case, by taking the integral © between limits corresponding to /+-4 instead of J, is noticed, for the first time so far as we are aware, by Mr. De Morgan in the same paper. ,% dogt pls nse . oe. ee ee ae Sa Sa Sein BO trp A ni is fF AG SO IRE } perenne ner saee Wi) H -———— = a Ea nn ” a ST TET pO KTS Sa ae Fe cr EE ST LT ATS eae INVOLVING LARGE NUMBERS. 153 and g of F; we have then a given probability R that the number of occurrences of E will fall within the limits h’pserV (2h'pq). Now, for p and q substitute the ratios observed in the first set of experiments, namely, m--/A and nh, and the limits corresponding to R become ye a —— 5 V(2h’mn), which, therefore, are the true limits on the hypothesis that the chance of E in a single trial is m+. But as this chance is not certain, but only presumed, the limits require to be extended in order that R may preserve the same va- lue. Confining our attention to ©, the first term of the expression for R (the second may be disregarded in the pre- sent approximation), let 4’==m’ 4-n! and m’: n’=m : n, then © is the sum of the terms of the binomial (p+q)” from that in which the exponent of p is m’4J/ to that in which the exponent is m’/—J. Now, when p and g are given a prior?, the chance of m’ times E and n’ times F in A’ trials is P, ; and when p and qg are only presumed from the results of previous trials, the chance of the same combination is 113 and (100) 1 is less than P, in the ratio of 1 to V/(1+6). In like manner, the chance of each of the other combinations of E. and F included in the integral © will be less in the case of p and g presumed, than in the case of pand q given, in the same ratio of 1 to ./(1+6). But it has been seen (93) that when /’ is a large number, the terms of the de- velopement of (p-++q)” which are nearest the greatest term, diminish at first very slowly; and, further, that only a small number of terms on. each side of the greatest are required to be taken, since Z is less than 4/m’ or /”’ (95); we may therefore, without sensible error, assume © to be proportion- al to the number of terms included in the summation, or were a en en SS SSS rn et el) it hie iy { fe ; i ; a f fa 3 ee 1] ; hag | ; ini | ia) } a | ia St en A tty st cy ees ; SSeS SS eee SS ee 154 SOLUTION OF QUESTIONS that the value of © will not be changed:if we include in the summation a number of terms greater in proportion as the value of each individual term isless. Hence it follows that the limits must be increased in the ratio of 4/ (1+6) to l, and the value of © corresponding to r will give the proba- bility that the number of events E, in h’ future trials, will be included between Bees / (2h'mn(1 +46)). h h 103. The following question may be proposed as an ex- ample of the application of the last formula. Out of a given number: of individuals taken at the age A, it has been ob- served that m are alive at the age A+-a ; required the pro- bability that out of 4’ other individuals taken at the same age A the number who survive at the age A-La will be includ- ed between m'=tl, the ratio of m’ to h’-being the.same as that of m to h. To solve this question, we have to find + from the equation Z= ; /(2h’mn(1+6)); and the corresponding value of © in the table, will give the required probability.. From the table given in the article MorTatrry, vol. xv. p- 555, it appears that out of 5642 individuals taken at the age 30, the number surviving at the age 50, according to the Carlisle Table, is 4397. Taking those numbers as an example, we have 5642, m=—=4397, m=1245; and as- suming also 4’ 5642, whence ¢=1 and V(14+0)=4/2, the equation of the limits becomes /—=r x 62°30, Let it be proposed to determine Z from the condition ©=}. In this ease the table gives 7=:4769, and we have consequently —=29:7. Hence it appears, that if it has been observed that of 5642 individuals taken at the age of 30, 1245. die Lee I 7 cseieetnmeeeenadieansanumamrnmantiteramamniare ne ee ee en i atta PE i a: er os ~. oo INVOLVING LARGE NUMBERS. 155 before reaching the age of 50, it is an even wager that out of 5642 other individuals also taken at the age of 30, and sub- jected to the same chances of mortality, the number who die before reaching the age of 50 will lie between 1245-30, that is, between 1215 and 1275. & phpeat 4 104. The following experiment recorded by Buffon, in his Arithmetique Morale, affords an example of the ap- plication of the preceding formule to the determination of the probable existence of a physical cause from the results of a large number of observations. A piece of © money was tossed 4040 times successively, and the result was head 2048 times, and tail 1992 times. Supposing the piece to have been perfectly symmetrical, the most pro- bable result would have been the same_number-of heads and tails. Let it now be proposed to assign the probability afforded by the experiment that the piece was not symme- trical, and that its form or physical structure was such as to render head an event, a priori, more probable than tail. In this case A=4040, m—=2048, n=1992; and by (102) we have the probability R (or ©, neglecting the correc- tion) that p, the unknown chance of head, is comprised T 2mn m 2048 7 Be a Bis ae h ~ 4040 between the limits moa h h r s2mn i a ee assume + X.011124==.00693, we shall have the probability © that p is comprised between the limits .50693=&.00693, that is, between two limits of which the least is .5, or one- half. This assumption gives 7=.00693 --.011124—=.623 s. and the corresponding value of © is found from the table —.62170. Now if p lie between the above limits, its value: is evidently greater than 4; but the probability of its lying: =.50693, and r X .0111243 therefore if we eel a eS « 5 Se . ee k | i ae 156 SOLUTION OF QUESTIONS Va H * 4 i > ° e eye ° Baie I between those limits is not the whole probability that p is ie BOE IE! : . wig reater than 1; for there is a chance of its exceeding the ti i th g 2 o 1 greatest limit, in which case also its value will be greater nf ? than §. The probability that p is not comprised between the assumed limits is 1—.62170 =.37830; and if it is not comprised between these limits, there is an equal chance of its being greater,than the greatest limit, or less than the least; the probability of its exceeding the greatest limit is conse- | quently 3 X .37830=.18915. Hence the whole probability u : that pis greater than .5, or that the chance of head is i | greater than that of tail is .62170-+.18915=.81085; and uy | the odds are therefore 81 to 19, or rather more than 4 to l iH that the piece was not perfectly symmetrical. i 105. The formule which have been demonstrated in the a present section are immediately applicable to the determina- it a | tion of the probable limits of the gain or loss which may t a arise from undertaking a great number of risks with a given expectation in respect of each. The following question has important practical applications. A is interested in a great number of similar enterprises, in each of which E or F must necessarily happen. When E happens he receives the sum a, and when I* happens he pays the sum 8; required the probability that his gain or loss shall be comprised within given limits ? Let p be the chance of the event EB, ¢ that of F, and h the number of entérprises. Suppose E happens m times, and IF’ times; the sum to be received will be ma, and the nen eae ohare peeeasy eee a Tacs i ' | sum to be paid will be 8, and therefore his gain. will be Hd ; ma—np. Let m=hp, n=hq, then m times E and n times: ii | Fis the most probable result, and in this case the gain i i ma—n3 becomesh(pa—q3). Find + froml4-L=rV (2hp7), ti then (98) @ is the probability that the. number of occur- ed SS ae re oe Brey } t { | 7 i ; | i : + i : } SB are ane nh coe SMR ON moe Sy) » ook eA nll he Ping ge Be A Spree pe aaa at aia eter Se a INVOLVING LARGE NUMBERS. 1572 rences of E will lie between the limits /pa=/.. But if E happens Ap—/ times, and consequently F hq+/ times, the corresponding benefit is (hp—l)a—(hq + l)8=h( pa—q3)— Ua+ 8); and if E happens hp+/ times, and F hq—/ times, the benefit is (hp +Da—(hq—l)B=h(pa—q3) + a-+8) s whence @ is the probability that his gain, that is, the diffe- rence between what he receives and what he pays, will be in- cluded within the limits h(pa—q3)==/(a +8) both inclusive. 106. The following conclusions follow immediately from this solution. (1). If pa be greater than q3, so that ‘A has a mathema- tical advantage (however small) in each risk, the risk may be repeated a sufficient number of times, or 2 may be taken a sufficiently high number, to give a probability as nearly equal to certainty as we please, that A’s gain shall exceed any given sum, however great. (2). Let there be two players A and B, whose chances of gaining a game are respectively p and q, and let 8 be the sum staked upon each game by A, and a the sum staked by B, then pa is the mathematical expectation of A in re- spect of a single game, and g8 that of B; and if pa be greater than q3 (however small the difference) the game may be repeated so often as to give rise to a probability approaching as nearly to certainty as we please, that A’s gain shall become equal to the whole of B’s capital, and, consequently, that B will be ruined. (3). If the mathematical expectations of the two players be equal, then pa—q3=0, and the most probable individual result of a large number of games, is that the gains and losses on either side shall be thesame. Butif/besupposed constant, then r is inversely proportional to”, and consequently the game may be repeated until ©, the probability that the egret seca er ee = Se oN REI POT aD TO NC NEE haere ements Aere ermenm eS [SSS = T58 SOLUTION OF QUESTIONS. gain or loss ((a+-8) shall be comprised within given limits, shall become as small as we please. Hence 1—®, the pro- bability that the gain or loss shall zot be comprised within given limits, may be rendered as great as we please ;. and it follows that although the play: may be on:terms of perfect equality, it may be continued until a probability shall be obtained, approaching as nearly to certainty as we please, that one of the two players shall be ruined. (4). The number of games which must be played, to afford a given amount of probability that one of the parties shall lose the whole of his fortune, depends on the magni- tude of the stakes (a+-8); but whether the stakes be large er small, the final result is the same. When the stakes are small, a greater number of games must be played. 107. As an example of this class of problems, we may take the following question: A and B engage in play with equal chances of winning, and stake five sovereigns on each game; how many games must they undertake to play in order that it may be two to one that one of them shall lose at least 100 sovereigns ? Here p=3, q=4, a=5, 8B=5, and 1 (a4 8)=100, whence l=10. The equation /+-4=14/(2hpq) therefore becomes 10‘5=r/(h+2), whence h=2 x (10°5)?—7%. ’ Now, the odds being 2 to 1 against the limits of the gain or loss. not exceeding 100, the probability © of the limits not ex- ceeding 100 is 4=:33333, corresponding to which the table gives by interpolation r=-30458 ; substituting which in the above equation we find A=2376:8 ; so that if 2377 games are played, the odds are 2 to 1 that one of the players shall have gained at least 10 games more than half that number, and, consequently, that the other shall have gained at least 10: iess than half, or that one of them shall have gained at least. a ea ce Pease ee Oe Se OR OR ee ; ‘ Neg eps acre ee an ee “et eg ae i Sires ce ce INVOLVING LARGE NUMBERS. 