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 penny ’the betting is even’ on head or tail.  Still, this assumption rests upon another, that the die is perfectly fair, or that the head and tail of a penny are exactly alike; and this is not true.  With an ordinary die or penny, a very great number of trials would, no doubt, give an average approximating to 1/6 or 1/2; yet might always leave a certain excess one way or the other, which would also become more definite as the trials went on; thus showing that the die or penny did not satisfy the mathematical hypothesis.  Buffon is said to have tossed a coin 4040 times, obtaining 1992 heads and 2048 tails; a pupil of De Morgan tossed 4092 times, obtaining 2048 heads and 2044 tails.

There are other important cases in which probability is estimated and numerically expressed, although statistical evidence directly bearing upon the point in question cannot be obtained; as in betting upon a race; or in the prices of stocks and shares, which are supposed to represent the probability of their paying, or continuing to pay, a certain rate of interest.  But the judgment of experts in such matters is certainly based upon experience; and great pains are taken to make the evidence as definite as possible by comparing records of speed, or by financial estimates; though something must still be allowed for reports of the condition of horses, or of the prospects of war, harvests, etc.