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.
§§ 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
pym(1 − y)n + (1 − p)yn(1 − y)m;
(viz.
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
pym (1 − y)n/pym (1 − y)n + (1 − p) yn (1 − y)m,
which is equal to
pym−n/pym−n + (1 − p) (1 − y)m−n.