20 games more than the other, and consequently have gain- ed at least 100 sovereigns. It is to be carefully observed that this question supposes the account between A and B not to be balanced until 2377 games have been played. If the condition of the play had been that it should cease as soon as A or B should have lost 100 sovereigns, the question would have been of an entirely different kind, and a much smaller number of games would have given the same probability of an equal loss. 108. The question just alluded to belongs to a class of problems connected with the Duration of Play, of extreme difficulty, and which have given rise to some of the most abstruse and refined researches in the modern analysis. In order to give an idea of the subject, we may take the fol- lowing question, which has been frequently considered. A and B, whose chances of winning a game are respec- tively p and q, play on these terms : A has m counters, and B has 2 counters; when A loses a game he gives a counter to B, and when B loses a game he gives a counter to A, and the play is to cease when one of them has lost all his counters. What is the probability that the play, which may go on for ever, shall be finished before more than h games shall have been played. To take a simple case, suppose each to have three coun- ters, and let the probability be required that the play shalt be concluded with or before the ninth game. As the play cannot end with less than three games, let the binomial (p+q)° be developed, and the terms P+ 3p?qt+3py +7 give the respective probabilities of all the cases which can arise in three games. The first term is the probability of A gaining all the three games, the last term is the proba- LE Bae ee Sse f > > a Poe WEP: a to nn a a ens gg ¥ een ene = - va art Sag as eres ms NT A IS armen Pe anoentyveen—t a an Se <= Cn I I I ~ TT Sa Learn cacecia comiseae oestmepaaiieanamn otek celediesecina ance matin ae aonemnmae ve sa OS a ae Bee ee SAS Re Re eet AAD NE 1 AIR aE Ep 160 SOLUTION OF QUESTIONS & bility of B gaining them, and the sum of the remaining two terms is the probability that neither will win all the games, or the chance that a fourth will be played. Now, if the fourth game be played, p is A’s chance of winning it, and q B’s chance; but these chances will only exist in respect of the fourth game, provided the play be not concluded with the previous one, the probability of which is 3p? g+3pq?. Multiplying, therefore, 3p°q+-3pq? by p+-q, the product 3p? q+ Op? q? + 3pq° gives the respective probabilities of the different ways in which the four games may be gained by A and B, except- ing the two ways in which the play would have terminated with the third game. But the play cannot end in any of these ways; for, taking the first term for example, if B gains a counter before A gains three, the play cannot ter- minate until A gain back that counter, and three others besides, so that five games must be played. In fact, it is obvious that there is no way of gaining an odd number of counters in an even number of games, or vice versa. The last product therefore expresses the chance of the 5th game being played; and by reason of p-+q=1 it is equal to 3p"q -++ 3pq’, the chance of the 4th being played, as it obviously ought to be, since the play cannot terminate with the 4th. Again, if the th game be played, p is A’s chance of gain- ing it, and q B's chance of gaining it; multiplying there- fore the last product by p+gq,’the different terms of the result, namely, Spig-++ 9p°q" + 9p*q? + 3pq' give the respective probabilities of all the cases which can arise hy the 5th game. The first term is the probability of A gaining 4 games and B gaining 1, and the last term is the INVOLVING LARGE NUMBERS. 161 probability of B gaining 4 and A gaining 1. These terms therefore are the probabilities of the play ending in favour of A and B respectively with the 5th game, and the sum of the other two terms is the probability that the play will not terminate with the 5th game, or the chance of another game being played. By pursuing the same reasoning it will be evident that on rejecting the two extreme terms of the above product, and multiplying the remainder by p+q, there will result the probabilities of the different ways in which six games may be played without the one player gaining all the counters of the other. But as the play cannot termimate with the 6th game, multiply again by p+q, and the result pg +27p Pp +27p g +9p will indicate the probability of the different cases that can arise out of the 7th game. Rejecting the two extreme terms, which give the respective probabilities of the play being concluded in favour of A or B, and multiplying the remaining two first by p+q to obtain the different proba- bilities in respect of the 8th game, and again by p+ 4, as the play cannot terminate with the 8th, we have the product 27 PP +8) pe +8 per +27 Pe of which the first and last terms give the respective chances of A and B winning at the 9th game, and the sum of the other two terms the probability that the play will not be concluded by the 9th. If we now collect the terms which have been set aside in the successive products, and denote by @ and b the respective probabilities of A and B gaining at the 9th game, or sooner, we shall have | 162 SOLUTION: OF QUESTIONS: | a=pP +3 pig +9 py + 27 p'g’, | b=P +3 q'pt+-9 gp +27 GP; where the law of the series is evident. It is easy to see that this process may be applied whatever be the number of counters which A and B have at the commencement, and whatever be the number, A, ef games to which the play is limited. The general rule is as follows: of the two numbers m and n, let m be that which is not less than the other. Raise p+-q to the power a, and reject the first term (which gives the chance of A winning 2 games in succession), and also the last if m=n. Multiply the remainder (h—n) times in succession by (p+ q), rejecting at each multiplication the first or last term of the product when it gives a combination which would terminate the play in favour of A or B; the sum a of the terms rejected: from the left-hand: side of the dif- | ferent products gives the probability in favour of A, and the sum of the terms rejected from the right-hand side the probability in favour of B. As the coefficients of the suc- cessive products are obviously formed by adding the coeffici- ent of the corresponding term in the preceding product to that of the term immediately before it, the products may , be written down at once without the trouble of multiplica- St RR OS O_O Tn cee Re SRR eT SF = «lO eR. TNR sm an tomer ce emcee pen a Ah OES os Be ee t= fetes. | Ne as sh BL tion; but it is evident that when m, n, andh are large num- mene i bers, it would be quite impracticable to sum. the series | formed of the rejected terms by the ordinary methods. From the manner in which the series are derived, they are called recurring series ; a general theory of which was first H given by Demoivre in his Doctrine of Chances, and forms | the most remarkable portion of that work. bie aitreer ely renee 5 6S OE ee ee ae t Ta Se ol Peat 109. The general problem is reduced to an equation of finite differences as follows: Let Yx, Yepresent A’s expec- CTR NNR PLIES IO ty RCA SBE het en P Nyt, a ‘ . | REP IESE Te one ee EY ane a rae 2 ioe y Rime I - oh 1 5A a PE SSS STE a a NT OG IES I CA ace eae ge Gy ear OLE TRE TIONS | 08 BOE RG FAS oe RO atts ae ae = ———— ar TNE: ——— = a = a INVOLVING LARGE NUMBERS. 163 tation when 2 games have been played, and he has still ¢ counters to win, or B has ¢ counters in his hand. If A gain the next game the value of his expectation will become Y +1, +1, and the chance of his gaining it 1s p; therefore his expectation in respect of that event is pY 241, 1+ On the other hand, if A loses. the next game his expectation will become ¥:+41,¢4+1 and the chance of losing it is q3 therefore his expectation in respect of that event 1s qY241, e+» Hence, according to the principles laid down in (32), Ys, = PYe+i, tA {y2t', t41 a linear equation of finite differences, with three independ- ent variables. It is therefore on the integration of an equa- tion of this kind that the. problem. of the duration of play ultimately depends, but the subject is of much too compli- cated a nature to admit of its. being satisfactorily explained in this place. We must therefore content ourselves with referring the reader to the treatise on generating func- tions, which forms the first part of the Théorie Analytique of Laplace. 1 Lagrange, in vol. i. of the Memozrs of the Society of Turin, was the first who shewed that the investigation of the general term of a recurring series depends on the integration of a linear equation of finite differences. In vols. vi. and vii, of the Memoires présentés & P Academie des Sciences of Paris, Laplace proposed a general method for the summation of recurring series by the integration of such equa- tions, and in the latter volume gives a number of examples of their use in the more complicated questions in the theory of chances, amongst which is the problem enunciated in (108). The subject was afterwards resumed by Lagrange in the volume of the Berlin Memoirs for 1775, where he has given a more direct method than that of Laplace, for the integration of the class of equations in question, and also applied it to the solution of the principal problems proposed in the works of Montmort and Demoivre. A general solution of the problem in the text is given by Ampere in a Tract entitled Con- sidérations sur la Théorie Mathématique du Jeu, (Lyons, 1802.) FE OG ren Soe H 1 oe Ri , A t i It ul 4 is} 7 ia: iy! : ay { it tk q i } : t 5 it i SPOTS i ME 164 RESULTS OF DISCORDANT OBSERVATIONS, SECTION IX. - OF THE MOST PROBABLE MEAN RESULTS OF NUMEROUS DISCORDANT OBSERVATIONS, AND THE LIMITS OF PRO- BABLE ERROR. 110. In the preceding section we have considered a class of questions which apply to events depending on con- stant causes, and supposed to be of such a nature that they necessarily happen or fail in each experiment, and have given formule by which approximate results can be ob- tained when the numbers involved are so large that they cannot be conveniently treated, or cannot be treated at all, by the ordinary methods of calculation. We come now to a more difficult problem, namely, to investigate the pro- bable result of a large number of observations which have reference not to the simple occurrence or failure of a cer- tain event, but to the magnitude of a thing, susceptible, within certain limits, of a very great or an infinite num- ber of different values, equally or unequally probable, the chance of any particular value being also supposed to vary in each experiment. On account of its immediate applica- tion to the determination of the most probable values of astronomical and physical elements from the results of ob- servation, this is, perhaps, in reference to practical utility, the most important question in the theory. 111. Let A represent a thing of any sort (as a line, or AND LIMITS OF PROL sE ERROR. an angle, or a function of any quantity) which may have every possible value within given limits, or which may be constant in itself, but of such a nature that its real mag~ nitude can only be observed within certain limits of accu- racy, and suppose a great number of observations to be made. ‘The object is, in the first place, to assign the pro- bability that the sum of the observed values shall fall within given limits, supposing the chances of the different values of A to be known a@ priori ; and, in the second place, when the law of the chances is un!:nown, to determine from the observations themselves the most probable mean value of A, and also the limits within which there is a given amount of probability that the difference between such mean value, and the true but unknown value of A, shall be contained. 2. Let a and 6 be the limits of the possible values of A, x a value of A between a and 6, and P the probability that the sum of the values of A given by # observations will be s exactly, s being a given quantity between ha and hb. Assume the values of A to be equidiffer ent, pe multiples of a certain constant «, and make ; at==a, Be), o-e=s, tee, where a, 8, and o are whole numbers (which may be posi- tive or negative), and zis also a whole number proportional to x, and varying between the limits 7a, and i=, and which, therefore, may be positive or negative, or zero. If the different values of A are supposed to be equally proba- ble, the chance of obtaining any given one of them, as 2, in a single trial is unit divided by the number of possible values, or equal to 1+-(@—a-+1) ; FS and if we assume an indeterminate quantity w, then (20) the number of combi- nations which give the sum of A values of A equal to ce is the coefficient of that term of the multinomial sa namininiTiam —— 4 166 RESULTS OF DISCORDANT OBSERVATIONS, a0" oh ae ape yee w®)”, (or of the developement of (Zw‘)' from i=a to =) in which the exponent of wis o; and consequently the pro- bability P that the sum of the values of A will be s ex- actly is that coefficient divided by (8—a+1)". 113. If the chances of the different values of A are un- ot OE tet etn in tat i eam = = “ a oa eaten ~ ~ mile equal, and also vary in each trial, let 7, be the probability of the observed value of A being 2 in the first trial, py the probability of its being # in the second, ps that of its being x in the third, andsoon. Now when h=1, or when there is only a single trial, then s=”=7e, and we have P=p!. If h=2, then, assuming Ze to be the value of A in the first trial, and 7c its value in the second, (2 and 7’ being any two tid] I | numbers between a and 8), the two observations may give a ie the sum of the two values equal to ce in as many different ways as it is possible to satisfy the equation 7+7’=03 and a consequently, according to the theory of combinations, P is the coefficient 0. that term of the product (arranged ac- “a cording to the powers of w) of the two series represented by Sp w* and Sp jw, in which the exponent of w is equal to oe. In like manner, if h=3, then the sum of the ob- j HH served values of A may be equal to ce in as many different mi ways as the equation ¢+2’ +2”=o admits of different so- £ pene a feet Sn ee re ST Jutions, and consequently P is the coefficient of the term of the developement of the product Sp,w. pow s Sp. s in which the exponent of w is equal to ce. Generally, when OME PSP A ANTE the number of observations is h, the probability P of the eS Pa sum of the observed values of A being s, or ce, exactly, is the coefficient of w*’ in the developement of the product earn ry aes f i it tk t es 1, 0% pi a 7 | ) | j « f Me 2. | BI oho BD 10 ie SDD oe. o's s as sD a : t ee ee —————E—— SE the sums = including all values of i from 7=a to i=. h . AND LIMITS OF PROBABLE ERROR. 