EXAMPLE 5 Probabilities for Rolling Two Dice

Figure 8.7 (page 347) displays the 36 possible outcomes of rolling two dice. For casino dice, it is reasonable to assign the same probability to each of the 36 outcomes in Figure 8.7. Because all 36 outcomes together must have probability 1 (Rule 2), each outcome must have probability .

Suppose we want to determine the probability of rolling a sum of 5. Because there are four ways to roll a sum of 5, the addition rule for disjoint events (Rule 4) says that its probability is

Similarly, we can find the probabilities for the other possible sums and, in this way, get the full probability model (sample space and assignment of probabilities) for rolling two dice and summing the spots on the sides facing up. The result is shown in Table 8.3.

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The model in Table 8.3 assigns probabilities to individual outcomes. Note that Rule 2 is satisfied because all the probabilities add up to 1. To find the probability of an event, just add the probabilities of the outcomes that make up the event. For example:

Suppose we want the probability of rolling an even number. We could find this probability by finding the sum of the following:

But a faster way would be to use the complement rule (Rule 3):

For an example of Rule 5, let event be “sum is odd” and event be “sum is a multiple of 3.” Suppose we want . Earlier, we calculated . You can verify that and . Now we are ready to apply Rule 5: