A System of Logic: Ratiocinative and Inductive, 7th Edition, Vol. II
John Stuart Mill · en
In addition to the question, What is the number of coincidences which,
on an average of a great multitude of trials, may be expected to arise
from chance alone? there is also another question, namely, Of what
extent of deviation from that average is the occurrence credible, from
chance alone, in some number of instances smaller than that required
for striking a fair average? It is not only to be considered what is the
general result of the chances in the long run, but also what are the
extreme limits of variation from the general result, which may
occasionally be expected as the result of some smaller number of
instances.
The consideration of the latter question, and any consideration of the
former beyond that already given to it, belong to what mathematicians
term the doctrine of chances, or, in a phrase of greater pretension, the
Theory of Probabilities.
CHAPTER XVIII.
OF THE CALCULATION OF CHANCES.
§ 1. "Probability," says Laplace,[17] "has reference partly to our
ignorance, partly to our knowledge. We know that among three or more
events, one, and only one, must happen; but there is nothing leading us
to believe that any one of them will happen rather than the others. In
this state of indecision, it is impossible for us to pronounce with
certainty on their occurrence. It is, however, probable that any one of
these events, selected at pleasure, will not take place; because we
perceive several cases, all equally possible, which exclude its
occurrence, and only one which favours it.
"The theory of chances consists in reducing all events of the same kind
to a certain number of cases equally possible, that is, such that we are
_equally undecided_ as to their existence; and in determining the number
of these cases which are favourable to the event of which the
probability is sought. The ratio of that number to the number of all the
possible cases, is the measure of the probability; which is thus a
fraction, having for its numerator the number of cases favourable to the
event, and for its denominator the number of all the cases which are
possible."