167 Assume wie’ —(¢ being the base of the Napierean logarithms), and let the above product be denoted by X. We shall then have X=sp,e" v=, sp,eY— : Save Mie. cd peon— , Now since P is the coefficient of the term of the develope- ment of this product which contains the factor oft i if we conceive the developement effected we shall have X=Pe@ NA 4 preNAL &e. a series in which all the terms are of the same form. Mul- tiplying both sides of the equation by Psi —I, we get xe bt — pay pro —VIN AT &c. Now by a well known theorem in trigonometry, (ALGEBRA, art. 269), fb? —)n/=T — cos (oft Ot ASE sin (o/c); substituting therefore this value, and multiplying by dé, the equation becomes Xe-eb 1 9—Pd+ P’ fcos (o’—c)é+ /—I sin (o'—o)6} dd+ &e, The factor which multiplies P’ in this equation will evi- dently become zero when integrated from €=—~x to €&=-+-n, (x being the semicircumference to radius 1), the positive and negative elements of the integral being equal, and con- sequently destroying each other. The same thing also takes place with respect to the following terms, which are all of the same form. Integrating therefore between those limits, and observing that,/dé—=2z, we find Heder Adit Seu Salat T —T 114. This value of P denotes the infinitely small chance that the sum of the values of A in A trials will be s exactly. Let » and y be two integer numbers between ha and hf, and let Q denote the probability that s will be comprised TONE TIOIO REIN I 2 at er eee oe TEE tre a an ee eee .. ete FO etn 168 RESULTS OF DISCORDANT OBSERVATIONS, between the two limits pe and ve, (these limits being in- cluded between ha and hb); then Q will be found by sub- Stituting successively nw, p-1, 1+2,...v for o in the above value of P, and taking the sum of all the resulting terms. This substitution gives the following series multiplied by X under the sign of integration : patina Pam a a OO nL a Cem J=T, On multiplying the series now found by of IT ob aa (=2,/—] sin 3), all the terms of the product, excepting the first and the last, destroy each other, and the sum of the terms becomes simply ea IAT 4) ASI therefore on making the substitution, and performing the multiplication now indicated, and dividing by 2a aii sin46, we obtain for the value of Q the equation Q= 1 ee fet +) ar Yd , 4z,/—l f__.. sin 38 115. In order to simplify the expression for Q, let the number of possible values of A within the given limits be conceived to be infinite, in which case the constant e be- comes infinitely small, and therefere, since the limits are finite, » and » infinitely great. Let the following substi- tutions also be made: pe=p—o, ve=p+d, O=ez, 8 being positive in order that » may be greater than p, agreeably to what has already been assumed. On substitut- ing these expressions in the above equation, the limits of the new variable z will be =& infinity; for e having been supposed infinitely small, z must become infinitely great when =a. Now since p and » are infinitely great, p—t and v-+-3 become sensibly » and v, whence we have ' AND LIMITS OF PROBABLE ERROR. 169 e—— 3) bp OF) 4 TL vz (o? = —e~2z,/=1) = 29, / —Jsin6éz. Again, by reason of 6=ez, we have dé=edz ; and « being infinitely small, 6 must be a very small arc, therefore 36 may be taken for sin 16, whence dé sin }6=2dz-z. By means of these transformations the expression for Q becomes Q— —{* e Xe VV —1 sin hace ro ey z and denotes the probability that the sum of the h values of A will lie between ~o=6. 116. It is now necessary to assign a value to the product denoted by X. Since the number of possible values of A between a and 0 has been supposed infinite, the chance of obtaining any given one of them, as a, in a single trial, is infinitely small. Assuming this chance to be a function of x, and to vary in the different trials, let it be represented by $,# in respect of the mth trial. In order to preserve continuity in the values of A, this must be understood as signifying that ¢,adz is the infinitely small chance that the value of A given by the zth observation will lie between xand «+-dzx. The function ¢,7, therefore, represents the law of the facility of the different values of A. It is posi- tive for all values of x between a and 8, and vanishes for all values of x less than a or greater than 6; and it is im- portant to remark, that whatever number 7 may be, the integral /,«dx taken from x=a to x=6 is always equal to unity; for since every observation gives a value of A be- tween a and 6, the sum of all the probabilities in respect of each observation must be unity or certainty. From this assumption, then, we have ¢,ad¢v=p,, $,vdv=py....s. ; ia ¢,vdx—=p,, whence the sums =p,cun — (113) are changed | I Ba a an one mae Sane caboe aca Ae et ome + OR oR ar eee gee oe ee NL ET Ne gre Ras E be em eae ee DE Ie PRR SN an cre pe tren an al phere ee Sagopa sas setiepge amar a ee it } ) 170 RESULTS OF DISCORDANT OBSERVATIONS, into definite integrals; and therefore, since 6=ez, r=, and consequently 76==«2, we obtain for the value of X, X= fer, de few NAG adie... ce ld nddz, the limits of the integrals heing a==a and z=6. By reason of &*V—!= cos zx-+-./—1 sin zx, each of these integrals may be expressed in terms of the cosine and sine of zz. The zth, for instance, becomes /¢,x cos za.dx sng A —1 /¢,« sin zx.dx. Now since f{¢,de=1, (from x=a to x=b), and ¢,x can have only positive values, each of the integrals is less than 1; whence we may assume Sf onz cos zx.dx = Ry, cos rn3 Son x sin zx.dz = Rysin ry 5 R,, being a positive quantity, and 7, an angle having al- ways a real value. This gives foe NA =! pn, xdx=R,(cos7,—r/—I1 sin = Raa J—l ; whence substituting successively for 2 the numbers 1, 2, 3...A, and for the sake of brevity making YoR, KR, Ro... Ry YT AT og cccceecsceee Thy we get X= Ye" 1; and the expression for Q becomes dz ! ean TF! vel y—¥2) N= gin az. 6 or mae Zz 117. The integral in this last expression is equivalent to two others, namely ve cos(y—z) sin dz. a oh AU —1 /Ysin (y—z) sin dz hed Zz Zz Now, on attending to the nature of the quantities repre- sented by Y and y, it will be manifest that according as z is positive or negative, 7,, and consequently y is positive or negative, while Y is positive in all cases, since R,, is always AND LIMITS OF PROBABLE ERROR. lft positive. Hence cos (y—w2) is always positive, and the ele- ments of the first of the above integrals having thus the same value and the same sign for the same value of z, whether z be positive or negative, the value of the integral from — oo to + is the double of its value from 0 to «o. On the other hand, since y and z have both the same sign, sin (y—vyz) is positive or negative according as z is positive or negative, and the elements of the integral into which it enters being equal for —z and +z, but having contrary signs, destroy each other, and the integral from x=—o to to x= vanishes. The expression for Q is therefore transformed into dz 2 DO Q=- he Y cos (y—wz) sin dz. —. iy 0 x ~ 118. The formula now found cannot in general be integrat- ed by any of the known methods, but in the present case the quantities denoted by Y and y are such that an approxi- mate value of Q may be obtained, which will always be more nearly equal to the true value as h, the number of observations, is increased. On adding the squares of the two quantities represented by R,, cos 7, and R,, sin 7', we get R,’=(/¢$,% cos zx.dx)?+(fo,x sin zx.dx)?. If z=0, this becomes R,=/¢,«dx, whence by (116), R,=1. When z has a real value, then it may be shewn that R, is less than 1 ; for let x’ be any value of A different from x, then as 2’ can only vary from as a to 6, we have obviously, SPnx’ cos zz'.dx'=f$,x cos zx.dzx, and /9,2’ sin'zz’.dx’=/) ,x sin zx.dzy and the above equation may be put under this form, RR? =/Onx cos 2x,dr.fp,x' cos 2x ‘dz! +/6xx sin 27.dx./0nx' sin zx'dz, whence R?,=/[b,0¢,x' cos 2(a—x’ \dxedx’. a te SS + a ORRIN ae tnt PI se nM en OCP ITC. aa . ~ rR er ee ( é % ; { 7 : » Su a a es eT Fee eR Ne er ent ae ee NE eNO eT a ar al wh ON ces Si IE in I sopra se aynameerat ry oa 172 RESULTS OF DISCORDANT OBSERVATIONS, Now, excepting the case in which z=0, this double integral is always less than //},7,«’daxdx’, or less than (/o,vdx)%, and consequently R,, is less than /¢,vdz, that is, less than unity. Since, then, it has been shewn that R,, is equal to unity when z=0, and less than unity for all other values of z, and since Y is a quantity of the order R,’, it follows that Y must diminish with great rapidity when 2, or its equal 6-~:« differs sensibly from 0, and even for very small values of z becomes insensible when / is a large number. We may therefore assume Ye”, an expression which is equal to unity when 6=0, and diminishes rapidly as @ is in- creased, and becomes zero when 6 is infinite. 119. For the sake of abridging let us assume h,=fap, cde, kh’, =ferp,cdx, k/=fx>>,xdx, &c. (the integrals in respect of x being always from w=a to x=b). From known formule we have elge wind Cae cos 2x#== | —__——— —— &c.3 sin 272 zx—_—— + &e.; 2 +5 “354 , 2.3 é substituting these series for cos zx and sin 2x in the inte- orals {p,« cos zx.dx and fp,x sin zx.da, and also k,, k’,, k’”,, &c., for the values they have now been assumed to repre- sent, then, from (116) we have R,, cos 7,== 1. 2 k! A: | a Bees FF n o— oe nt 5 : 4 ae . a . z° R, sin 7,=2h,— ar hk” + &e. and it will be seen presently that all the terms involving higher powers of z than the cube may be neglected. Add- ing together the squares of these two equations, we get R,?=1—z7(k',—F?,,) 4-24f— &c.; whence R,=1—42? (k’,—h?,) +24 f'— &e. ; J’ being independent of z. On dividing the second by the AND LIMITS OF PROBABLE ERROR. 173 first, there results tan r,=zh,—}2°h’’, 4+. 425k, kh’, — &c. whence by reason of 7,= tan 7,—+4 tan*r, + &c., r,=zk,—}z> (k",—3h,k’, + 2h,* ) + &e. If, therefore, we make C= 3 (R'—R?,)s Ine (Rp — 3h BR 4 2R, 5 )s the values of R,, and r,, become respectively RS 1 —27e, +24 f'— &a 3 rach, —z 9,4 &e Now, by hypothesis (116) Y=R, xR, xR,......R,3 therefore log Y== log R,== log (1—2?e, +- 24 f’— &c.)= —z 127e,—~24( S’—den)— &e.} (by reason of the formula log R,=R,—1—4(R,,—1)? + &c.) But we havealsoassumed ay = oe; hence log Y==—6?, and consequently OP f2%c,—z! (f’ —4e,) — &e.$ In like manner, since yor tro tts +7, ==7,, therefore y= zk, —Zz5g, + &c. Now, the sums © include all values of c,, k,, g¢,, from n? ‘w=1 to n=A; let the mean values of those quantities, therefore, be denoted by ¢, &, g, that is to say, let Le,—he, 3k,z=hk, 39,=hg, and make also hf”’==s(f' —4c,), and we have 6?=27hc— zthf’”’ + &c. By reverting the series the value of z is found rs | J (hey * Phe /Chey + *° But the second term of this series, being divided by h//, in terms of 6; namely z= and / being by supposition a large number, is very small in comparison of the first, and may be neglected as insen- sible. All the succeeding terms of the series are divided by higher powers of #, and may therefore be rejected a _for- tiort. Confining the approximation, therefore, to terms of the order 1~,//, and rejecting all those into which / or its powers enters as a divisor, we have z—=§+4/(he), and likewise dz——z—=dd-~6. From (116) we have also y=3r,=z2h,—z°29,+ &c., 5 yaa eh’ VALE Nn et Se I — in a= alee i th Ae pc ty ae ~ ee : Emote ae > Pitt ied EPITOME Se cate Seema Dugeoteamantiaters ii Seamer so | 174 RESULTS OF DISCORDANT OBSERVATIONS, therefore in gonsequence of the above transformations, y= zhk—z*hg+- &c.; and on substituting for z its value just found in terms of 6, y=h6/(h-+-c)—gl? +en/(he), and consequently y— Wz = (hk—w) 0 +/ (he) —g6> e—/ (he). In order to deduce from this an expression for cos (y—z), let w and v denote any two arcs, then by trigonometry, cos (u—v)—=cos u cos v-+ sin usin v. Suppose v to be-small, and let its cosine and sine be developed in series and sub- stituted in this equation ; it will become v2 v3 cos (u—v)=cos u— Wha ut+ &c. +v sin u— res sin u+ &e. whence, making u=(hk—)0~4/ (he), v=765 +-c/(he), and rejecting as before terms of the order 1+h, we have S({ YymaLz = f ‘fp whee cos(y—p J=g08 4 (hh Y aay t+ a sin | ik ey yen + ThE ts If we now substitute the values of Y, 2, dz, cos (y—pz) found in the last three paragraphs in the value of Q (117) we obtain the following expression in which the largest terms omitted are of the order 1+, and which therefore is more accurate as / is a higher we Viz. 2 a . a= f, a “cos } (hk—v) a ey * sin shy 29 Dh é : 66 3 a eee é€ sin { (hk) V/ (he) } sin Whey do. 120. As no restriction has yet been made with respect to_ the value of yy, excepting that it isa mean between peand ve, and therefore included between ha and hb (115), let us now assume y=/k, This gives cos (hk—y)=1, and sin (hAk--)=0; and the equation becomes O= 8 —" dé wars “ e sin / (hic) ° ri AND LIMITS OF PROBABLE ERROR. [176 which is the probability that the sum of the observed va- lues of A will fall between A==5. 