Question Video: Using the Addition Rule to Determine the Probability of an Event Involving Mutually Exclusive Events Mathematics

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given that 𝑃(𝐴 ∪ 𝐵) = 0.93 and 𝑃(𝐴 − 𝐵) = 0.39, find 𝑃(𝐵).

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Video Transcript

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given the probability of 𝐴 union 𝐵 is equal to 0.93 and the probability of 𝐴 minus 𝐵 is equal to 0.39, find the probability of 𝐵.

In this question, we are told that the probability of 𝐴 or 𝐵, denoted 𝐴 union 𝐵, is equal to 0.93. We are also told that the probability of 𝐴 occurring and 𝐵 not occurring, denoted the probability of 𝐴 minus 𝐵, is equal to 0.39. We are asked to find the probability of 𝐵 occurring. As the two events 𝐴 and 𝐵 are mutually exclusive, we know that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. It also means that the probability of 𝐴 and 𝐵 occurring, denoted the probability of 𝐴 intersection 𝐵, is equal to zero.

We can represent the probability of these two events on a Venn diagram as shown. Since the events are mutually exclusive, we can see from the diagram that the probability of 𝐴 minus 𝐵, that is, the probability of 𝐴 occurring and 𝐵 not occurring, is equal to the probability of 𝐴. We know this is equal to 0.39. We also know that the the probability of 𝐴 union 𝐵 is equal to 0.93. And since the events are mutually exclusive, this equals the probability of 𝐴 plus the probability of 𝐵. We have 0.93 is equal to 0.39 plus the probability of 𝐵. Subtracting 0.39 from both sides of this equation gives us the probability of 𝐵 is equal to 0.93 minus 0.39, which is equal to 0.54.

If 𝐴 and 𝐵 are mutually exclusive events, the probability of 𝐴 union 𝐵 is equal to 0.93, and the probability of 𝐴 minus 𝐵 is 0.39, then the probability of 𝐵 is equal to 0.54.

Whilst it is not required in this question, it is worth noting that the probability of neither 𝐴 nor 𝐵 occurring is equal to 0.07. And this can be represented in the completed Venn diagram as shown. The sum of all the probabilities in the Venn diagram must equal one.