ge recurrence of double sixes, or double aces,
approximate to one in thirty-six. Such a law of average recurrence is
the basis of Probability. Chance being the fact of coincidence without
assignable cause, Probability is expectation based on the average
frequency of its happening.

Sec. 2. Probability is an ambiguous term. Usually, when we say that an
event is 'probable,' we mean that it is more likely than not to happen.
But, scientifically, an event is probable if our expectation of its
occurrence is less than certainty, as long as the event is not
impossible. Probability, thus conceived, is represented by a fraction.
Taking 1 to stand for certainty, and 0 for impossibility, probability
may be 999/1000, or 1/1000, or (generally) 1/_m_. The denominator
represents the number of times that an event happens, and the numerator
the number of times that it coincides with another event. In throwing a
die, the probability of ace turning up is expressed by putting the
number of throws for the denominator and the number of times that ace is
thrown for the numerator; and we may assume that the more trials we make
the nearer will the resulting fraction approximate to 1/6.

Instead of speaking of the 'throwing of the die' and its 'turning up
ace' as two events, the former is called 'the event' and the latter 'the
way of its happening.' And these expressions may easily be extended to
cover relations of distinct events; as when two men shoot at a mark and
we desire to represent the probability of both hitting the bull's eye
together, each shot may count as an event (denominator) and the
coincidence of 'bull's-eyes' as the way of its happening (numerator).

It is also common to speak of probability as a proportion. If the
fraction expressing the probability of ace being cast is 1/6, the
proportion of cases in which it happens is 1 to 5; or (as it is,
perhaps, still more commonly put) 'the chances are 5 to 1 against it.'

Sec. 3. As to the grounds of probability opinions differ. According to one
view the ground is subjective: probability depends, it is said, upon the
quantity of our Belief in the happening of a certain event, or in its
happening in a particular way. According to the other view the ground is
objective, and, in fact, is nothing else than experience, which is most
trustworthy when carefully expressed in statistics.

To the subjective view it may be objected, (a) that belief cannot by
itself be satisfactorily measured.