and, therefore, on causation. The inductive evidence
underlying an estimate of probability may be of three kinds: (a) direct
statistics of the events in question; as when we find that, at the age
of 20, the average expectation of life is 39-40 years. This is an
empirical law, and, if we do not know the causes of any event, we must
be content with an empirical law. But (b) if we do know the causes of an
event, and the causes which may prevent its happening, and can estimate
the comparative frequency of their occurring, we may deduce the
probability that the effect (that is, the event in question) will occur.
Or (c) we may combine these two methods, verifying each by means of the
other. Now either the method (b) or (_a fortiori_) the method (c) (both
depending on causation) is more trustworthy than the method (a) by
itself.

But, further, a merely empirical statistical law will only be true as
long as the causes influencing the event remain the same. A die may be
found to turn ace once in six throws, on the average, in close
accordance with mathematical theory; but if we load it on that facet the
results will be very different. So it is with the expectation of life,
or fire, or shipwreck. The increased virulence of some epidemic such as
influenza, an outbreak of anarchic incendiarism, a moral epidemic of
over-loading ships, may deceive the hopes of insurance offices. Hence we
see, again, that probability depends upon causation, not causation upon
probability.

That uncertainty of an event which arises not from ignorance of the law
of its cause, but from our not knowing whether the cause itself does or
does not occur at any particular time, is Contingency.

Sec. 5. The nature of an average supposes deviations from it. Deviations
from an average, or "errors," are assumed to conform to the law (1) that
the greater errors are less frequent than the smaller, so that most
events approximate to the average; and (2) that errors have no "bias,"
but are equally frequent and equally great in both directions from the
mean, so that they are scattered symmetrically. Hence their distribution
may be expressed by some such figure as the following:

[Illustration: FIG. 11.]

Here _o_ is the average event, and _oy_ represents the number of average
events. Along _ox_, in either direction, deviations are measured. At _p_
the amount of error or deviation is _op_; and the number of such
deviations is represented by the line or ordinate _p