Each performance of an n-trial binomial experiment results in some whole number from 0 through n as the value of the random variable X, where ** f. We want to study the probability function of this random variable. For example, we are interested in the number of bull's-eyes, not which shots were bull's-eyes. A binomial experiment can produce random variables other than the number of successes. For example, the marksman gets 5 shots, but we take his score to be the number of shots before his first bull's-eye, that is, 0, 1, 2, 3, 4 (or 5, if he gets no bull's-eye). Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

The constancy of p and the independence are the conditions most likely to give trouble in practice. Obviously, very slight changes in p do not change the probabilities much, and a slight lack of independence may not make an appreciable difference. (For instance, see Example 2 of Section 5 -- 5, on red cards in hands of 5.) On the other hand, even when the binomial model does not describe well the physical phenomenon being studied, the binomial model may still be used as a baseline for comparative purposes; that is, we may discuss the phenomenon in terms of its departures from the binomial model.

A binomial experiment consists of ** f independent binomial trials, all with the same probability ** f of yielding a success. The outcome of the experiment is X successes. The random variable X takes the values ** f with probabilities ** f or, more briefly ** f.