# 11.4.1. Calculating Probability

If and are two events of a random experiment, in which , , and , determine the probability of occurrence of ONLY one of the two events.

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### Example

If and are two events of a random experiment, in which , , and , determine the probability of occurrence of ONLY one of the two events.

### 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 is =, where the notation means the non-occurrence of or the non-occurrence of , or both. Since , it follows that . Since and are complementary, then 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 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. Now use this fact to find .

The probability of only one of the two events and occurring is the difference of and . Next, find the difference of and .

Thus, the probability of occurrence of only one of the two events is .

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