176 



INDUCTION. 



different : the credibility of his assertion is but a small frac- 

 tion of what it was in the former case ; the reason of the dif- 

 ference being as follows. 



The witnesses of whom we are speaking must, from the 

 nature of the case, be of a kind whose credibility falls materially 

 short of certainty : let us suppose, then, the credibility of the 

 witness in the case in question to be -^ ; 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 sup- 

 pose that there have taken place a sufficient number of draw- 

 ings 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 thousand 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 an- 

 nouncement, that false announcement 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 

 therefore requires a much greater amount of testimony to render 

 it credible.* 



* Not, however, as might at first sight appear, 999 times as much. A 

 complete analysis of the cases shows that (always assuming the veracity of the 

 witness to be -j^) in 10,000 drawings, the drawing of No. 79 will occur nine 

 times, and be announced incorrectly once ; the credibility therefore of the 

 announcement of No. 79 is T 9 7 ; while the drawing of a white ball will occur 

 nine times, and be announced incorrectly 999 times. The credibility therefore 

 of the announcement of white is-n^re* an( ^ the ratio of the two 1008 : 10 the 

 one announcement being thus only about a hundred times more credible than 

 the other, instead of 999 times. 



