Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/127

Rh 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 $1⁄2$ 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