# 11.4.1. Calculating Probability

In a school graduation celebration, the probability that the governor attends is 0.82, and the probability that the general education director attends is 0.45. If the probability that both of them attend is 0.31, find the probability that both of them will NOT attend.

• A0.04
• B0.14
• C0.51
• D0.96

### Example

In a school graduation celebration, the probability that the governor attends is 0.82, and the probability that the general education director attends is 0.45. If the probability that both of them attend is 0.31, find the probability that both of them will NOT attend.

### Solution

In this case, event is the governor attending the school graduation celebration, and event is the general education director attending. The notation refers to the probability of either event or event occurring, or both. If and are two non-mutually exclusive events, then , where the notation refers to the probability of events and occurring together. First, use this fact to determine .

Since and are complementary, the sum of their probabilities is equal to 1. Now use this fact to find the probability of .

De Morgan's First Law states that , where the notation means the non-occurrence of and the non-occurrence of . Since , it follows that . Thus, the probability that neither the governor nor the general education director will attend the school graduation celebration is 0.04.

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