[3] The generalized algebraical form of this result is as follows. Let p be the à priori probability of an event, and x be the credibility of the witness. Then, if he asserts that the event happened, the probability that it really did happen is
px/px + (1 − p)(1 − x);
whilst if he asserts that it did not happen the probability that it did happen is
p(1 − x)/p(1 − x) + (1 − p)x.
In illustration of some remarks to be presently made, the reader will notice that on making either of these expressions = p, we obtain in each case x = 1/2. That is, a witness whose veracity = 1/2 leaves the à priori probability of an event (of this kind) unaffected.
If, on the other hand, we make these expressions equal to x and 1 − x respectively, we obtain in each case p = 1/2. That is, when an event (of this kind) is as likely to happen as not, the ordinary veracity of the witness in respect of it remains unaffected.
[4] Todhunter's History, p. 400. Philosophical Magazine, July, 1864.
[5] “When therefore these two kinds of experience are contrary, we have nothing to do but subtract the one from the other, and embrace an opinion, either on one side or the other, with that assurance which arises from the remainder.” (Essay on Miracles.)
[6] Considerations of this kind have indeed been introduced into the mathematical treatment of the subject. The common algebraical solution of the problem in § 5 (to begin with the simplest case) is of course as follows. Let p be the antecedent probability of the event, and t the measure of the truthfulness of the witness; then the chance of his statement being true is pt/pt + (1 − p)(1 − t). This supposes him to lie as much when the event does not happen as when it does. But we may meet the cases supposed in the text by assuming that t′ is the measure of his veracity when the event does not happen, so that the above formula becomes pt/pt + (1 − p)(1 − t′). Here t′ and t measure respectively his trustworthiness in usual and unusual events. As a formal solution this certainly meets the objections stated above in §§ 14 and 15. The determination however of t′ would demand, as I have remarked, continually renewed appeal to experience. In any case the practical methods which would be adopted, if any plans of the kind indicated above were resorted to, seem to me to differ very much from that adopted by the mathematicians, in their spirit and plan.
[7] Laplace, for instance (Essai, ed.