Page:A philosophical essay on probabilities Tr. Truscott, Emory 1902.djvu/133

Rh together, and by the product of the probabilities $1⁄10$ and $3⁄10$ that the witnesses deceive; one will have thus $1⁄330000$ for the probability of the event observed in the second hypothesis. Now one will have the probability of the fact attested or of the drawing of number 2 in dividing the probability relative to the first hypothesis by the sum of the probabilities relative to the two hypotheses; this probability will be then $2079⁄2080$, and the probability of the failure to draw this number and of the falsehood of the witnesses will be $1⁄2080$.

If the urn should contain only the numbers 1 and 2 one would find in the same manner $21⁄22$ for the probability of the drawing of number 2, and consequently $1⁄22$ for the probability of the falsehood of the Witnesses, a probability at least ninety-four times larger than the preceding one. One sees by this how much the probability of the falsehood of the witnesses diminishes when the fact which they attest is less probable in itself Indeed one conceives that then the accord of the witnesses, when they deceive, becomes more difficult, at least when they do not have a secret agreement, which we do not suppose here at all.

In the preceding case where the urn contained only two numbers the à priori probability of the fact attested is $1⁄2$, the resultant probability of the testimonies is the product of the veracities of the witnesses divided by this product added to that of the respective probabilities of their falsehood.

It now remains for us to consider the influence of time upon the probability of facts transmitted by a traditional chain of witnesses. It is clear that this probability ought to diminish in proportion as the chain