121. The last step in this investigation is to reduce the integral now found to a known form, which may be accom- plished as follows: Let w be a new variable, then by means ’ ° * p Uu pase a= U6 Sas of the trigonometrical formula cos u=Je Vv hte v 7 fe "00s u.dé=%4 ape ent dak fo NA a, But —@°4 ud, /_j] = — zu" —(6—3gu,/—1)’s assume therefore, eae (whence dv = dé), then 2 fe OF gg — ete db Ee fe dv. When 6=0, then v= ey, —1, and when @ is infinite, v is infinite; therefore, if the integral in respect of 6 be taken from 6=0 to 6 =, the integral in respect of v must + * — Le aaa pale , be taken from v = —3u ae 16 © =) O&O: If we now suppose « to be negative, we shall have in like Mill Jatt «ae | tee fens at ahi manner 3 fe a 'da=ze"™ fe dv, the limits in this case being from v = ate te tov=o. Hence jibe cos u6.d6 = 1e—™ ‘(fe foe dv fe~ dv) But the sum of the two integrals on the right-hand side of this equation, the first being taken from v = —zu, /—j : to infinity, and the second from v = + 3u,/__] to infinity, is obviously the double of Cm ° dv from v=0 to v= a, Or | (96) equal to 4/7; and we have therefore . 2 CO —42 —lus é So e § cos ub.dé=1f/me* . Let both sides of the equation be multiplied by dw, and integrated from uw=0 to u==d+/(he)=w’ ; then observ- ing that f‘cos (ué)dé = sin (ué)+-0, we shall have aad dé wu’ oat ef orn ee. 7 fe ot the ve JG ee tee Ay 6 OSES SST LE ETE Re ete) eae A ei po Tan RR AS oe = Se ee SS, ee Ro ne a ee em nce wee ae Se eee roi ninceneneeaaee SS — Se ey ~ Tage een eae ade 176 RESULTS OF DISCORDANT OBSERVATIONS, Comparing this equation with that in (110), we find Q=( bs Vm) fe? du, Now, let w==2¢, and let r be whaté becomes when u=w’=S+-4/(he) ; then 5? =F" dia ae b--/(he)=2r, or 6==2r4/(he), and we have, finally, 2 a 2 . Q= 5. fe "dt, or, Ql ae e—q, 7 0 a for the probability that s, the sum of the observed values of A, will be comprised between the limits ~—é and +6, thatis, between hka=2r4/(he); or, that thearithmetical mean of all the observations, namely s--h, will lie between k>=214/(c+h). 122. The expression now found for Q is that which in (96) was denoted by ©, and of which the table gives the values corresponding to the different values of r. The ge- neral result of the investigation is, therefore, that whatever be the nature of the function ¢,a@ which represents the law of the facility of the different values of A, if a large num- ber of observations be made, the sum of the values of ‘A; divided by the number of observations, approaches continu- ally to a certain special quantity & (which is the true mean value of A) as the number of observations is increased, and that by multiplying the number of observations, a probabi- lity © may always be obtained, approaching as nearly to cer- tainty as we please, that the difference between the arith- metical mean or average of the observations and the true mean value of A, will be comprised within limits which may be made as small as we please. The analysis employed in the preceding articles, (113 to 121), for the purpose of establishing this very impor- tant result, belongs to Poisson, and is given in nearly the same form in the Recherches sur la Probabilité des Juge- “a » AND LIMITS OF PROBABLE ERROR. IV ments, chap. iv., and in the Additions to the Connaissance des Tems for 1832. We have preferred it to the method followed by Laplace in the Théorie Analytique, as being somewhat simpler and also more general. 123. In order that the limits 2r4/(he) may be real, it is necessary that the special quantity ¢ be positive, a condition which has hitherto been assumed. Now, since cc, —A, it is obvious that ¢ will be positive if ¢,—=3(2’,—?,) be po- sitive. On writing for 4’, and &, their values (119) we have 2¢,=/x'h,ada—( fxd, rdx)’, the limits of the integrals being always from za to r—=5. But it is evident that no change will be made in the values of these definite integrals (the limits continuing the same), by substituting in them any other of the possible values of A, asx’. We have therefore /2’?,'d«’=/xp, «dx, and since in all cases f,2’dz'=1, the above equation may be other- wise written 2¢,—=/x* pn, cde [O,x'da’—_xo, ada fu’? x dx, whence 2¢,=//0,0,0'(x2°—a«a')dudx’, - or 2Cr=[O nthe (x ?—2'x)dadx’. Adding together the two last equations, there results 4¢,=[]0,«p,x' (a—a’)’dadx’, a quantity which is necessarily positive, and can never be zero so long as x can have different values. 124. The special quantity & to which the average of the values of A continually approaches, is connected with the centre of gravity of the area ofa curve by the following rela- tion. Let 2 and y be the co-ordinates of a curve, of which the equation is y=¢, ; then the element of the area is $,vdx. But (116) 9, «dz is the infinitely small probability that the value of A in the zth observation will lie between x and x-+dx; therefore the element of the area of the curve 178 RESULTS OF DISCORDANT OBSERVATIONS, represents this probability, and the curve itself represents the law of the probability of the different values of A in re- spect of the mth trial. In like manner, the curve whose co- ordinates are x and (1+-/)5¢@,, represents the law of the mean probability of A in respect of the whole series of observa- tions. Now, if 2; be the absciss of the centre of gravity of any curve whose co-ordinates are x and y, the well known formula of mechanics gives «,=/yxdx—+/ydx ; therefore, applying this formula to the curve of the mean probability, and making the whole area (/ydzx from w=a to e=b)=1, the absciss of the centre of gravity is 1,.=(1+h)3/xo, edz. But this is the quantity denoted by # (119) ; hence the spe- cial quantity to which the average of a large number of ob- servations indefinitely approaches is the absciss of the cen- tre of gravity of the area of the curve which represents the law of the mean chances of A. 125. It has been assumed in the foregoing analysis that A is susceptible of an infinite number of values, increasing continuously from a to &. The results, however, are easily adapted to those cases in which the number of possible va- lues of A is finite. Suppose A to be a thing susceptible of only 2 different values, represented by @,, dq, Az esevery, and let the chances of these values, which may be different in the different trials, be respectively y,, y; Va erseeeYy in respect of the zth trial. Now, suppose ?,2% to be a dis- continuous function, which vanishes for all values of x, of which the difference from one or other of the above values of A exceeds an infinitely small quantity ¢; then the whole integral /?,cdx from a—=a to x=6, will be made up of a series of d partial integrals /0,adx taken between the limits a;—Ke, the sum of which will be unity, since one or other of the values of A must necessarily be given by the trial. But AND LIMITS OF PROBABLE ERROR. 1v9 the integral /?,7dx between the limits ake is the expres- sion of the chance that the value of A given in the ath trial will lie between ake ; whence for those limits /?,,71dr=y,. Now the difference z—a; must be infinitely small, since it cannot exceed «; we may therefore substitute a, for x, and a,” for «” under the sign of integration, when the limits are a,;—e,so that for those limits we have /29,.«dx—=a, [0,¢dx= ya, On writing for z all the different numbers 1, 2, 3......A, and observing that the A partial integrals thus formed make up the whole integral /x?,ada from =a to r=, and that therefore their sum is #,, we have, in respect of the wth trial, | Dee Oe PV oo HY 55 rvevee fy, G,, In like manner, for k’,=/x’0,«dx (from a to 6), we have Ry Ay AY eGo? FY 55 eee $Y? 5 so that the two special quantities A and k’ become kh =(1--h)E(y 1G, Hohe B55 vcveee 7% ); R= ALA)E(y1 4)? $7944? +545” » eb 7,4," )s the sums = extending to all the / values of , or to all the trials, the chances denoted by 7,, y., &c. being supposed to vary in the different trials. 126. When the chances of the different values of A are equal and constant, then y,=1-+), and the above values ofk and k’ become R=(1=A)(a, +g fa, ores $A), h’=(1+A)(a,7 +4," + a5” .0-... a, *), so that & is the arithmetical mean of the possible values of A, and #/ the mean of the squares of those values. On this hypothesis, ‘therefore, & and k’ may be computed a@ priors, and consequently the lirhits determined within which there is a given probability @ that the average of / observations will fall, thelimits being A=2r \/(c+-h,) wherec=43(h’—h*). LE gee OP. wiersne yo SSA Sh Sea ——— SE : ssa cc ae 2 paver Figen RRR ANAT TITRE eI NL TN IN Sena ~~ = -_— ee ES seas aoe abe 180 RESULTS OF DISCORDANT OBSERVATIONS, When the chances of the different values of A are un- equal, but constant in the different trials, then A=h,, and k’=h’,, and we have h=114, +790. +505 «0... +7,%, 2 ‘ 2 W=y7,a, +704." +y34,"...... +47,4,. In this case the special quantity % to which the average of the observed values continually approaches, is the sum of the possible values, each multiplied into its respective proba- bility; and 2’ is the sum of the products of the squares of those values into their respective probabilities. 127. Resuming the consideration of the general formula in (121), we shall now give an example of its application when the function which represents the law of facility of the different values of A is supposed to be known a priori. Of all the hypotheses which may be made respecting the law of facility, the simplest is that which supposes the chances of all the possible values of the thing observed to be equal, and to remain constant during the series of trials. This supposes ?,7=9x=a constant; whence JSeadz, be- tween the limits =a and x=}, becomes (6—a) ox. But between those limits we have also JSexdx=1; therefore ?r=1+-(6—a). From this value of $x it is easy to deduce the special quantities & and 2’. On the present hypothesis k=k, and k’=k’, ; therefore, the limits of the integral being 8. baits “=a and x=6b, we have h=fxoudu= CEE Pee ba 262g) 2(6+-a), whence #?=1(b-+-a)?. Inlikemanner kh’ fx? gudx ada bs ; becomes jog (OF +4040") ; whence c—=3(k/—h?) =} (0? +ba+a?)—1 (64a)%. Hence by (121) we have AND LIMITS OF PROBABLE ERROR. 18] the probability © that the average value of A given by h observations, or the sum of the values of A divided by their number, will lie between L(b--a) =e eV {H(b2 $bafa2)—h (B44)? Vh. 128. This formula may be applied to the following ques- tion. Of the comets which have been observed since the year 240 of our era, the parabolic elements of 138 have been computed, and the mean inclination of their orbits to the ecliptic is found to be 48°55’. Now, supposing every possi- ble inclination of an orbit to be equally probable, let the probability be demanded that the mean inclination of 138 orbits will not differ from 45° (the mean of the possible in- clinations) more than 5° in excess or defect. In this case the limits of the possible values of the phe- nomenon are 0 and 90°. We have therefore a0, 690°, h=138, and the above limits of the error of the average, become 45°==7 X 90°+4/(6 X 138). In order that the limits may not exceed 5°, we have to determine r from the equation rt X 90° + 7 (6 X 138) =5°, which gives r=17 23 ; whence 7==1°6 very nearly. The tabular value of © corresponding to 7T=1'6 is -97635, or nearly 44; and the odds are therefore 41 to 1 that on the supposition of all inclinations being equally probable, the mean inclination of 138 comets would fall between 45°=5°, that is, between 40° and 50°. The mean of the inclinations actually com- puted falls within those limits (being 48° 45’); there Is therefore a very great probability that whatever may be the nature of the unknown causes which determine the positions of the cometary orbits, it is not such as to render different inclinations unequally probable. If the question had been to assign the limits within which it is as probable that the mean of the inclinations will fall SS a ee . k Po en Te eee RE pred Pepe stems wort ee —— So Lee carretera teat gltye tne = a a 2a aga ad i ioe sae tae: : PE -snige came Akan ete om 182 RESULTS OF DISCORDANT OBSERVATIONS, as not, we should have had e=4, and consequently (from the table) 7==-476936, and the limits would have been 45°>=90° x *476936+/(6 x 138), which is found on cal- culation to be 45°>=1°5. On the supposition, therefore, that all inclinations are equally probable, it is one to one that the mean of 138 inclinations will fall between 432° and and 465°, or at least not exceed those limits. 