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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 Af.

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.

To summarize:

A binomial experiment consists of Af independent binomial trials, all with the same probability Af of yielding a success.

The outcome of the experiment is X successes.

The random variable X takes the values Af with probabilities Af or, more briefly Af.