i
e A e a2 b S
and we have
T» 'R 0 3
... <U+2)i> '(U+7E_> g <U~ ;>t1<U“2;>1_' -
The factors towards the beginning and end may be all taken as 15
because the terms of the binomial increase rapidly in value from
either end when 7= oo, and we shall have the true limit for X by
taking an indefinitely great number of factors on either side of U,
which number, however, may be infinitely less than 7.
As the sth factors before and after U may be expressed thus
(s being always very small compared to »)—
¢ \T e \t.«
<U(}s)'7[j/| S’ <LTU—31'U/ 3
andas... To+T,+T+¢,+4,. .. =1, we have
€
X=TerU
Now we have seen in art. 8 that
(oo 2Tt Ty—t;—2p— . . )
Sl e ek
Ts=Te 2pqr " 2pqr 5 hi— Ee %0 2pqr
§2 P —
9 0
29T pgr’
Hence Te—t;=Te—
g s
4 STS—Sis:LT)‘(A?['F-; € 2)3,1,- .
2 .
Hence the exponent of ¢ above becomes
B 9 2
St 4 S lE e
B =€ 3par
rU " pq Es:O o ReRn S
8 being the extreme limit for s.
s . i x2
If we put « =\7/" and xe " 9pe = ¢(),
the above sum is
1 2 s e
¢(D)+¢<F>+¢(F>+ e ¢<7,7%;>:7'£ E "p(x)dx .
Now it is easy to prove that
o 22
-[ x’e” K dx is finite;
and much more is it so when the superior limit is finite.
Hence the exponent of ¢ becomes
€ g Fouat e )—Q Xy
P LA e ) 8 %
U pg U° pg
where K is finite; so that the exponent becomes infinitesimal
when r=o0,
The limit therefore towards which X tends is
X=U=z-ge=a+pe,
that is, the mathematical value of the fortune.
The very important applications of probability to annuities and
insurance are to be found in the articles on those subjects, to which
therefore we refer the reader.
IV. PROBABILITY OF TESTIMONY.
26. We have here to treat of the probability of events attested
by several witnesses of known credibility, or ‘which have several
different probabilities in their favour, derived from different inde-
pendent sources of information of any kind, of known values.
A witness may fail in two ways: he may be intentionally dis-
honest, or he may be mistaken ; his evidence may be false, either
because he wishes to deceive, or because he is deceived himself,
However, we will not here take separate account of these two
sources of error, but simply consider the probability of the truth of
a statement made by a witness, which will be a true measure of the
value of his evidence. To estimate this probability in any given
case is not an easy matter ; but if we could examine a large number
of statements made by a certain person, and find in how many of
them he was right, the ratio of these numbers would give the pro-
bability that any statement of his, taken at random, whether past
or future, is a true one.
27. Suppose a witness, whose credibility is p, states that a fact
occurred or did not occur, or that an event turned out in one way,
when only two ways are possible. If nothing was known a prior:
as to the probability of the fact, or if its real facility was %, it is
clear that the probability that it did occur is p. For if a great
1 The question now before us is quite different from that of the chance of an
event happening or having happened which may happen in different ways, in
which case we add the separate probabilities. Thus if there are but two horses
in a race, of equal merit and belonging to one owner, his chance of winning is
4+1=1. But suppose I only know that one of the two is his, and, besides, some
one whose credibility is } tells me he has won the race; here I have two separate
probabilities of } each for the same event ; but it would clearly be wrong to add
them together,
FROBABILIT Y
7
number N of trials were made (either really as to the event, if its
facility is known to be %, as in tossing a coin ; or as to it and other
cases resembling it as to our ignorance of the real facility, if such
is the state of things) in 4N the event happens, and out of these
the witness asserts in 4pN cases that it did happen. Now, out of
the whole number, he asserts in 4N cases that it happened, as there
is no reason for his affirming oftener than he denies (or, it may be
said, he affirms in 1pN cases where it did happen, and in (1 -p)N
cases where it did not). Hence, dividing the whole number of cases
when it happens and he affirms it by the whole number of cases
where he affirms it, we find 1pN+1N'=p.
We have entered at length on the proof of what is almost self-
evident (perhaps indeed included in the definition) in this case,
because the same method will succeed in other cases which are not
so0 easily to be discerned.
28. Let usnow consider the same question when the o Priori pro-
bability of the fact or event is known. Suppose a bag contains n
balls, one white and the rest black, and the same witness says he
has seen the white ball drawn ; what is the chance that it was
drawn ?
A great number N of trials being made, the number in which
; : b L bl
the white ball is drawn is 7N, and out of these he states it in
n-1pN cases. Out of the remaining (1-n-1) N cases where a
black ball was drawn, he says (untruly) that in (1-p) (1-2-1) N
cases it was white.
Now, dividing the number of favourable cases, viz., those where
he says it is white and it is so, by the whole number of cases, Viz.,
those where he says it is white, we have for the probability
required
= n-1p P
= v Iip+(1-n-)1l-p) n-1-m-2)p° ° °
This holds for any event whose & priori probability is n-1.
If » be very large, this probability will be very small, unless p
is nearly =1; and, indeed, if we go back to the common sense
view, it is clear we should hesitate to believe a man who said he
had drawn the white ball from a bag containing 10,000 balls, all
but it being black. It may be observed that if n=2, @w=p, as in
art. 27.
We have thus a scientific explanation of the universal tendency
rather to reject the evidence of a witness than to accept the truth
of a fact attested by him, when it is in itself of an extraordinary or
very improbable nature.
29. Two independent witnesses, A and B, both state a fact, or
that an event turned out in a particular way (only two ways being
possible), to find the probability of the truth of the statement.
Supposing nothing is known @ priori as to the event in question,
let a great number N of trials be made as to such events ; the
number of successes will be 4N ; out of these the witness A affirms
the success in $pN cases; out of these the witness B affirms it,
too, in 4pp'N cases.2 Out of the 4N failures A affirms a success
in §(1-p)N cases; and out of these B also affirms one in
(1 -p)(1-p')N cases. Hence, dividing the favourable cases by the
whole number, the probability sought is
2
VS e v
P’ +(1-p)1-p)
where p, p’ are the credibilities of the two witnesses.
This very important result also holds if » be the probability of
the event derived from any source, and ' the credibility of one
witness, as in art. 28 ; or if p and p’ be any independent proba-
bilities, derived from any sources, as to one event.
30. We give another method of establishing the formula (27).
Referring to art. 13, the observed event is the concurrent evidence
of A and B that a statement is true. There are two Aypotheses
—that it is true or false. Antecedent to B’s evidence the pro-
babilities of these hypotheses are p and 1-p (art. 27), as A has
said that it is true. The observed event now is that B says the
same. On the first hypothesis, the probability that he will say this
is p’ ; on the second, it is 1—p". Hence by formula (12) the pro-
bability e posteriori of the first hypothesis, viz., that the joint
statement is true, is, as before,
pp’
o E(I=PRE=Pp}
31. If a third witness, whose credibility is ", concurs with the
two former, we shall have to combine p” with = in formula (27);
hence the probability =’ of the statement when made by three
witnesses is
; z:'-Z)l/ i ])Z)/])/I
T T (-1l +A-p)A-PA-p")
and so on for any number.
(26).
Pe g ol
(28);
2 Here we are assuming the independence of the witnesses, If B, for instance,
were disposed to follow A's statements or to dissent from them, he would affirm
thie success here in more or less than 4 pp'N cases,
i Es ;
XIX - 8
