GROUNDS OF DISBELIEF. 447 



ness makes, nine on an average are correct, and one incorrect. Let us 

 noNV 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 announcement. 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 ev- 

 ery 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.* 



To make this argument valid it must of course be supposed, that the 

 announcements made by the witness are average specimens of his 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 reporting 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 received as completely representing all cases of coincidence. Laplace 

 has so contrived his example, that though black answers to 999 distinct 

 possibilities, 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 cause of falsehood or error 

 operating in favor of any of them, which would not operate in the same 

 manner if there were only one. Alter this supposition, and the whole ar- 

 gument falls to the ground. Let the balls, for instance, be numbered, and 

 let the white ball be No. 79. Considered in respect of their color, there 

 are but two things which the witness can be interested in asserting, or can 

 have dreamed or hallucinated, or has to choose from if he answers at ran- 

 dom, viz., black and white ; but considered in respect of the numbers at- 

 tached to them, there are a thousand; and if his interest or error happens 

 to be connected with the numbers, though the only assertion he makes is 

 about the coloi", the case becomes precisely assimilated to that of the thou- 



* 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 ^y) in 10,000 draw- 

 ings, the drawing of No. 79 will occur nine times, and be announced incorrectly once ; the 

 credibility, therefore, of the announcement of No. 79 is ^j^ ; while the drawing of a white ball 

 will occur nine times, and be announced incorrectly 999 times. Tiie credibility, therefore, of 

 the announcement of white is j^Qg, and the ratio of the two 1008 : 10 ; tlie one announcement 

 being thus only about a hundred times more credible than the other, instead of 999 times. 



