4. Statistical Method.—The alternative to the unitary method is the statistical method. Suppose that an event (or fact or set of conditions, etc.) C might occur in a large number N of cases, and that whenever it might occur it would necessarily be asso ciated with one of two or more mutually exclusive events E, E" . . .; "mutually exclusive" meaning that on each occasion only one of these events can occur. Of the total number N of cases, suppose that E would occur in pN cases, E' in p'N cases, E" in p"N cases . . . . Then p, p', p" are said to be the probabilities of C being associated with E, E', E" . . . . It is clear that = 1.
In reference to cases of this kind we should use the word " probability" rather than chance," since the question is not necessarily as to the happening of some future event. The definition can be applied equally to the probability of a future or a present or a past event. The numerical probability is in each case the result of grouping the C-cases into classes according as they are associated with E or with E', etc., and taking the prob abilities to be proportional to the numbers in the different classes. If, for instance, in the case considered in sec. I, we say that the probability of the red ball being drawn is 1/5, we mean that in the long run (sec. 8), or on the average, the red ball will be drawn about once in five times.
If an event E occurs in m cases out of n, m is called the of occurrence of E, and the ratio mht is its relative frequency. Thus the probability of an event is the same thing as its relative frequency when the number of cases considered is very large.
5. Observations on the Definition.—The following are points in reference to the definition.
(i.) We say that the red ball will be drawn " about" once in five times, because we can never know the exact probability of any event; and it is, indeed, doubtful whether there is any exact probability of anything. We may throw a die too,000 times, and note the number of cases in which a 6 occurs; but if we throw another ioo,000 times we shall get a slightly different result. The study of variations of this kind forms part of the theory of error, which, again, is part of the general theory of frequency distributions.
(ii.) We proceed, however, as if any probability with which we are dealing has a definite value. For practical purposes, therefore, we should not use actual observations as the means for determining a probability unless we have made a fairly large number of observations; until we have done so, we may have to depend on a priori considerations.
(iii.) The determination of the probability of an event, on the statistical basis, involves a process of classification. A man comes to have his life insured for a year. We require, at least, to know his age. Knowing this, we put him into a certain class or category, namely, the category of men of that age; and we find that, of men of that age, 99 out of roo, on the average, survive for a year. We therefore say that his chance of surviving is 99/1oo. But this does not imply that there is such a thing as chance, in the ordinary sense; nor is it a statement as to the individual, as distinct from the rest of the class. We do not treat him as an individual, but as one of the class, taken at random. Further information may alter the chance. We find that he will be working in an unhealthy climate: this means a further limi tation, defining more strictly the class in which he is to be placed; and so on. The ultimate probability assigned to him is the probability of survival for a year in the class in which he is ultimately placed.
6. Illustrative Example.--As a basis for illustration of the statistical definition, let us take a concrete case. A man being called short or tall according as his height is under or over 674 in., suppose that, in a community in which every man has one son who attains maturity, the statistical relation between height of father and height of son, in a representative (sec. 15) i,000 pairs of father and son, is given in Table I. This table, if our supposition is correct, provides us with various statements as to probability, of which the following are examples.
(i.) The probability that a father is tall is 535/1,000. (This is a short way of saying: if a father is taken at random, the probability that he will be one of the tall fathers is 535/1,000. The phrase " at random" is considered in sec. 9.) (ii.) The probability that a son is tall is 661/1,000.
(iii.) The probability that a tall father has a tall son is (iv.) The probability that a short son has a tall father is 7. Relation Between the Two Methods.—To some extent the two methods—unitary and statistical—supplement one another.