1 ⁄ 999 , it will then be ½, ⅓, etc., according to the interest that he will have in announcing its drawing. Supposing it to be 1 ⁄ 9 , it will be necessary to multiply by this fraction the probability 999 ⁄ 1000 in order to get in the hypothesis of the falsehood the probability of the event observed, which it is necessary still to multiply by 1 ⁄ 10 , which gives 111 ⁄ 10000 for the probability of the event in the second hypothesis. Then the probability of the first hypothesis, or of the drawing of number 79, is reduced by the preceding rule to 9 ⁄ 120 . It is then very much decreased by the consideration of the interest which the witness may have in announcing the drawing of number 79. In truth this same interest increases the probability 9 ⁄ 10 that the witness will speak the truth if number 79 is drawn. But this probability cannot exceed unity or 10 ⁄ 10 ; thus the probability of the drawing of number 79 will not surpass 10 ⁄ 121 . Common sense tells us that this interest ought to inspire distrust, but calculus appreciates the influence of it.
The probability à priori of the number announced by the witness is unity divided by the number of the
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 ½ 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 ½. 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 ⅓.
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
to each hypothesis, and whose common denominator is the sum of these probabilities, one will have 9⁄1008 for the probability of the first hypothesis and of the drawing of a white ball, and 999⁄1008 for the probability of the second hypothesis and of the drawing of a black ball. This last probability strongly approaches certainty; it would approach it much nearer and would become 999999⁄1000008 if the urn contained a million balls of which one was white, the drawing of a white ball becoming then much more extraordinary. We see thus how the probability of the falsehood increases in the measure that the deed becomes more extraordinary.
We have supposed up to this time that the witness was not mistaken at all; but if one admits, however, the chance of his error the extraordinary incident becomes more improbable. Then in place of the two hypotheses one will have the four following ones, namely: that of the witness not deceiving and not being mistaken at all; that of the witness not deceiving at all and being mistaken; the hypothesis of the witness deceiving and not being mistaken at all; finally, that of the witness deceiving and being mistaken. Determining à priori in each of these hypotheses the probability of the event observed, we find by the sixth