ness 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 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 announcement, 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.[1]
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 argument 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 random, viz., black and white; but considered in respect of the numbers attached 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 color, 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 910) 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 910); 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 91008, and 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.