129. On the same hypothesis of an equal probability of all possible values, if we suppose the mean value of A to be 0, we have then a=—d, and gx becomes 1--2 a, whence the limits corresponding to a given value of © (127) become O==27b: V(6h). Let ©=4, whence 7=-476936, and suppose 4=600. With these values the limits become O>='0166 nearly ; that is to say, it is an even wager that the average of 600 observations will not differ from the true mean value of A more than the sixteen-thousandth part of a or b, what is the greatest possible difference. 130. As a second ‘hypothesis, suppose the chance of a given value of A to decrease uniformly as the magnitude in- creases from 0 to =a; then ov will be found as follows . Let ¢x=8 when a=0; we have then by the hypothesis x: B=(a—wx):a, whence ?x=(a—x)B--a, and conse- quently /prdx=Br—ex?-+2a, which, from 7=0 to v=--a, becomes 36a. But /¢axdx from x=—a to «= +a is ] (errors beyond those limits being supposed impossible), therefore from z=0 to w= +-a, JS¢adz=4t, and consequently $pa=t,orB=1—+-a. Hence pxr=(a—x)-+a", from which the value of ¢ is easily deduced, that of 2 being 0, as in the former case. 131. Although the function @z which represents the law of facility of the different values of A is in general unknown, its form may be assigned if we assume that jt is subject E~ AND LIMITS OF PROBABLE ERROR. 183 to certain conditions, which, from the nature of the thing, must be very nearly, if not absolutely true, in most practical cases: Ist, That the chance of an error dimi- nishes as the magnitude of the error increases, and for er- rors beyond a certain limit vanishes altogether ; and, 2d, that positive and negative errors, of equal magnitude, are equally probable. The last condition is equivalent to the assumption that the average of the observed values is the true mean value. For simplification, we suppose the chance of an error of a given magnitude to remain constant in all the trials. . 132. Let 2, 2’, v”, &c. be a series of values of A, the sum of which is s, and the number /, and make m=s—A, then m is the arithmetical mean or average, which by hypothesis is the true value of the phenomenon A. Let x—m= A, a’ —m=A’, 2’—m=A", &c., so that A, A’, A”, &c. are the errors of 2, 2’, 2”, &c. Now, the most probable single er- ror is 0; and the probability of obtaining an error of a given magnitude A in any observation is obviously the same as that of obtaining a given value of x; therefore Qa= Q(x—m)=A; so that PA is the probability of a single error being exactly A. In like manner, the probability of an error ror =A’ is 9A’; and if we take P to denote the probability of a given system of errors, A, A’, A’’, &c., then the errors being supposed independent of each other, we have (7) P=9A . 9A’ . PA”, &e. Let this system be assumed to be the most probable result of the observations, then P is a maximum, and its differen- tial co-efficient zero. Taking the logarithms of both sides of the equation, differentiating, and making d log.@A= 0’ AdA, and dP-—dA=0, we obtain 0=9’A+0'A’+9'A""-+, &e,, 184 RESULTS OF DISCORDANT OBSERVATIONS, an equation which may be otherwise written OA QA’ QA” QO=A nage. ae +A" sar +: &e. This is the conditional equation of the most probable sys- tem of errors. But the hypothesis of the average being the true value, furnishes this other equation, 0=(a—m) + (a!—m) +(2”—m) 4, ke. or, which is the same, O=A + A’/--A’’ 4, &c.; and on com- paring this with the above conditional equation, it is evi- dent that they can only be both true simultaneously on the hy : supposition of a ee ae =, &c. Hence it follows that ?’A~-A is independent of any particular value of A, or is equal to a constant, which we shall call K. We have then Q’A _7@ . log. PA x A AdA The integral of this expression is log. ~PA=3KA? + const., which, making the last constant =log.H, and passing to num- I bers, gives PA= He? K 4’, It now only remains to deter- mine the two constants H and K. With respect to K, as we suppose the most probable value of A to be 0, and that gA diminishes as A increases, it is obvious that K must be negative. Assume 3K =-_y, and the formula becomes gA=He—144, For the determination of H we have the equation /gAdA=1, the limits of the integral being —a/ and +a’, where a’=1(b—a), a and b being the limiting values of z. But it is to be observed, that as all values of A ex- ceeding the limits = a’ are supposed to be impossible, or at least to be so improbable that it is unnecessary to take - account of them, the value of the integral JSeAdaA from A=—a’' to A=-}-d’ will not be altered by extending the limits from — infinity to 4. infinity. We have therefore AND LIMITS OF PROBABLE ERROR. 185 S *pAda=1. Let A=t-+/y, then dA=dt+4/y, and gA—He—7*—=He-*, on substituting which in the last equation, and observing that from ¢=—® to¢= -+- @ we have fePdt=/r (96), we find (H+/y7)/7=1, and H=/(y+7). Whence, finally, QA=V/(y_ne—V™", 133. The general properties of the function now found may be illustrated by means of a curve line. Let aNb be a curve of which 9A is the ordinate corresponding to the absciss A. Let AB be its axis, and MN its greatest ordi- nate. Suppose the origin to be placed at M, draw PQ an ordinate at any point P, and pq indefinitely near to PQ, and make MB=—=a’, MA=—a’, MP=A, and PQ=9A; then as was shewn in (124), PQqp, theelement of the area, repre- sents thechance of an error lying between A and A+dA, that is,of an error greater than MP but less than Mp. Now, ifgA== J (y-_r)e—7™* the function will not be changed by chang- ing A into —A; therefore 9A =?(—A), and the curve is symmetrical on both sides of MN, as it obviously ought to be according to the hypothesis ; foron making MP’ =MP, then positive and negative errors of equal magnitude being equally probable, we must have P’/Q’=PQ. Again, since ey? diminishes rapidly as A increases, the curve at a short dis- tance from MN must approach very near to its axis AB; put as the function only vanishes when A is infinite, the curve will not meet the axis at any finite distance from MN. This eurye, therefore, can only represent approximately the law of 186 RESULTS OF DISCORDANT OBSERVATIONS, facility, inasmuch as it is supposed that errors beyond a cer- tain limit are impossible; but on account of the rapid dimi- nution of the ordinate at a short distance from MN, the chance of an error exceeding a small value of A, as MB, becomes insensible. Hence the limits of the integrals in respect of A may be extended without sensibly altering their values from A=Ka! to A==Eo. 134. It is now necessary to find the special quantities , 4’,ande. Substituting A for 2, and observing that as the law ofthe chancesis here supposed to remain constant, we havg k=k,, k'=k',, the formule in (119) become kA=fAgada, h’=f A?@AdA. Hence on making PA==/(y-+-m)e—Y4’, we have —y it y 1 west. 1 uy Ravn (2 Z ya oie + fa. Ti ee Y When A becomes infinite, this becomes 0, therefore from A=—o to A=-+ 0, k=0. This is an obvious conse- quence of the symmetry of the curve, for the centre of gra- vity is necessarily in the straight line MN. With respect to k’ we may proceed thus. We have =f arpa 6 Moon =V/ (y+) f aze—74? dA. But from the principles of the differential calculus, d . Ae—yo* —e—y4 *dA—2yA2%e—v4 "dA, therefore, integrating and transposing, Wh 2e—vAt ga | pp—yA? se ie e—vA?® da, oy a Now, from A = —o to A = + o, the term of this equa- tion which is not under the sign of integration vanishes, and Sew da= =/(r+-y) (from (96), on substituting ¢ for yA), therefore (A2e—74" da — (1 + 27)a/ (a 7); and consequently #’=]—2y, arenas i a rt odin es Oe TRENT a ee OP EE erate Ce ste fin AND LIMITS OF PROBABLE ERROR. 187 In (119) we assumed c= 3(h’—A?) ; therefore in the pre- sent case c=3h’, whence c=1+4y, or y=1+4e. 135. The expressions which have now been found for the function which represents the probability of an error, and the limits corresponding to an assigned degree of pro- bability, are given in terms of the indeterminate constant y (or c), which depends on the nature of the observation, and therefore, where instruments are requisite, on the goodness of the instrument and the skill of the observer. This con- stant is called by Laplace the modudus of the law of facility. It cannot, in general, be assigned a priori ; but if we assume that positive and negative departures from the mean are alike probable, which is the most plausible hypothesis the nature of the thing admits of, an approximation to its value, in respect of observations of a given kind, may be deduced with great probability frum the results of a large series of observations of the same kind already made. We now pro- ceed to give the analysis by which this is accomplished, fol- lowing the method of Poisson. The approximation is carried only to quantities of the order 1+4//; terms having A for a divisor are neglected on account of their smallness, 4 being supposed a large number. 136. In the expression for Q in (119), suppose y=8, and consequently ~y—d=0, y+ d==26, and write also z for 6+,/(he); the equation then becomes 2 ad 2 68 Q—- cE e—* cos (hkz—8z) sin 62. — T.J O 6 20 cs -E 47 ne/(he) J o and Q is the probability that s, the sum of the values of A given by all the observations, will lie between 0 and 28. If therefore, we suppose 6 to be variable, the differential of —© sin (hkz—6z) sin 62 « 67dé, . 188 RESULTS OF DISCORDANT OBSERVATIONS, this expression taken with respect to 8, will express the infinitely small chance of the sum of the values being 25 exactly. Differentiating, and observing that if « and v denote any two arcs, the trigonometrical formulz give sin (u—v) sin v-+- cos (w~—v) cos v=3 cos (2v—u), ——~ cos (u—v) sin v +- sin (w—v) cos v=— sin ( 2v—u), we shall find = di= — ati cos (28z—hkz) = 2gd8 og rey (hey o Let ¢ be a variable quantity, and assume 26=hk +. 2tV (he), whence di=diW (he), and let the corresponding value of 27 be denoted by qdt, we shall have, on substituting a ah (262—hhz) 267d). these values, and replacing zg by 6-+-W (he), gdt= 2dt 7 e.8) eo) aoa 2gdt =o . 6) d§— 36. S, e~" cos (2¢6) d moa EN) e~ "sin (246) 65 dé The two integrals in this equation are found from the formula in (121). Writing 2¢ for w, that formula gives e ¢) oR e—* cos (20)di=li/n.e”; Oo and if this last equation be differentiated in respect of ¢, three times in succession, the result will be a0 ; J, Ser sin (226) 65 dé—=1 /x( 3te—t2__23 e-t) ; 0 whence, if we make V— (3¢—2¢5), we shall have saat OS ha 2c/ (he) qdt= (1+4/7) (1—V) e-#dt, where V is a quantity containing only uneven powers of 4, and of the order 1-+-/h, so that when multiplied by another of the same order, the product will be of the order 1h, AND LIMITS OF PROBABLE ERROR. 189 and will therefore be rejected in the present approxima- tion. This value of qd¢ is the probability that s will be pre- cisely 28 or hk+-2t,/(he), or it is the infinitely small pro- bability of the equation shk4+-2tV/ (he). 137. In order to apply this result to the determination of the probable limits in terms of observations actually made, it is necessary to remark that the analysis by means of which it has been obtained is grounded on the very general supposition that the thing to be measured may be any function whatever of the quantity obsérved ; for the in- finitely small chance of a particular value of the function is evidently the same as that of the corresponding value of the quantity, and is consequently ¢,vdx. Let X therefore be a function of 2, and let K, C, T, be what &, 2, ¢ become when X is substituted for 2, the above equation then be- comes SX =AK42TA/(hC), the symbol = including all the / values of X; and the pro- bability of this equation is an expression of the same form as that which is represented by qdé. 138. Hitherto no restriction has been made with respect to px ; we now introduce the hypothesis that positive and negative departures from the mean of equal magnitude are equally probable, and consequently that the curve repre- senting the law of facility is symmetrical, but shall sup- pose the chances of a particular value, or a particular error, to vary in the different trials. Let the origin be transferred to the centre of gravity, the absciss of which =&, and let e—k —A, x'—k= A’ &c. We have then by (132) ¢a=ga, fupudx=f(Ap ada, fx’oudt=fa’p Ad A, the integra- tion in respect of 4 being from —o to +4 ». The special 190 —s- RESULTS OF DISCORDANT OBSERVATIONS, quantities k and k' then become k=(1+h)E fad, Ada =0, A =(1+-h)E Ah, Ad A, whence c=(1+2h)3fad, Ada. ' The object is now to eliminate o,A, and determine e¢ in terms of the observations. 