Some experiments are composed of repetitions of independent trials, each with two possible outcomes. The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another. We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models. We begin with a few examples of binomial experiments.

A trained marksman shooting five rounds at a target, all under practically the same conditions, may hit the bull's-eye from 0 to 5 times. In repeated sets of five shots his numbers of bull's-eyes vary. What can we say of the probabilities of the different possible numbers of bull's-eyes?

In litters of eight mice from similar parents, the number of mice with straight instead of wavy hair is an integer from 0 to 8. What probabilities should be attached to these possible outcomes?

When three dice are tossed repeatedly, what is the probability that the number of aces is 0 (or 1, or 2, or 3)?

More generally, suppose that an experiment consists of a number of independent trials, that each trial results in either a ``success'' or a ``non success'' (``failure''), and that the probability of success remains constant from trial to trial. In the examples above, the occurrence of a bull's-eye, a straight haired mouse, or an ace could be called a ``success.'' In general, any outcome we choose may be labeled ``success.''