Ten students act as managers for a high-school football team, and of these managers a proportion p are licensed drivers. Each Friday one manager is chosen by lot to stay late and load the equipment on a truck. On three Fridays the coach has needed a driver. Considering only these Fridays, what is the probability that the coach had drivers all 3 times? Exactly 2 times? 1 time? 0 time?

Note that there are 3 trials of interest. Each trial consists of choosing a student manager at random. The 2 possible outcomes on each trial are ``driver'' or ``nondriver.'' Since the choice is by lot each week, the outcomes of different trials are independent. The managers stay the same, so that ** f is the same for all weeks. We now generalize these ideas for general binomial experiments.

For an experiment to qualify as a binomial experiment, it must have four properties: (1) there must be a fixed number of trials, (2) each trial must result in a ``success'' or a ``failure'' (a binomial trial), (3) all trials must have identical probabilities of success, (4) the trials must be independent of each other. Below we use our earlier examples to describe and illustrate these four properties. We also give, for each property, an example where the property is absent. The language and notation introduced are standard throughout the chapter.

For the marksman, we study sets of five shots (** f); for the mice, we restrict attention to litters of eight (** f); and for the aces, we toss three dice (** f).