139. Let X,==a be the observed value of A in the 2th ob- servation, then d,—A=A is the true error of the observa- tion. Let the function denoted by X in (137)be (A,—k)? =A’, and the corr esponding value of K (since in this case, K=(1--h) 3 f X¢,,dx) becomes K=(1+h) 2 fA?d, Ada. ~ Comparing this with the value of ¢ found above, we have K==2e; therefore on substituting these values of X and K in the equation (137), and assuming ¢’ and e’ to be the values of T and C when X=(1,—A)?, we get = (yn —k)? =2he 4-2t's/(he’), whence c=(1+2h)30—,—h)?—'U" (1) (U being a quantity of the order 1--4/h) ; and the probabi- lity of this equation is q'dt’=(1+/7)(1—V)e—t” dt’, where V’ is a function containing only uneven powers of d’, and of the order 1A. . In the equation (137) suppose K=«=),, and Jet ¢” and e” be the corresponding values of T and C, then since on this supposition K=£, the equation becomes DA, =k. 2t’’/ (he’’), whence k=(1-+h)=n,—t"'U", (2) | (U” being of the order 1+4/h); and the probability of this equation is q"dt"=(1+-4/7)(1—V" et de’, where V”, like V’ and V, contains only uneven powers of é” and is of the order 1+4/h. 140. The two equations (1) and (2) may be regarded ie ote AND LIMITS OF PROBABLE ERROR. 191 as two distinct events, having the respective probabilities now assigned to them, and therefore the probability of their being true simultaneously is the product of their respective probabilities, and is accordingly (neglecting the product V’V" which is a quantity divided by /), q'q’'dt' dt" =(1-+)(1— VV" ee dt! dt”. Let the value of & given by equation (2) be substituted in (1), and the expression now given will accordingly be the probability of the resulting equation, namely, I 2 55 2On— = DA, + t’U')*—t’U”. Let m=(1—A)=X,, then m is the average or arithmetical mean of the observed values, and \,—m the reputed error of the observation. The last equation will then become c= (1+2h) = (X,—m +¢t'U')? —t’U"; or, rejecting (¢’U’)? which is of the order 1-~~A, e=(1-+-2h)= f(x, —m)? +2(0,—m)t’ ut oe Ue For the sake of abridging let us also assume po (1A)z(A,—m)’, v= (1 +h)EQ,—m)t’, so that p» is the mean of the squares of the errors, or mean square of the errors, and the equation becomes c=dy+rvU’— UV", (3) the probability of which is q’q’’dt'dt’’. 141. Now, by (121), we have the probability © that sh, or SA,,--h =m the arithmetical mean of the observed va- lues of A, will fall within the limits A==27V(c-+-h). Sub- stituting in those limits the above value of ¢, and observing that (bu--vU’—2”/U")? = / (4) NOU! —'U") + &e., and that U’ and U” being of the order 1--,/h, when di- vided again by ,// are to be rejected, the limits become h==27,/ (Sp+h), or ka==1/ (2n-+h), and the probability of these being the true limits is © mul- 192 RESULTS OF DISCORDANT OBSERVATYONS, tiplied into the probability of the equation c=hyu-4rU'— ’U"’; and is therefore (140) (+7)0(1 —V’—V" ete try", 142. The expression now obtained js the infinitely small probability of the limits Aq=r J (2u-+h) of the average m, in respect of the particular value of s, for which we have de- duced the equation (3). But for every value of s between the limits 0 and 28, there will be an equation correspond- ing to (3); therefore, in order to have the whole proba- bility of those limits, the integral of the expression must be found for all values of ¢’ and é’. From the nature of the expressions e~*? and ¢—t” as well as the consideration that errors beyond a certain magnitude, though possible, are wholly improbable, it is evident that the integration may be extended without sensible error from — 0 to 03 and since the functions V’ and V” contain only uneven powers of ¢' and ¢’”’, the terms into which they enter, disappear in the integrations between those limits. (See Lacroix, Calcul Diff: et Integral, tom. iii. p- 506). Now, from ’#=—= —q@ to t’=+4 © we have ( 96) fe-P*'dt'= J w; and SOO di sie therefore 1 , —/2 =/9 Py YOA—V'—V" ee dt'dt”=0. The result of the preceding analysis is therefore that on the hypothesis of positive and negative errors of equal mag- nitude being equally probable, and on rejecting terms di- vided by h (the number of the observations may be always SO great as to render such terms insensible), we may sub- stitute $u for cin the limits of the error to be apprehended, without sensibly altering the probability, and consequently : “ef 2 —# there is the probability o= ae Re “dt that the true LP AND LIMITS OF PROBABLE ERROR. 193 mean value 4 of the phenomenon A will lie between the limits m=2r/ (duh), or mer, (2n~h), which contain only quantities given by observation. - On this hypothesis we have also (138) e=(1—2h) 3fA*,AdA, or, supposing the law of facility to remain con- stant during the trials, c—}fA°pAda, therefore p=/A* pAda; that is to say, the mean of the squares of the actual errors may be taken for the sum of the products of the squares of the possible errors multiplied by their respective probabilities. It is important to remark that as the obser- vations become more numerous, the quantity p, the mean of the squares of the errors, converges more and more to a constant quantity, and finally becomes independent of the number of observations. — 143. The limits now found may be otherwise expressed. By hypothesis, m=(1-+-h)=i, =the arithmetical mean of the observed values, and p=(.1--/)3(A,—m)’= the mean of the squares of the reputed errors. Now Q,—m)?= h.2—2r,m--m?, and (1+h)=2A,m=2m(1+-h) LDA, == 2m? 5 therefore uo (1-+-h)2),2—m’, that is to say, the mean of the squares of the observations minus the square of the mean. Hence the limits, corresponding to a given proba- bility ©, of the difference between the average of all the observations and the true value, are expressed by either of these formulze —tr,/ {(2x mean square of errors+-h}, —r,/{2x mean square of obs——(mean of obs.)? } + VA; h being the number of observations, and the relation be- tween © and 7 being given by the table. Generally speak- ing, the first of these formule is the most convenient for calculation. K a Bae en sia a ES a a ee a a mae : — Le OE Dae en Oe ga ieee tenner nei Ri rae aT Te SET Son te ay eat TRESS ER Es = SS aad x pe ner epee in oe oe ——— serene iil UM EEE —— = aes i % : i t . iy : ' i 194 RESULTS OF DISCORDANT OBSERVATIONS, 144. Let Z be the limit of the error to be feared in tak- ing the average of the observations as the true result, then l=r,/(2u-+h), and r==1,/(h+-2y.) Now when ¢ is con- stant, that is, for a given probability ©, the determination will be more exact in proportion as J is a smaller number, _and the precision will therefore be proportional to ,/(A--2y). Hence ,/(4+2y) is called by Gauss the measure of the precision of the determinations Suppose two series of ob- servations to have been made for the determination of an ele- ment, the comparative accuracy of the results will depend on two things, the number of observations in each series, and the amount of the squares of the errors in each. If the num- ber of observations is the same in both series, the precision of each result will be inversely as the square root of the sum of the squares of the errors, and the presumption of accuracy is in favour of that result with respect to which the sum of the squares of the errors is less than in the other. On the other hand, if the mean square of the errors is the same in both series, then the observations are alike good in both, and their relative values of the two results are di- rectly as the square roots of the number of observations in each series. Hence, in order that one determination may be twice as good as another, it must be founded on four times the number of equally good observations. These considera- tions are very important, in comparing tables of mean va- lues of whatever kind, for example, of the probabilities of life at the different ages, and in estimating, risks which de- pend upon them. ( 145. Astronomers employ the terms, weight, probable error, and mean error, of a result, to denote certain func- tions of #, the mean square of the errors. The square of the quantity which measures the precision of the result, is AND LIMITS OF PROBABLE ERROR. 195 called the weight of the determination. Denoting the weight by w, we have therefore w=a=ha-Qp=h? -2E(dj-—m)?, or the weight is equal to the square of the number of ob- servations divided by twice the sum of the squares of the errors.- Substituting this in the expression of the limits, we have 7=r--,/w, and r=/,/w ; that is to say, for a given probability ©, the limits of the error to be apprehended in taking the average as the true result are reciprocally pro- portional to the square root of the weight. When obser- vations of different kinds, or results deduced from observa- tion, are compared with each other, their relative weights (supposing the number of observations the same) are inversely as y, and are expressed numerically by taking the weight of a certain series of observations as the unit of weight. 146. The probable error of the determination is that which corresponds to the probability @=3. For @=4 we have r==:476936; whence r,/2=°674489, arid the formula ma=r,/ (2h) becomes m==:674489 /(u--h) ; whence probable error —-674489 / (uA). 147. The mean error of the result of a large number of observations may be deduced from the general formula in (136) as follows. That formula gives gdt=(1+,/7) (1—V)e—*dé for the probability that the sum of the ob- served values will be 25=hk-+-2¢,/(he) exactly. Di- viding the sum by h, qd¢ is also the probability that the average value given by all the observations will be exactly h+-2t,/(e+h). Now, on the hypothesis that positive and negative departures from the mean are equally probable, and supposing the origin of the co-ordinates to be trans~ ferred to the centre of gravity of the curve of mean proba- RYN ae RE RR a ace A CR Im seca = a E Sn giao pe St ae en re rwetin ee ne mertaten TS ee aa rn met eae . aes odin = ans en aid 196 RESULTS OF DISCORDANT OBSERVATIONS, bility, we have A=0, and qdt=(1+- ,/x)(1—V) edt is the infinitely small chance of the average error being 2¢,/(c+-h) exactly. Multiplying therefore this error into the chance of its taking place, and integrating the product from t=0 to ¢= 0, we shall have the mean error, or mean risk of all the possible average errors affected with the positive sign. Now, observing that V represents a quantity divid- ed by ,/, and therefore when multiplied by 2¢,/(ce+-h) becomes of the order 1-+, and may consequently be re- jected, the product of the average error 2¢,/ (c--h) into its probability is 2,/(c+-h) x te~“dt ; and since fiePdi= 3 fe? =}e-?, which from t=0 to t= & becomes simply 4, the integral of the above product from t=0 to tao is /(e-+7h). Substituting for c its value (142) =, this re- sultbecomes ,/ (u-+22h) ; whence on computing ,/(1+-27) we obtain | mean error of series =.398942,/ (uh). This is the mean error or mean risk in respect of posi- tive errors alone, or on the supposition that negative errors are not taken into account. But as positive and negative errors are equally likely, the mean error in respect of nega- tive errors is the same quantity, whence the mean error in respect of errors of both kinds is .797884,/ (u--h). This is usually called the average error. The mean error differs from the probable error in this respect, that it depends on the magnitude of individual errors, as well as on the proportion in which errors of different magnitudes occur. The proba- ble error is independent of the magnitude. 148. When the quantity » (the mean square of the er- rors) has been found from a series of observations, the precision, weight, probable error, and mean error, of a com- i sae AND LIMITS OF PROBABLE ERROR. 