4i6 



INDUCTION. 



only has been drawn out, then if an 

 eye-witness affirms that the number 

 drawn vas 79, this, though the 

 chances were 999 in 1000 against it, 

 is not on that account the less 

 credible ; its credibility is equal to 

 the antecedent probability of the wit- 

 ness's veracity. But if there were in 

 the box 999 black balls and only one 

 white, and the witness affirms that 

 the white ball was drawn, the case 

 according to Laplace is very different : 

 the credibility of his assertion is but 

 a small fraction of what it was in the 

 former case ; the reason of the differ- 

 ence being as follows : — 



The witnesses of whom we are 

 speaking must, from the nature of 

 the case, be of a kind whose credi- 

 bility falls materially short of cer- 

 tainty : let us suppose, then, the 

 credibility of the witness in the case 

 in question to be ^'^ J that is, let us 

 suppose that in every ten statements 

 which the witness makes, nine on an 

 average are correct and one incorrect. 

 Let us now suppose that there have 

 taken place a sufficient number of 

 drawings to exhaust all the possible 

 combinations, the witness deposing in 

 every one. In one case out of every 

 ten in all these drawings he will 

 actually have made a false announce- 

 ment. But in the case of the thou- 

 sand tickets these false announcements 

 will have been distributed impartially 

 over all the numbers, and of the 999 

 cases in which No. 79 was not drawn, 

 there will have been only one case in 

 which it was announced. On the 

 contrary, in the case of the thousand 

 balls, (the announcement being always 

 either "black" or "white,") if white 

 was not drawn, and there was a false 

 announcement, that false announce- 

 ment must have been white ; and 

 since by the supposition there was a 

 false announcement once in every ten 

 times, white will have been announced 

 falsely in one -tenth part of all the cases 

 in which it was not drawn, that is, in 

 one-tenth part of 999 cases out 'of every 

 thousand. White, then, is drawn, on 

 an average, exactly as often as No. 79, 



but it is announced, without having 

 been really drawn, 999 times as often 

 as No. 79 ; the announcement there- 

 fore requires a much greater amount 

 of testimony to render it credible.* 



To make this argument valid it 

 must of course be supposed that the 

 announcements made by the witness 

 are average specimens of hia general 

 veracity and accuracy, or at least 

 that they are neither more nor less so 

 in the case of the black and white 

 balls than in the case of the thousand 

 tickets. This assumption, however, 

 is not warranted- A person is far 

 less likely to mistake who has only 

 one form of error to guard against, 

 than if he had 999 different errors to 

 avoid. For instance, in the example 

 chosen, a messenger who might make 

 a mistake once in ten times in report- 

 ing the number drawn in a lottery, 

 might not err once in a thousand 

 times if sent simply to observe 

 whether a ball was black or white. 

 Laplace's argument, therefore, is 

 faulty even as applied to his own 

 case. Still less can that case be re- 

 ceived as completely representing all 

 cases of coincidence. Laplace has so 

 contrived his example, that thougli 

 black answers to 999 distinct possi- 

 bilities, and white only to one, the 

 witness has nevertheless no bias which 

 can make him prefer black to white. 

 The witness did not know that there 

 were 999 black balls in the box and 

 only one white ; or if he did, Laplace 

 has taken care to make all the 999 

 cases so undistinguishably alike, that 

 there is hardly a possibility of any 



* Not, however, as might at first sight 

 appear, 990 times as much. A complete 

 analysis of the cases shows that (always 

 assuming the veracity of the witness to 

 he -^r) in 10,000 drawings, the drawing of 

 No. 79 will occur nine times, and be an- 

 nounced incoiTectly once ; the credibility, 

 therefore, of the announcement of No. 79 

 is -fo ; while the drawing of a white ball 

 will occur nine times and be announced 

 incrrectly 999 times. The credibility, 

 therefore, of the announcement of white 

 is yjj'ny, and the ratio of the two 1008 : 10 ; 

 the one announcement being thus only 

 about a hundred times more credible than 

 the other, ius'ead of 999 times. 



