Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/122

112 numbers in the urn; it is changed by virtue of the proof into the veracity itself of the witness; it may then be decreased by the proof. If, for example, the urn contains only two numbers, which gives $1⁄2$ for the probability à priori of the drawing of number 1, and if the veracity of a witness who announces it is $4⁄10$, this drawing becomes less probable. Indeed it is apparent, since the witness has then more inclination towards a falsehood than towards the truth, that his testimony ought to decrease the probability of the fact attested every time that this probability equals or surpasses $1⁄2$. But if there are three numbers in the urn the probability à priori of the drawing of number 1 is increased by the affirmation of a witness whose veracity surpasses $1⁄3$.

Suppose now that the urn contains 999 black balls and one white ball, and that one ball having been drawn a witness of the drawing announces that this ball is white. The probability of the event observed, determined à priori in the first hypothesis, will be here, as in the preceding question, equal to $9⁄10000$. But in the hypothesis where the witness deceives, the white ball is not drawn and the probability of this case is $999⁄1000$. It is necessary to multiply it by the probability $1⁄10$ of the falsehood, which gives $999⁄10000$ for the probability of the event observed relative to the second hypothesis. This probability was only $1⁄10000$ in the preceding question; this great difference results from this—that a black ball having been drawn the witness who wishes to deceive has no choice at all to make among the 999 balls not drawn in order to announce the drawing of a white ball. Now if one forms two fractions whose numerators are the probabilities relative