197 ing observation of the same kind are found by supposing h=1 in the above expressions, and are respectively precision . : =J/(1+2p) weight : : =1]+2yu probable error . =.674489 / a4 F mean error ° =.39 8942 ,/ p. 149. The preceding formule give the limits of the error to be feared in determining the value of a quantity from a series of observations, when the thing to be determined is that on which the observations are immediately made. We have now to apply the formulz to the cases in which the quantity sought is not observed itself, but is a function of several others, which are separately determined by obser- vation. The following problem is important : Let ube a given function of a number of unknown quan- titities, 2, 2’, x”, &c.; it is required to assign the limits of the probable error in the determination of U, and the weight of the result, when values of x, 2’, x”, found from observations independent of each other, and respectively at- fected with the probable errors gn/ ps eV 2's eN Bs &e. (e= 674489) are adopted instead of the true but unknown values of those quantities. Let u=f(a, 2’, x’’, &c.) be the given function, A, A‘, XN’; &c. observed values of x, 2’, x’, &c. and make \—a=e, W—a’==e’, N’—ax"=e!’, &c. so that e, e’, e’’; &c. are the errors of observation, supposed to be so small that their du du ® du ‘ wae. ign 4 OT ea pte squares may be rejected. Make Fe = Fgh =" deel eg &c., then a, a’, a’’, are given quantities 5 and on substitut- ing ae, a’ +e', x” +e” for x, x', x”, respectively, in the equation u=f(2, x’, #”, &c.), and supposing w to become ae 5 eet es Sh SAS ps OFT ear fa i { % 198 RESULTS OF DISCORDANT OBSERVATIONS, u-|-E when the substitutions are made, so that E is the corresponding error of u, we have, on expanding u by Taylor’s theorem, E=ae-+a'e'+ a’e’ + &c. in respect of a single observation of each of the quantities. Taking the square of both sides of the equation, we have Ei? ae? +ae-+ae!24 &e, +2aa’ee'+2aa"ee”4+2a'a"ele’ + &e, Now since positive and negative errors are supposed equal- ly probable, the sums of the products ee’ » ee”, e', e”, &c. or their mean values, become each =0; therefore ZE* =a" ke? fal? Se? 4¢" Se! 4. &e. Taking the mean value of each of these sums, and observing that » the mean value of Se? is independent of the num- ber of observations (142), and assuming M to be the mean value of ZE?, we get M=07 nba?! tal! 4. &e. This equation contains the solution of the problem, for all the functions of the error are givenin terms of M. The probable error is -674489,/M. 150. Let W be the weight of the determination, and w, w', w"’, &c. the weights corresponding to p, p’, u!’, &e. then by the definition of weight, w is reciprocally propor- tioned to », and W to M; and we have by substitution, q’!* a? a’? Wa1+(— ++ 4 &c.) If the weights are supposed all equal, this becomes w Mss aa”? tal"? b &e, _ Suppose the errors e, e’ e’, &c. to be respectively multi- plied by numbers proportional to the Square roots of the weights, (which is equivalent to supposing all the observa- * AND LIMITS OF PROBABLE ERROR. 199 tions to have the same degree of precision measured by J (pw)), then the value of M becomes Ma? pw-+-al pw’ aly!" 4 &e- But w being reciprocally as p, we have pwo=p/w’ =p’ 'y!', &ec. ==1, therefore W 1 = ea; aC TT a a? +a +a’? + &e. : ee Fax ane ose : = Soar see cee — UE eS RE Cec Bs ec Sinners Fe a a nes (Ftc Spefnsa amin bE weet So LT geen ret tec a OMT = FUR EME Se = = SS: ag We 200 OF THE METHOD SECTION X. OF THE METHOD OF LEAST SQUARES. 151. In the determination of astronomical and physical elements from the data of observation, the thing which is actually observed is for the most part not the element which is sought to be determined, but a known function of that element. Thus, if V be a given function of X deter- mined by the equation V=F(X), the quantity observed may be a value of V, whilst the element sought to be de- termined is X. If the observation could give the value of V with absolute accuracy, then X would also be absolutely known ; but as all observations are affected with certain errors of greater or less amount, owing to the imperfections of instruments or of sense, or the ever varying circum- stances under which they are made, an exact value of X cannot be found from any single observation ; and in order to obtain the utmost precision, it is necessary to employ a great number of observations, repeated under every variety of circumstance by which the result can be supposed to be affected. 152. The observed quantity V, instead of being a func- tion of a single element X, may be a function of several elements X, Y, Z, &c.; for example, V may be the posi- tion of a planet, in which case it is a function of the six elements of the orbit, for the determination of which the OF LEAST SQUARES. 201 observation is made. Each observation gives rise to an equa- tion of this form, V=F(X, Y, Z, &c.) ; therefore when the number of equations is just equal to the number of un- known quantities, the problem is determinate ; and suppos- ing F to be an algebraic function, the values of X, Y, Z, &c. may be found by the ordinary methods of elimination. If the number of equations is less than the number of un- known quantities, the problem is indeterminate; but if greater, it may be said to be more than determinate, inas- much as the equations may be combined in an infinite number of ways, each distinct combination giving a diffe- rent value of the elements. It therefore becomes a ques- tion of the utmost importance to the perfection of the sciences of observation, to assign the particular combination which gives the most advantageous results, or values of X, Y, Z, &c. affected with the smallest probable errors. 153. As approximate values of the elements are in all cases either already known, or can be easily found, the ob ject of accumulating observations is the correction of the approximate values. Let V be the true value of the thing observed, V, an approximate value, however found, X the true value of the element sought, X, an approximate value, corresponding to Vj, so that we have the two equations V=F(X), V.=F(X,); also, let the observed value of V in any observation be L, and make = V—L, ~&V,—L, then v is the true but unknown error of the observation, and / its reputed error, that is to say, the difference between the computed value of the function and the result of the observation. Now if we assume x to represent the true correction of the approximate element, so that X=X,+4+ 2, then, on substituting X,4-x for X in the function F, we Siena teeta ace nee oa ae 202 ‘OF THE METHOD get V=F(X,+-2) ; whence, expanding the function by Taylor’s theorem, and rejecting terms multiplied by a? and higher powers of x, because a is a very small quantity dV, t: dX Let us now denote the differentia] coefficient, which is a known quantity, bya; then, observing that V—V,—v—J, the equation becomes v=/-- az ; that is to say, the true error of the observation is a linear function of the correction O the element. 154. In like manner, when there are several elements, X, Y, Z, &c.3; on making af il sit = Zz com fi &c. a single observation furnishes the equation vl ax by +cz+ &e., and a series of observations, whose errors are respectively v, v', v", &c. gives a system of linear equations equal in number to the number of observations ; namely, vl ax + by + cz4. &e. Vl Laat by tez+ &e. (1) Ol 4a at byt clzd. &e. &c. and the object is to give such values to X, y, Z, &c. that the errors v, v', wv’, &c. in respect of the whole of the observa- tions, shall be the least possible. The equations being supposed independent of each other, if their number is just equal to that of the unknown quantities, the errors v, v', vo", &c. can be made all zero; but if, as is usually the case, there are more equations than unknown quantities, it is impossible by any means whatever to annihilate the whole of them, and therefore all that can be accomplished is to find the system of values of a, ¥, 2, &c. which most nearly, OF LEAST SQUARES. 203 and with the greatest probability, satisfies the whole of the equations. If the observations are not all equally good, the equations are supposed to be each multiplied by a number proportional to the square root of the presumed weight of the observation on which it depends, in order that they may all have the same degree of precision. 155. As the question is to find the most probable values of a, y, 2, &c. the first thing necessary is to express each of these elements in terms of the observations. Suppose kh, k’, k!’, &c. to be a system of indeterminate quantities, independent of « y, 2, &c. and let the first of the above conditional equations be multiplied by %, the second by #’, the third by #”, and so on; then adding the products, if hk, k’, k'’, &c. be determined so as to make the coefficient of # equal to unit, and those of y, z, &c. each equal to 0 ; that is to say, so as to satisfy the equations | ha + hd +-k’a" 4 &e. =1 kb + kb! +k" + &c. =0 (2) he’ + i'c! +. k''e”’ + &c =0 &c. we shall then have a=K--hu- fh’ hv" 4 &c. where K is a quantity independent of », v’, v”, &c. Hence 2 is found =K, with an error =hv+h’v’ --h"v! 4-&c.; and the weight of the determination, by the formula in (150), is ] ke 4h? 4h!’ ? &ee consequently greater in proportion as kh? +h? kh’? 4. &e. is smaller ; and hence of all the possible systems of inde- terminate coefficients, R, h’, k’’, &c. which satisfy the equa- tions (1), the system which gives the most probable value of x, or the most advantageous result, is that for which kh? th’? 4h’? +&c. is an absolute minimum, The weight of the determination is rR 204 OF THE METHOD 156. We have now to find, in terms of known quanti- ties, values of the indeterminate coefficients hk, hk’, hk’, &c, which satisfy the condition of the minimum. For the sake of abridging, let us denote the aggregate of the products aa-a'a'+a"a" 4. &c, by S(aa), that of ab 4. a'b! +a!'b" 4. &c. by S(ab), and so on, and also assume §=av-+a’e" -a!/v" +. &e. | n==bv+b'e! by" 4. 8c. (3) “4 (=cv--clv! belo” +. &e. . | On substituting in these equations the values of x, v, v"" &c. given by the equations (1), there results =S(al) + xS(aa) +yS(ab) 4-28(ac) +4. &c. n=S(61) + 2S(ab) +yS(66)+4-2S(bc)4- &e. (4) (=S(cl) 4+ 2S8(ac) +yS(bc) 4-28(cc)4- &e. a system of equations equal in number to the number of elements 2, y, z, &c. and from which, consequently, those elements would be determined absolutely if the observa- tions were perfectly exact, that is, if the errrors v, 2, v’”, &c. were individually zero, and consequently £, 7, & &c., ' were each zero. On eliminating y, z, &c. from the last sys- : tem, the value of z is given in terms of é, 7, G and known | quantities by a linear equation of the following form : eA fet gn het &e. (5) where f, 9, h, &c. are co-efficients independent of 2, y, z, &c. and also of &, 7, 6 &c. If we now substitute in equation (5) the values &, », ¢ &c. given by equations (3), and also assume a=fa+tgbthe+ &e. a’=f'd' +-9'b! +h'c! 4. &e. (6) alfa! 4b"! 4 Wel! 4. &e., we shall have by addition e=A-pav-+ae! tay’ &e.: whence it appears that a, a’, a’, &c. are a system of multi- OF LEAST SQUARES. 905 pliers by which y, 2, &c. are eliminated from equations (1); they must therefore satisfy the equations (2), whence adda’ 4+-d’a"-+-&e. =1 ab--a’b’-a"b’ 4+ &c. =0 (7) ac+-a’c’ a’! 4+ &c. ==0. Subtracting these from the equations (2) we obtain 0=(h—a)a+-(h’—a')a’ 4 (kh —a" a" + &e. O=(h—a)b +(k —a )b! 4 (2 —a"")b" 4 &e- O=(k—a)e + (k—a Jc! +(k"—al" ce" + &e., on multiplying which respectively by f, 9 h, and adding the products, we get by reason of the equations (6), O=(k—a)a-+- (k’—a’)a! 4 (k—a"" a” 4. &e. This equation may be put under the form ? + #’ 2 Rl”? te & eC. mn? ee’ 2 fear! 2-4 80, + (R—w) 2+ (fh —ee’)? + (Rk —a") 2 +.&e. from which it is evident that 2? + 2/2 +-k’2 4 &c. will be a minimum when f==za, A’==a’, k'’=d’’, &c. Hence it fol- lows that the most probable value of # which can be de- duced from the equations (1), is z==A; and by (150) the weight of the determination is 1+-(aa+-a’a’--a"a" + &c.)= ]-+S(aa). This quantity S (aa) is equal to f the co-efficient of € in the equation (5); for on multiplying the first of equations (7), by f, the second by g, and the third by &, and adding the products, we obtain by reason of equations (6), aa--a’a’ baa” + &e, ap 157. The method explained in the two last paragraphs of determining the most advantageous combination of a system of linear equations, of the form of those in (154), is given by Gauss in his Theoria Combinationis Observationum erroribus minimis obnoxie, (Gottingen, 1823). The prac- tical rule to which it leads is as follows: Having given a near value V of a function of several elements, X, Y, Z, — a a re ‘ v a hg he pees 206 OF THE METHOD &c. and also a series L, L’/, L’, &c. of observed values of V, make (V—L),/w=2, (V—L’),/w'=v’, (V—L”) Jw’, =v", &c. and form the equations in ( 1). From these equa- tions (4) are easily deduced; and from these, again, by elimination, are found the values of 2, ¥y, 2, &c. the correc- tions of the approximate elements X, Y, Z, &c., in equa- tions of the form (5), which, for the sake of symmetry, may be thus written : w=A-+(aa)E-+(a8)n + (ay) C4 &e. Y=B + (a8)é+- (88) + (By)(-+4 &e. 2=C-+(ay)é + (By)n (yy) &e: then the most probable values of z, Y; %, &c. are respectively A, B, C, &c.; the weights of the determinations respectively aay’ (aay? on &c. ; and the probable errors of the se- veral determinations are p,/(aa), pa/ (88), pr (yvy)> Ses, where p='476936. 