EXAMPLE 14 Finding the probability of the complement of an event

If is the event “observing a sum of 4 when the two fair dice are rolled,” then your roommate is interested in the probability of , the event that a sum of 4 is not rolled. Find the probability that your roommate does not roll a sum of 4.

Solution

Which outcomes belong to ? By the definition, is all the outcomes in the sample space that do not belong in . The following outcomes are included in .

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Figure 12 shows all the outcomes, except the outcomes from in the two-dice sample space. There are 33 outcomes in and 36 outcomes in the sample space. The classical probability method then gives the probability of not rolling a sum of 4 to be