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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 Af, the set of pairs Af, 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 Af ; ;

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.

Random variables.