158. The values of 2, y, z, &c. now deduced are obtained immediately, by supposing the sum of the squares of the errors of observation to be a minimum. Thus, forming the squares of the equations (1), and making Q=v? 4.o/? 4 v’’? 1. &c., the differentiation of © in respect of each of the variables 2, y, z, &c. produces the quantities denoted in (156), by & 7, & &c. that is to say, it gives ae = 26, a 2h, Ze =26, &c. therefore if © be a minimum, &, n, ¢& become severally zero, and the equations (4) give by elimination, a=A, y=B, z=C, where A, B, and C denote the same quantities as above. Now from equation (5) the general value of x is w= A + (aa)é+ (a8) + (ay) and the most probable value being x= A, it follows that OF LEAST SQUAUES. 207 the most probable values of the corrections x, ¥, 2, are found by making the differential coefficients of @ equal to zero, that is, by making v?--v’?-v'”? 4 &c. an absolute mini- mum. Hence this method of combining equations of con- dition is called the method of least squares ; and it follows from the preceding analysis, that it gives the most probable values of the corrections, or the most advantageous results. 159. As an example, let us suppose there is only one un- known element X, of which X, is known to be an approxi- mate value, and L, L’,; L’’, &c. are observed values, the weights of which are respectively propartional to 2, w’, w", &ec. and that it is required to determine the most probable value of X from the observations, and also the weight of the determination. Make (X—L)./w=v, (X,—L) /w=/, and let x be the correction of X, so that X—=X,—z. On sub- stituting thisin (X—L)/w=v, we have (X,—2—L)f/w=r, or v=1,/w—x/w. Each observation gives a similar equa- tion, and the equations (1) in (154) consequently become Vl /w—2fw VV 0! —aV Wal w!—av w",", &e. therefore, multiplying each by the coefficient of itsown x, we have €=S(lw)—zS (w), whence consequently a—=S(fw) S(w)—£-+-S(w), thatis tosay, the most probable value of x is th ho Uw! 4- Uw" + &e. ww +w' +t &e. and the weight of the determination is proportional to the reciprocal of w+w’+w’+ &c. Since X—L—a—H, we have also yee Lw+L/w+L’w-+ &e. Wee w’ fw!’ &e. whence this proposition: If a series of values of an element ? 208 OF THE METHOD 4 are found from observations which have not all the: same degree of precision, the most probable value of the element is found by multiplying each observation by a number pro- portional to its weight, and dividing the sum of the products by the sum of the weights; and the comparative weight of the result is unit divided by the sum of all the weights. If the weights be all equal, and the number of the obser- vations be /, then X=(L+L/+L/’4 &c.)-+h; that is to say, the average of a series of equally good observations gives the most probable value. The average may, there- fore be considered as a particular casc of the method of least squares. 160. To illustrate the method of proceeding when there are several elements to be corrected from the observations, we shall take the following numerical example from Gauss (Theoria Motus). Suppose there are three elements, and that three observations, of equal weight, have given the equations +—y -+-22=3, 3x 4-2y—5z=5, 4x +y+4z2=21; and that a fourth observation, of which the relative weight is one-fourth, or its precision one-half, of that of the others has given —2x4-6y+6z=28. The first step is to reduce this last equation to the same standard of weight with the others, for which purpose it must be multiplied by 3; it, . then becomes —x + 3y4.3z—=14. Now, asa, y, and z can- not be determined so as to satisfy four independent equa- tions, we suppose each observation, or equation, to be affect- ed with an error v, and accordingly obtain the following system of equations, corresponding to equations (1), viz. : v= —3 4+ 27—y 4-22 v =—5.43a4 2y—dz of == 2] + 427+y+42 VY" =—14—x4 37 432, Seemed OF LEAST SQUARES. 209 from which the most probable values of x, y, and 2 are to be deduced. Let each equation be multiplied by the co- efficient of its own 2, taken with its proper sign, namely, the first by 1, the second by 3, the third by 4, and the fourth by —1; the results added together give the value of &, namely, &=—88 + 272+ 6y. In like manner, let the first be multiplied by —1, the second by 2, the third by 1, and the fourth by 3, the sum of the products will give ».- Lastly, let the equations be multiplied respectively by the coefficients of 2, and the sum of the products made equal to ¢; we have then the following equation’ corresponding to the equations (4) £=—88 + 27x + by +0 n= —70+ 62 + liy +2 —=—107 +0 ++ 542. From these we get by elimination 19899x==49154 4 809&—324n + 6¢ 737y=2617—12é +-54n—€¢ 397 982==76242 + 12&—54y + 14736, whence (157) A, B, C, the most probable values of 7, Y, 2; are respectively 49154 2617 76242 — 2919* 9-470, B= = 3'551, C= 19165 A= roggg te 110 Ba 737 351 so798 and the relative weights w, w’, w”, are respectively Bl 9899 4. ADE Ba ie » 39798 _ °="809 se Oe 80 gg I SO eae ead i whence the probable errors ('476936-+ ,/w) are respectively 096, -129, -092. } The method of least squares, to which modern astrono- my is indebted for much of its precision, was first proposed by Legendre, in his Nowvedles Méthodes pour la Determi- nation les Orbites des Cométes, (Paris, 1806,) merely as a ne aiioanesia - sionaae simone ~~ - —t 5 om sa i cane eal a = AACA NRE AR ta ; sve arameth atten ppaoeea = Neen nc iinnie ae ninneincaaiaae anaemia ed Bey i = pe ye ee mee vo Nang is opi —— SG Goa ce eae area — ns Tae emepmeeenge tee pap ley 210 OF THE METHOD means of avoiding inconvenience and uncertainty arising from the want of a uniform and determinate method of combining numerous equations of condition, and without reference to the theoty of probability. The same method, however, had previously been discovered by Gauss, and a demonstration of it, deduced from the general theory of chances, was given by him in his Theoria. Motus, (1809.) It may be shewn in various ways, that this method of com- bination gives values of the unknown quantities affected with the smallest probable errors ; but it is to be observed, that all the demonstrations are subordinate to the hypo- thesis, that positive and negative errors of equal magni- tude are equally probable, or that the average of a large number of results gives the most probable value, and con- sequently that the function which represents the probabi- lity of an error has the form assigned to it in (132). The limits of this article will not permit us to enter into further details respecting the applications of the method of least squares. On the general theory of the probable errors of results deduced from observation, and the most advantageous methods of combining equations of condition, the reader may consult the Théorie Analytique des Proba- bilités of Laplace ; the Theoria Motus of Gauss ; the The- oria Combinationis Observationum, and the Supplementum Theorie Combinationis, &c. (Gottingen, 1828) of the same author ; the Recherches sur la Probabilité des JSugements, with the two Memoirs of Poisson in the Connaissance des Tems for 1827 and 1882; and three masterly papers, by Mr. Ivory, in the Philosophical Magazine for 1825. In the volumes of the Berliner Astronomisches Jahrbuch for 1833, 1834, and 1835, M. Encke has treated the subject at great length, and given a number of formule calculated OF LEAST SQUARES. 211 to facilitate the labours of the computer. We may also refer, in conclusion, to a very recent and remarkable dis- quisition on the theory of probable errors, by the celebrated astronomer Bessel, forming Nos. 358 and 359 of Schuma- cher’s Astronomische Abhandlungen, Altona, October 1838. TABLE OF THE VALUES OF THE INTEGRAL Ba) 2 : "e bn re =— e ATS 0 a/ Tr, s for-intervals each ='01, from r=0 to 7==3, with their first and second differences. © ae Se - - oe rs <= tree SAE ae m2 ror RES SSPE , - — 8 emi ee ET a os Fa al, ‘a a a — *, “evs nm se - _ ness = oe ae a ——— a pti? cae = . SRI ahaa oR SEE = OS TS = Soren: f | ae 0563718 | 112497 0676215 | 112362 0788577 | 112204 0900781 | 112025 1012806 | 111824 1124630 | 111600 1236230 | 111354 1347584 | 111087 1458671 | 110799 | < 1569470 | 110489 "1679959 | 110158 1790117 | 109806 1899923 | 109434, 2009357 | 109041 2118398 | 108627 *2227025 | 108193 2336218 | 107740 2442958 | 107267 2550225 | 106775 *2657000| 106263 °2763263 | 105734 *2868997 | 105185 *2974182 | 104618 *3078800 | 104034, 3286267 *3389081 3491259 *3592785 3693644 3793819 *3893296 3992059 “4090093 *4187385 -4283922 4379690 4474676 4568867 4662251 4754818 4846555 4937452 5027498 -5116683 *5204999 5292437 °5378987 5464641 5549392 5633233 *S716157 *5798158 102814 102178 101526 100859 100175 99477 98763 98034 97292 96537 95768 94986 94191 93384 92567 91737 90897 90046 89185 88316 87438 86550 85654 84751 | 83841 82924 82001 81071 6116812 6194114 *6270463 6345857 *6420292 °6493765 6566275 °6637820 -6708399 -6778010 6846654 *6914330 6981038 -7046780 *7111556 "7175367 °7238216 ‘7300104 °7361035 “7421010 *7480033 *7538108 *7595238 “7651427 -7706680 *7761002 *7814398 -7866873 *79 18432 *7969082 -8018828 "8067677 *8115635 8162710 *8208908 *8254236 *8298703 *8342315 -8385081 TABLE, CONTINUED. A 78251 77302 76349 75394 74435 73473 72510 71545 70579 69611 68644 67676 66708 65742 64776 63811 62849 61888 60931 59975 59023 58075 57130 | 56189 | 55253 54322 53396 52475 51559 50650 49746 48849 47958 ATOTS 46198 45328 44467 43612 4.2766 41927 © *8427008 *8468105 *8508380 *8547842 *8586499 *8624360 8661435 8697732 °8733261 8768030 *8802050 *8835330 8867879 8899707 “8930823 *8961238 -8990962 9020004 9048374 9076083 -9103140 °9129555 9155339 -9180501 9205052 -9229001 9252359 9275136 "9297342 9318987 -9340080 °9360632 "9380652 "9400150 °9419137 9437622 9455614 9473124 -9490160 *9506733 A 41097 40275 39462 38657 37861 37075 36297 35529 34769 34020 33280 32549 31828 31116 30415 29724 29042 28370 27709 27057 26415 25784 25162 24551 23949 23358 22777 22206 21645 21093 20552 20020 19498 18987 18485 17992 17510 17036 16573 16118 0:9522851 9538524 *9553762 *9568573 9582966 -9596950 - *9610535 9623729 9636541 9648979 *9661052 *9672768 9684135 9695162 9705857 ‘9716227 9726281 9736026 9745470 ‘9754620 9763484 -9772069 ‘9780381 9788429 9796218 *9803756 -9811049 9818104 “9824928 "9831526 9837904 9844070 ‘9850028 9855785 ‘9861346 9866717 °9871903 ‘9876910 -9881742 ‘9886406 TABLE, CONTINUED. 9890905 9895245 9899431 9903467 9907359 ‘9911110 "9914725 9918207 9921562 ‘9924793 9927904 "9930899 9933782 9936557 "9939226 994.1794 9944263 ‘9946637 9948920 9951114 9953223 9955248 9957195 9959063 ‘9960858 ‘9962581 °9964235 9965822 9967344. 9968805 “9970205 9971548 9972836 9974070 9975253 ‘9976386 9977472 ‘9978511 9979505 9980459 0-9981372 9982244 *9983079 ‘9983878 9984642 9985373 ‘9986071 9986739 9987377 9987986 -9988568 9989124 9989655 -9990162 9990646 -9991107 2°36) -9991548 -9991968 2°38] 9992369 9992751 9993115 2-41) -9993462 9993793 9994108 2-44) -9994408 2-45) -9994694 2-46) -9994966 2°47) -9995226 2-48) -9995472 2°49) -9995707 2-50} -9995930 2°51; 9996143 ‘9996345 9996537 9996720 ‘9996893 *9997058 2°57| -9997215 2-58) +9997364 2°59| -9997505 TABLE, CONTINUED. © 2.60| -9997640 9.61| -9997767 2.62| -9997888 2-63| -9998003 2-64| -9998112 2:65| 9998215 2-66| -9998313 2-67| -9998406 2.63| 9998494 2.69| -9998578 | 70| -9998657 ‘T1| -9998732 | -72| +9998803 73| -9998870 74| -9998933 | ‘75| -9998994 ‘76.| +9999051 ‘17| -9999105 ‘18| -9999156 ‘79| -9999204; 2-80} 9999250 2-81 -9999293 282 | -9999334 ‘9999372 -9999409 2-86| -9999476 2°87! -9999507 2-88] -9999536 2:89| -9999563 2°90| -9999589 2-91] -9999613 2°92) -9999636 2:93} -9999658 9999679 2:95) -9999698 2°96| -9999716 2°97 | -9999733 3:00| -9999779 2-85| -9999443 | 215 U9 aD OO OO 09 POP LS S ee Or & Cr Or TD ADDN bm GO Re OO 09 Sot OO Ot DW In the note, read ERRATA. Analytique des Probabilites. ilémentaire. — 49, line 13, read beginning with. Traité — 43, for adopt, read adapt. — 89, line 9, for m—m, read m+m’. LY ‘= ~ . S o BR ro os o Cy a” .@ %» ~~ S °e i) 8 S S a a) ~) ww mj a) ‘~ S a) Ry 3 a) oe ~~ Q & om) =< o &a S fas Sakae eT a mr eet ternary teste ace A = eS I A 9 Sc crete ew weer ee tin ls AeEcagnt alan chi aL gee age Sol seeepmneng acme nenia< Site PEARY ee eer OP n: SR n . te 6 < Ch _ ot ; | | or for the use of others. THE TRAVELERS INSURANCE COMPANY Actuarial Department LIBRARY No. ve. This book is to be returned within two months for renewal It is to be held in any case only as _ long as needed. Books are subject to recall by the librarian at | any time. DATE DATE Return book to librarian’s desk—not to shelf. 8630. 4-24-14. ee URBANA BURGH eaceepeee eecak IVERSITY OF ILLINOIS is geentt C001 A TREATISE ON PROBABILITY EDIN beatets andl iy lita’ 519.2G138T Aeesreetecs eee SI en anaes en eon AW SEWN Daw Te