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FIGURE 3.13 Tree diagram for the television viewing-income example, showing joint and marginal probabilities adding the probabilities on the corresponding sub-branches. This approach is particularly useful when there are more than two attributes of interest. For example, the advertiser might be interested also in the age of the head of the household or whether there are children in the household.

Since the events A\, A;, . . . , Ah are mutually exclusive and collectively exhaustive. their marginal probabilities must sum to 1. The same is true of the events £?,, B2. . . . , bk. For the television viewing-income example, this is illustrated in Table 3.2, where the row and column totals both sum to 1, and in the tree diagram of Figure 3.13, where the sum of the probabilities on the main branches is also 1. It also follows that the joint probabilities summed over all event combinations add to 1.

In many applications, the conditional probabilities are of more interest than the marginal probabilities. For example, an advertiser will be less concerned with the total size of the viewing audience for a show than with the chance that a family that is likely to be in the market for a particular product is watching. The conditional probabilities can be obtained by direct application of the definition of conditional probability introduced in Section 3.6. Thus, the probability of A-, given bf is

for any pair of events A,- and br Similarly p(A, n b,) P(Bj)

Once the joint and marginal probabilities are known, the conditional probabilities then follow. For example, the probability that a randomly chosen family occasionally watches the show given that its income is in the middle range is

/'(Occasionally watch |Middle income) = -, „—:--—--