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.

However, where statistical evidence is obtainable, no one dreams of
estimating probability by the quantity of his belief. Insurance offices,
dealing with fire, shipwreck, death, accident, etc., prepare elaborate
statistics of these events, and regulate their rates accordingly. Apart
from statistics, at what rate ought the lives of men aged 40 to be
insured, in order to leave a profit of 5 per cent. upon L1000 payable at
each man's death? Is 'quantity of belief' a sufficient basis for doing
this sum?

Sec. 4. The ground of probability is experience, then, and, whenever
possible, statistics; which are a kind of induction. It has indeed been
urged that induction is itself based upon probability; that the
subtlety, complexity and secrecy of nature are such, that we are never
quite sure that we fully know even what we have observed; and that, as
for laws, the conditions of the universe at large may at any moment be
completely changed; so that all imperfect inductions, including the law
of causation itself, are only probable. But, clearly, this doctrine
turns up