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