CHANCE, AND ITS ELIMINATION.
Empirical laws are certain only in those limits within which they have
been _observed_ to be true. But, even within those limits, the
connection of two phenomena may, as the same effect may be produced by
several different causes, be due to Chance; that is, it may, though
being, as all facts must be, the result of some law, be a coincidence
whence, simply because we do not know all the circumstances, _we_ have
no ground to infer an uniformity. When neither Deduction, nor the Method
of Difference, can be applied, the only way of inferring that
coincidences are not casual, is by observing the frequency of their
occurrence, not their absolute frequency, but whether they occur _more_
often than chance would (that is, more often than the positive frequency
of the phenomena would) account for. If, in such cases, we could ascend
to the causes of the two phenomena, we should find at some stage some
cause or causes common to both. Till we can do this, the fact of the
connection between them is only an empirical law; but still it is a law.
Sometimes an effect is the result partly of chance, and partly of law:
viz. when the total effect is the result partly of the effects of casual
conjunctions of causes, and partly of the effects of some constant cause
which they blend with and modify. This is a case of Composition of
Causes. The object being to find _how much_ of the result is
attributable to a given constant cause, the only resource, when the
variable causes cannot be wholly excluded from the experiment, is to
ascertain what is the effect of all of _them_ taken together, and then
to eliminate this, which is the casual part of the effect, in reckoning
up the results. If the results of frequent experiments, in which the
constant cause is kept invariable, oscillate round one point, that
average or middle point is due to the constant cause, and the variable
remainder to chance; that is, to causes the coexistence of which with
the constant cause was merely casual. The test of the sufficiency of
such an induction is, whether or not an increase in the number of
experiments materially alters the average.
We can thus discover not merely _how much_ of the effect, but even
whether _any_ part of it whatever is due to a constant cause, when this
latter is so uninfluential as otherwise to escape notice (e.g. the
loading of dice). This case of the Elimination of Chance is called _The
discovery of a residual phenomenon by eliminating the effects of
chance._
The mathematical doctrine of chances, or Theory of Probabilities,
considers what deviation from the average chance by itself can possibly
occasion in some number of instances smaller than is required for a fair
average.