Supposing evidence to be ultimately nothing but accumulated experience, the amount of it in favour of causation is incomparably greater than the most that has ever been advanced to show the probability of any other kind of event; and every relation of events which is shown to have the marks of causation obtains the support of that incomparably greater body of evidence.  Hence the only way in which causation can be called probable, for us, is by considering it as the upward limit (1) to which the series of probabilities tends; as impossibility is the downward limit (0).  Induction, ‘humanly speaking,’ does not rest on probability; but the probability of concrete events (not of mere mathematical abstractions like the falling of absolutely true dice) rests on induction 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: