18.28 The addition rule. Probability rule D states: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. This is sometimes called the addition rule for disjoint events. A more general form of the addition rule is: the probability that one or the other of two events occurs is the sum of their individual probabilities minus the probability that both occur. To verify this rule, suppose you roll two casino dice as in Examples 2. Refer to the outcomes in Figure 18.1 to answer the following.
(a) How many of the outcomes in Figure 18.1 have the sum of the spots on the up-faces equal to 6? What is the probability that the sum of the spots on the up-faces is 6?
(b) How many of the outcomes in Figure 18.1 have at least one of the up-faces showing a single spot? What is the probability that at least one of the up-faces has only a single spot?
(c) How many of the outcomes in Figure 18.1 have the sum of the spots on the up-faces equal to 6 and at least one of the up-faces showing a single spot? What is the probability that the sum of the spots on the up-faces is 6 and at least one of the up-faces has only a single spot?
(d) How many of the outcomes in Figure 18.1 have either the sum of the spots on the up-faces equal to 6 or have at least one of the up-faces showing a single spot? What is the probability that the sum of the spots on the up-faces is either a 6 or at least one of the up-faces has a single spot?
(e) The addition rule says that your answer to part (d) should be equal to the sum of your answers to parts (a) and (b) minus your answer to part (c). Verify that this is the case.
Note: The outcomes you were asked to count in part (c) are among those counted in parts (a) and (b). When we combine the outcomes in parts (a) and (b), we “double count” the outcomes in part (c). One of these “double counts” must be eliminated so that the combination corresponds to the outcomes in part (d). This is the reason that, in the addition rule, you subtract the probability that both occur from the sum of their individual probabilities.