PROBABILITY 777 and we have The factors towards the beginning and end may be all taken as 1, because the terms of the binomial increase rapidly in value from either end when r= 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 r. As the sth factors before and after U may be expressed thus (s being always very small compared to r) TT * - Ue ?-U and as . . . Ue Vu 2 . . . =1, we have Now we have seen in art. 8 that sfl r> ?!_, Pri, - ?=? T 8 = Te -2i>q>- IPV , t, = fe tpqr tpgr Hence sT - s *-^~ T > e 2*?r- Hence the exponent of c above becomes rU pq ^=0 r "- 1 " 1 $! being the extreme limit for s. IP s " r2 ./ N If we put x = T- , and x 2 c~ 2^ (> = <p(x), /r the above sum is Now it is easy to prove that is finite; and much more is it so when the superior limit is finite. Hence the exponent of e becomes -^f rU ^ pq where K is finite ; so that the exponent becomes infinitesimal when r = co . The limit therefore towards which X tends is X = U = s - qe = 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 or 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. 1 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 priori 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 + 5=1. But suppose I only know that one of the two is his, and, besides, some one whose credibility is tells me lie 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. 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 iN the event happens, and out of these the witness asserts in ^>N cases that it did happen. Now, out of the whole number, he asserts in |N cases that it happened, as there is no reason for his affirming oftener than he denies (or, it may be said, he affirms in i?;N cases where it did happen, and in i(l -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 aflirms it, we find i/>N-^N=^>. 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 supceed in other cases which are not so easily to be discerned. 28. Let us now consider the same question when the a 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 1 A great number N of trials being made, the number in which the white ball is drawn is N, and out of these he states it in n~ 1 p~N cases. Out of the remaining (l-n~ l ) N cases where a black ball was drawn, he says (untruly) that in (1 -p) (1 -n 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 ~ l P P (OQ] - This holds for any event whose a priori probability is n~ l . If n be very large, this probability will be very small, unless jj 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, -zr=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 a 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 |N ; out of these the witness A affirms the success in |pN cases ; out of these the witness B affirms it, too, in ^;/N cases. 2 Out of the N failures A affirms a success in |(1 -j3)N cases ; and out of these B also affirms one in ^(1 -p)(l -/)N cases. Hence, dividing the favourable cases by the whole number, the probability sought is Pi / 97 > W = ,W_LM _.^n _,,M ^ /i where p, p are the credibilities of the two witnesses. This very important result also holds if p be the probability of the event derived from any source, and p 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 hypotheses 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/ ; on the second, "it is 1 -/. Hence by formula (12) the pro bability a posteriori of the first hypothesis, viz., that the joint statement is true, is, as before, PP 1 pp + (l-p)(i-p ) 31. If a third witness, whose credibility is p", concurs with the two former, we shall have to combine/ with -a in formula (27) ; hence the probability -a of the statement when made by three witnesses is asp ppp and so on for any number. 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 the success here in more or less than Ipp ^S. cases.
XIX. 98