In order to make it plain let us consider two urns, A and B, of which the first contains a million white balls and the second a million black balls. One draws from one of these urns a ball, which he puts back into the other urn, from which one then draws a ball. Two witnesses, the one of the first drawing, the other of the second, attest that the ball which they have seen drawn is white without indicating the urn from which it has been drawn. Each testimony taken alone is not improbable; and it is easy to see that the probability of the fact attested is the veracity itself of the witness. But it follows from the combination of the testimonies that a white ball has been extracted from the urn A at the first draw, and that then placed in the urn B it has reappeared at the second draw, which is very extraordinary; for this second urn, containing then one white ball among a million black balls, the probability of drawing the white ball is 1/1000001. In order to determine the diminution which results in the probability of the thing announced by the two witnesses we shall notice that the event observed is here the affirmation by each of them that the ball which he has seen extracted is white. Let us represent by 9/10 the probability that he announces the truth, which can occur in the present case when the witness does not deceive and is not mistaken at all, and when he deceives and is mistaken at the same time. One may form the four following hypotheses: 1st. The first and second witness speak the truth. Then a white ball has at first been drawn from the urn A, and the probability of this event is ½, since the ball drawn at the first draw may have been drawn either from the one or the other urn. Consequently the ball drawn, placed in the urn B, has reappeared at the second draw; the probability of this event is 1/1000001, the probability of the fact announced is then 1/2000002. Multiplying it by the product of the probabilities 9/10 and 9/10 that the witnesses speak the truth one will have 81/200000200 for the probability of the event observed in this first hypothesis. 2d. The first witness speaks the truth and the second does not, whether he deceives and is not mistaken or he does not deceive and is mistaken. Then a white ball has been drawn from the urn A at the first draw, and the probability of this event is ½. Then this ball having been placed in the urn B a black ball has been drawn from it: the probability of such drawing is 1000000/1000001; one has then 1000000/2000002 for the probability of the compound event. Multiplying it by the product of the two probabilities 9/10 and 1/10 that the first witness speaks the truth and that the second does not, one will have 9000000/200000200 for the probability for the event observed in the second hypothesis.