# 11.4.1. Calculating Probability

If and are two events of a random experiment, where , , and , determine the probability of occurrence of the event only.

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### Example

If and are two events of a random experiment, where , , and , determine the probability of occurrence of the event only.

### Solution

The events and are complementary, so the sum of their probabilities is equal to 1. First, use this fact, along with the fact that , to find the probability of .

De Morgan's Second Law states that , where the notation means the non-occurrence of or the non-occurrence of , or both. Since , it follows that . Also, since and are complementary, the sum of their probabilities is equal to 1. Now use this fact to find the probability of .

The notation refers to the probability of the occurrence of and the non-occurrence of . In other words, it refers to the probability of the occurrence of event only. Also, the notation refers to the probability of events and occurring together. For this reason, is equal to the sum of and . Next, use this fact to find .

Thus, the probability of the occurrence of event only is .

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