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# Rules for Counting

Some basic rules for counting can help us calculate probabilities. We begin with an example. Brazil (B), United States (U.S.), Mexico (M), Canada (C), and Japan (J) are in the final round of a volleyball tournament. How many ways can a winner and second place finisher be selected?

To systematically address this problem, we create a tree diagram where the possibilities for first place are represented by the initial 5 branches. For each choice of a first place finisher, there are four choices for second place. These are represented by the sets of 4 branches in the second stage. For instance, the leftmost branch of the tree diagram depicts the case where Brazil is first and the United States second and so on.

There are 5 × 4 = 20 terminal points or ways for these teams to finish first and second. Generalizing the tree diagram to k choices followed by m choices, we obtain the product rule for counting.

Product Rule

An experiment consists of two parts where the first part has k distinct possible outcomes and, for each outcome of the first part, there are m distinct possible outcomes for the second part. Then, there are k × m distinct possible outcomes to the experiment.

The product rule is readily extended to experiments with more than two parts. Suppose we are interested in the winner, second, and third place finishers in the volleyball tournament. ...

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