778 PROBABILITY As an example, let us find how many witnesses to a fact, the odds against which are 1,000,000,000,000 to 1, would be required to make it an even chance that the fact did occur, supposing the credibility of each witness to be # = & Let x be the number.
12 "~log9 so that thirteen such witnesses would render the chance more than an eveji one. 32. Let us now consider an event which may turn out in more than two ways, and let each way be equally probable a priori, and suppose a witness whose credibility is p states that it turned out in a certain way ; what is the chance that it did so ? Thus if a die has been thrown, and he states that ace turned up ; or if tickets in a lottery are numbered 1, 2, 3, &c., and he states that 1 was drawn ; to find the chance that he is right. Take the case of the die, and suppose a great number N of throws. In &N the ace turns up, and he says so in pN cases. In N the two turns up, and he is wrong in &(1 -j?)N cases out of these ; but he says ace in only of these, as there is no reason why he should give it more or less often than any of the five wrong numbers. In the same way for the other throws ; so that the whole number of cases where he says ace turned up is and, the number, out of these, when it actually turned up being &J0N, we find tJie chance it did turn up is p, the credibility of the witness. In any such case, this result will hold. We might indeed safely have argued that when the die is thrown a great number of times, any witness, whatever his veracity, will quote each face as often as any other, as there is no reason for one to turn up oftener than another, nor for him to affirm, rightly or wrongly, one rather than another ; so that he will say ace in &N of the throws, while he says ace in pN out of the &N cases where it does turn up. This result compared with art. 28 affords an apparent paradox. If a large number of tickets are marked 1,0,0,0,0,0 . . . . and a witness states that 1 has been drawn from the bag, we see from art. 28 that the chance he is right is very small ; whereas if the tickets were marked 1,2,3,4,5,6 .... and he states that 1 has been drawn, the chance he is right is p, his own credibility. However, we must remember that in the first case he is limited to two state ments, 1 and,0, and he makes the first, which is very improbable in itself; whereas in the other case, the assertion he makes is in itself as probable as any other he can make e.g., that 2 was the ticket drawn and therefore our expectation of its truth depends on his own crt libility only. 33. Suppose now that two witnesses A, B both assert that the event has turned out in a certain way, there being, as in art. 32, n equally probable ways. Both, for instance, say that in a lottery numbered 1,2,3,4,5 . . . . No. 1 has been drawn. A large number N of drawings being made, 1 is drawn in ?i- J N cases; out of these A says 1 in 71-^N cases, and out of these B also says 1 in n~ l pp N. No. 2 is drawn in 7i- J N cases ; here A is wrong in n~ -p)~S, but says 1 in only (n-lJ- n-^l-^N; and B will also say 1 in (1 -p )(n- I)- 1 of these ; that is, bol;h agree that 1 has been drawn in cases. So likewise if No. 3 has been drawn, and so on ; hence, when No. 1 has not been drawn, they both say that it has in cases. Hence the number of cases where they are right divided by the whole number of cases where they make the statement, that is, the probability that No. 1 has been drawn, is PP (29). If n be a large number the chance that they have named the ticket drawn is nearly certainty. Thus, it two independent witnesses both select the same man out of a large number, as the one they have seen commit a crime, the presumption is very strong against him. Of course, for the case to come under the above formula, it is supposed that some one of tiie number must be guilty. 34. In the same case, when the event may turn out in n ways not equally probable, as in a race between n horses A, B, C . . . . whose chances of winning area, b, c. . . ., so that a + 6 + c + . . . =1, if one witness whose credibility is p states that A has won, it is easily shown by the same reasoning as in art. 33 that the pro bability A has really won is ap --i- -,-,. . . (31). and if two witnesses say so, it is app -a)(-l) It is easily shown in formula (30) that if^>?i~ 1 the probability a is increased by the testimony, beyond a, its antecedent value. Thus, suppose there are ten horses in a race, and that one of them, A, has a chance 3 of winning, and that just after the race I learn that a black horse has won, black being A s colour ; now, if I know that ^ of racehorses in general arc black, this gives me a new chance (see art. 16) that A has won. Therefore from (30) the chance of the event is now T = f. 35. To illustrate the effect of discordant testimony. In art. 29 let A have asserted that the fact occurred, and let B deny it. It is easy to see that 1 p 1 is to be put for p , so that the probability that it did occur is if there had been an a priori probability a in favour of the fact this would have been X1 -P } ,OON rwi ^ (33). Thus if the credit of both witnesses were the same, p=p , and we find from (33) -a = a, so that the evidence has not altered the likelihood of the event. 36. Where the event may turn out in n equally probable ways as in art. 33, and the witness A asserts one to have occurred, say the ticket marked 1 to have been drawn, while the witness B asserts another, say the ticket marked 2 ; to find the chance that No. 1 was drawn, By the same reasoning as in art. 33 we find for the chance (34)
This result will also follow if we consider B s evidence as testi mony in favour of No. 1 of the value (1 -_?/)( - 1 )~ J . When the number of tickets n is very great, (34) gives p-pp " i / * 1 -pp 37. As remarked in art. 26, the methods we have given for de termining the probability of testimony apply to cases where the evidence is derived from other sources. Thus, suppose it has been found that a certain symptom (A) indicates the presence of a certain disease in three cases out of four, there is a probability f that any patient exhibiting the symptom has the disease. This, however, must be considered in conjunction with the a priori probability of the presence of the disease, if we wish to know the value of the evidence deduced from the symptom being observed. For instance, if we knew that f of the whole population had the disease, the evidence would have no value, and the credibility of the symptom per sc would be , telling us nothing either way. For if a be the a priori probability, -a that after the evidence, p the credibility of the evidence, we have found ap so that, if w = a, p = . If & and a are given, the credibility p of the evidence is deduced from this equation, viz., a + -ar - 2 38. Suppose now the probability of the disease when the symptom A occurs is ar (that is, it is observed that the disease exists in z?N cases out of a large number N where the symptom is found), and likewise the same probability when another independent symptom B occurs is -a . What is the probability of the disease where both symptoms occur ? Let a be the a priori probability of the disease in all the cases ; then the value of the evidence of B is, as explained above, P - ; ^ i a + TV - 2a-ar and this has to be combined with -a, which is the probability of the disease after A is observed. We find the probability (n)
required to be