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.
3d. The first witness does not speak the truth and the second announces it. Then a black ball has been drawn from the urn B at the first drawing, and after
having been placed in the urn A a white ball has been drawn from this urn. The probability of the first of these events is ½ and that of the second is 1000000⁄1000001; the probability of the compound event is then 1000000⁄2000002. Multiplying it by the product of the probabilities 1⁄10 and 9⁄10 that the first witness does not speak the truth and that the second announces it, one will have 9000000⁄200000200 for the probability of the event observed relative to this hypothesis.
4th. Finally, neither of the witnesses speaks the truth. Then a black ball has been drawn from the urn B at the first draw; then having been placed in the urn A it has reappeared at the second drawing: the probability of this compound event is 1⁄2000002. Multiplying it by the product of the probabilities 1⁄10 and 1⁄10 that each witness does not speak the truth one will have 1⁄200000200 for the probability of the event observed in this hypothesis.
Now in order to obtain the probability of the thing announced by the two witnesses, namely, that a white ball has been drawn at each draw, it is necessary to divide the probability corresponding to the first hypothesis by the sum of the probabilities relative to the four hypotheses; and then one has for this probability 81⁄18000082, an extremely small fraction.
If the two witnesses affirm the first, that a white ball has been drawn from one of the two urns A and B; the second that a white ball has been likewise drawn from one of the two urns A´ and B´, quite similar to the first ones, the probability of the thing announced by the two witnesses will be the product of the probabilities of their testimonies, or 81⁄100; it will then
be at least a hundred and eighty thousand times greater than the preceding one. One sees by this how much, in the first case, the reappearance at the second draw of the white ball drawn at the first draw, the extraordinary conclusion of the two testimonies decreases the value of it.
We would give no credence to the testimony of a man who should attest to us that in throwing a hundred dice into the air they had all fallen on the same face. If we had ourselves been spectators of this event we