We shall find a formula for the probability of exactly x successes for given values of p and n. When each number of successes x is paired with its probability of occurrence ** f, the set of pairs ** f, is a probability function called a binomial distribution. The choice of p and n determines the binomial distribution uniquely, and different choices always produce different distributions (except when ** f; then the number of successes is always 0). The set of all binomial distributions is called the family of binomial distributions, but in general discussions this expression is often shortened to ``the binomial distribution,'' or even ``the binomial'' when the context is clear. Binomial distributions were treated by James Bernoulli about 1700, and for this reason binomial trials are sometimes called Bernoulli trials.

Each binomial trial of a binomial experiment produces either 0 or 1 success. Therefore each binomial trial can be thought of as producing a value of a random variable associated with that trial and taking the values 0 and 1, with probabilities q and p respectively. The several trials of a binomial experiment produce a new random variable X, the total number of successes, which is just the sum of the random variables associated with the single trials.

The marksman gets two bull's-eyes, one on his third shot and one on his fifth. The numbers of successes on the five individual shots are, then, 0, 0, 1, 0, 1. The number of successes on each shot is a value of a random variable that has values 0 or 1, and there are 5 such random variables here. Their sum is X, the total number of successes, which in this experiment has the value ** f.