# 11.4.1. Calculating Probability

and are two events of a sample space of a random experiment. If the number of outcomes that tends to occur of the event equals 4, the number of outcomes that tends to occur of the event equals 13, and the number of outcomes that tends to occur of both and equals 2 and , then find .

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### Example

and are two events of a sample space of a random experiment. If the number of outcomes that tends to occur of the event equals 4, the number of outcomes that tends to occur of the event equals 13, and the number of outcomes that tends to occur of both and equals 2 and , then find .

### 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, Find the probability of event in terms of the total number of possible outcomes in the sample space as follows:

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, Now find the probability of event in terms of the total number of possible outcomes in the sample space.

The probability of both event and event occurring is equal to the number of outcomes favorable for both events divided by the total number of possible outcomes in the sample space, Next, find the probability of both event and event occurring in terms of the total number of possible outcomes in the sample space.

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 the total number of possible outcomes in the sample space.

Since the total number of possible outcomes in the sample space is 45, it follows that , and The notation refers to the probability of the occurrence of and the non-occurrence of For this reason, is equal to the sum of and Next, use this fact to solve for

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

Another way to write is Thus, since it follows that

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