The binomial description of frequency. In Chapter I attention was directed to the very simple process of finding the relative frequency of occurrence of an event or character among s cases in question. Let us now conceive of repeating the process of finding relative frequencies on many random samples each consisting of s items drawn from the same population. To characterize the degree of stability or the degree of dispersion of such a series of relative frequencies is a fundamental statistical problem.

To illustrate, suppose we repeat the throwing of a set of 1,000 coins many times. An observed frequency distribution could then be exhibited with respect to the number of heads obtained in each set of 1,000, or with respect to the relative frequency of heads in sets of 1,000. Such a procedure would be a laborious experimental treatment of the problem of the distribution of relative frequencies from repeated trials. What we seek is a mathematical method of obtaining the theoretical frequency distribution with respect to the number of heads or with respect to the relative frequency of heads in the sets.

To consider a more general problem, suppose we draw many sets of s balls from an urn one at a time with replacements, and let p be the probability of success in drawing a white ball in one trial.