No one will maintain that belief,
merely as a state of mind, always has a definite numerical value of
which one is conscious, as 1/100 or 1/10. Let anybody mix a number of
letters in a bag, knowing nothing of them except that one of them is X,
and then draw them one by one, endeavouring each time to estimate the
value of his belief that the next will be X; can he say that his belief
in the drawing of X next time regularly increases as the number of
letters left decreases?

If not, we see that (b) belief does not uniformly correspond with the
state of the facts. If in such a trial as proposed above, we really wish
to draw X, as when looking for something in a number of boxes, how
common it is, after a few failures, to feel quite hopeless and to say:
"Oh, of course it will be in the last." For belief is subject to hope
and fear, temperament, passion, and prejudice, and not merely to
rational considerations. And it is useless to appeal to 'the Wise Man,'
the purely rational judge of probability, unless he is producible. Or,
if it be said that belief is a short cut to the evaluation of
experience, because it is the resultant of all past experience, we may
reply that this is not true. For one striking experience, or two or
three recent ones, will immensely outweigh a great number of faint or
remote experiences. Moreover, the experience of two men may be
practically equal, whilst their beliefs upon any question greatly
differ. Any two Englishmen have about the same experience, personal and
ancestral, of the weather; yet their beliefs in the saw that 'if it rain
on St. Swithin's Day it will rain for forty days after,' may differ as
confident expectation and sheer scepticism. Upon which of these beliefs
shall we ground the probability of forty days' rain?

But (c) at any rate, if Probability is to be connected with Inductive
Logic, it must rest upon the same ground, namely--observation.
Induction, in any particular case, is not content with beliefs or
opinions, but aims at testing, verifying or correcting them by appealing
to the facts; and Probability has the same object and the same basis.

In some cases, indeed, the conditions of an event are supposed to be
mathematically predetermined, as in tossing a penny, throwing dice,
dealing cards. In throwing a die, the ways of happening are six; in
tossing a penny only two, head and tail: and we usually assume that the
odds with a die are fairly 5 to 1 against ace, whilst with a