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.  No one will maintain that belief, merely as a state of mind, always has a definite numerical value of which one is conscious, as 1/100 or 1/10.  Let anybody mix a number of letters in a bag, knowing nothing of them except that one of them is X, and then draw them one by one, endeavouring each time to estimate the value of his belief that the next will be X; can he say that his belief in the drawing of X next time regularly increases as the number of letters left decreases?