# 11.4.1. Calculating Probability

The probability that Adam succeeding mathematics is 0.5, the probability of him failing Physics is 0.44, and the probability of him succeeding in one of them at least is 0.87. Find the probability that Adam succeeding one of the two subjects at most.

• A0.81
• B0.13
• C0.19
• D0.68

### Example

The probability that Adam succeeding mathematics is 0.5, the probability of him failing Physics is 0.44, and the probability of him succeeding in one of them at least is 0.87. Find the probability that Adam succeeding one of the two subjects at most.

### Solution

In this case, event is Adam succeeding in mathematics, and event is Adam succeeding in physics. This means that event is Adam failing in physics. The events and are complementary, so the sum of their probabilities is equal to 1. First, use this fact to find the probability of .

The notation refers to the probability of either event or event occurring, or both. In other words, the notation refers to the probability of at least one of the events occurring. 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. Next, use this fact to find the probability of .

The notation refers to the probability of events and not occurring together. In other words, the notation refers to the probability of the occurrence of at most one of the events. Thus, the probability of Adam succeeding in at most one of the two subjects is 0.81.

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