THE CALCULATION OF CHANCES.
In order to calculate chances, we must know that of several events one,
and no more, must happen, and also not know, or have any reason to
suspect, which of them that one will be. Thus, with the simple knowledge
that the issue must be one of a certain number of possibilities, we
_may_ conclude that one supposition is most probable _to us_. For this
purpose it is not _necessary_ that specific experience or reason should
have also proved the occurrence of each of the several events to be, as
a fact, equally frequent. For, the probability of an event is not a
quality of the event (since every event is in itself certain), but is
merely a name for the degree of ground _we_ have, with our present
evidence, for expecting it. Thus, if we know that a box contains red,
white, and black balls, though we do not know in what proportions they
are mingled, we have numerically appreciable grounds for considering the
probability to be two to one against any one colour. Our judgment may
indeed be said in this case to rest on the experience we have of the
laws governing the frequency of occurrence of the different cases; but
such experience is universal and axiomatic, and not specific experience
about a particular event. Except, however, in games of chance, the
purpose of which requires ignorance, such specific experience can
generally be, and should be gained. And a slight improvement in the data
profits more than the most elaborate application of the calculus of
probabilities to the bare original data, e.g. to such data, when we are
calculating the credibility of a witness, as the proportion, even if it
could be verified, between the number of true and of erroneous
statements a man, _qua_ man, may be supposed to make during his life.
Before applying the Doctrine of Chance, therefore, we should lay a
foundation for an evaluation of the chances by gaining positive
knowledge of the facts. Hence, though not a _necessary_, yet a most
usual condition for calculating the probability of a fact is, that we
should possess a _specific_ knowledge of the proportion which the cases
in which facts of the particular sort occur bear to the cases in which
they do not occur.
Inferences drawn correctly according to the Doctrine of Chances depend
ultimately on causation. This is clearest, when, as sometimes, the
probability of an event is deduced from the frequency of the occurrence
of the causes. When its probability is calculated by merely counting and
comparing the number of cases in which it has occurred with those in
which it has not, the law, being arrived at by the Method of Agreement,
is only empirical. But even when, as indeed generally, the numerical
data are obtained in the latter way (since usually we can judge of the
frequency of the causes only through the medium of the empirical law,
which is based on the frequency of the effects), still then, too, the
inference really depends on causation alone. Thus, an actuary infers
from his tables that, of any hundred living persons under like
conditions, five will reach a given age, not simply because that
proportion have reached it in times past, but because that fact shows
the existence there of a particular proportion between the _causes_
which shorten and the causes which prolong life to the given extent.