Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/121

Rh this probability will be $9⁄10$; that is to say, the veracity itself of the witness. This is likewise the probability of the drawing of number 79. The probability of the falsehood of the witness and of the failure of drawing this number is $1⁄10$.

If the witness, wishing to deceive, has some interest in choosing number 79 among the numbers not drawn, —if he judges, for example, that having placed upon this number a considerable stake, the announcement of its drawing will increase his credit, the probability that he will choose this number will no longer be as at first, $1⁄999$, it will then be $1⁄2$, $1⁄3$, 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