# 11.4.1. Calculating Probability

is the sample space of a random experiment whose outcomes are equiprobable, where and are two events from , , and . Given that the number of outcomes that tends to occur of the event equals 6, and the number of possible outcomes of the experiment equals 12, find the probability of occurrence of both .

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### Example

is the sample space of a random experiment whose outcomes are equiprobable, where and are two events from , , and . Given that the number of outcomes that tends to occur of the event equals 6, and the number of possible outcomes of the experiment equals 12, find the probability of occurrence of both .

### 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 find .

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 Second Law states that , where the notation means the non-occurrence of or the non-occurrence of , or both. Since , it follows that .

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