The major question in this chapter is: What is the probability of exactly x successes in n trials?

In Chapters 3 and 4 we answered questions like those in the examples, usually by counting points in a sample space. Fortunately, a general formula of wide applicability solves all problems of this kind. Before deriving this formula, we explain what we mean by ``problems of this kind.''

Experiments are often composed of several identical trials, and sometimes experiments themselves are repeated. In the marksmanship example, a trial consists of ``one round shot at a target'' with outcome either one bull's-eye (success) or none (failure). Further, an experiment might consist of five rounds, and several sets of five rounds might be regarded as a super experiment composed of several repetitions of the five round experiment. If three dice are tossed, a trial is one toss of one die and the experiment is composed of three trials. Or, what amounts to the same thing, if one die is tossed three times, each toss is a trial, and the three tosses form the experiment. Mathematically, we shall not distinguish the experiment of three dice tossed once from that of one die tossed three times. These examples are illustrative of the use of the words ``trial'' and ``experiment'' as they are used in this chapter, but they are quite flexible words and it is well not to restrict them too narrowly.