During a round of target practice the sun comes from behind a cloud and dazzles the marksman, lowering his chance of a bull's-eye.

Strictly speaking, this means that the probability for each possible outcome of the experiment can be computed by multiplying together the probabilities of the possible outcomes of the single binomial trials. Thus in the three dice example ** f, ** f, and the independence assumption implies that the probability that the three dice fall ace, not ace, ace in that order is (1 6) (5 6) (1 6). Experimentally, we expect independence when the trials have nothing to do with one another.

A family of five plans to go together either to the beach or to the mountains, and a coin is tossed to decide. We want to know the number of people going to the mountains. When this experiment is viewed as composed of five binomial trials, one for each member of the family, the outcomes of the trials are obviously not independent. Indeed, the experiment is better viewed as consisting of one binomial trial for the entire family. The following is a less extreme example of dependence. Consider couples visiting an art museum. Each person votes for one of a pair of pictures to receive a popular prize. Voting for one picture may be called ``success,'' for the other ``failure.'' An experiment consists of the voting of one couple, or two trials. In repetitions of the experiment from couple to couple, the votes of the two persons in a couple probably agree more often than independence would imply, because couples who visit the museum together are more likely to have similar tastes than are a random pair of people drawn from the entire population of visitors. Table 7 -- 1 illustrates the point. The table shows that 0.6 of the boys and 0.6 of the girls vote for picture A. Therefore, under independent voting, ** f or 0.36 of the couples would cast two votes for picture A, and ** f or 0.16 would cast two votes for picture B. Thus in independent voting, ** f or 0.52 of the couples would agree. But Table 7 -- 1 shows that ** f or 0.70 agree, too many for independent voting.