# 9.8.1. Operations on Events

and are two events from a sample space . The number of outcomes that lead to the occurrence of the event is 9, the number of all possible outcomes of the random experiment is 33, , and . Determine the probability of occurrence of the two events together.

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### Example

and are two events from a sample space . The number of outcomes that lead to the occurrence of the event is 9, the number of all possible outcomes of the random experiment is 33, , and . Determine the probability of occurrence of the two events together.

### Solution

The probability of event occurring is equal to the number of favorable outcomes divided by the total number of possible outcomes in the sample space, . First, find the probability of event as follows:

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 determine .

Thus, the probability of events and occurring together is .

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