Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/132

122 of these numbers renders infinite ; and to observe that if the witnesses deceive they have the greatest interest, in order to accredit their falsehood, in promising an eternity of happiness. The expression of the probability of their testimony becomes then infinitely small. Multiplying it by the infinite number of happy lives promised, infinity would disappear from the product which expresses the advantage resultant from this promise which destroys the argument of Pascal.

Let us consider now the probability of the totality of several testimonies upon an established fact. In order to fix our ideas let us suppose that the fact be the drawing of a number from an urn which, contains a hundred of them, and of which one single number has been drawn. Two witnesses of this drawing announce that number 2 has been drawn, and one asks for the resultant probability of the totality of these testimonies. One may form these two hypotheses: the witnesses speak the truth; the witnesses deceive. In the first hypothesis the number 2 is drawn and the probability of this event is $1⁄100$. It is necessary to multiply it by the product of the veracities of the witnesses, veracities which we will suppose to be $9⁄10$ and $7⁄10$: one will have then $63⁄10000$ for the probability of the event observed in this hypothesis. In the second, the number 2 is not drawn and the probability of this event is $99⁄100$. But the agreement of the witnesses requires then that in seeking to deceive they both choose the number 2 from the 99 numbers not drawn: the probability of this choice if the witnesses do not have a secret agreement is the product of the fraction $1⁄99$ by itself; it becomes necessary then to multiply these two probabilities