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.[ TN: space] 1825, p. 149), says that if we saw 100 dies (known of course to be fair ones) all give the same face, we should be bewildered at the time, and need confirmation from others, but that, after due examination, no one would feel obliged to postulate hallucination in the matter. But the chance of this occurrence is represented by a fraction whose numerator is 1, and denominator contains 77 figures, and is therefore utterly inappreciable by the imagination. It must be admitted, though, that there is something hypothetical about such an example, for we could not really know that the dies were fair with a confidence even distantly approaching such prodigious odds. In other words, it is difficult here to keep apart those different aspects of the question discussed in Chap. XIV.[ TN: space] §§ 28–33.
8 In the first edition this was stated, as it now seems to me, in decidedly too unqualified a manner. It must be remembered, however, that (as was shown in § 7) this plan is really the best theoretical one which can be adopted in certain cases.
9 It is on this principle that the remarkable conclusion mentioned on p. 405 is based. Suppose an event whose probability is p; and that, of a number of witnesses of the same veracity (y), m assert that it happened, and n deny this. Generalizing the arithmetical reasoning given above we see that the chance of the event being asserted varies as
(viz.[ TN: space] as the chance that the event happens, and that m are right and n are wrong; plus the chance that it does not happen, and that n are right and m are wrong). And the chance of its being rightly asserted as *pym (1 − y)n*. Therefore the chance that when we have an assertion before us it is a true one is
which is equal to
But this last expression represents the probability of an assertion which is unanimously supported by m − n such witnesses.
10 The stress which Butler lays upon this notion of a scheme is, I think, one great merit of his Analogy.