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(𝐵) = 4 𝑃(𝐴) and 𝑃(𝐴 ∪ 𝐵) = 0.95, find 𝑃(𝐵).

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Video Transcript

Suppose 𝐴 and 𝐵 are two mutually exclusive events. Given that the probability of 𝐵 is four multiplied by the probability of 𝐴 and the probability of 𝐴 union 𝐵 is equal to 0.95, find the probability of 𝐵.

We begin by recalling our definition of mutually exclusive events. Two or more events are said to be mutually exclusive if they cannot happen at the same time. If these two events are 𝐴 and 𝐵, the probability of 𝐴 intersection 𝐵 is equal to zero. And the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵. In this question, we are told the probability of 𝐴 union 𝐵 is equal to 0.95. We are also told that the probability of 𝐵 is equal to four times the probability of 𝐴. This gives us the equation 0.95 is equal to the probability of 𝐴 plus four multiplied by the probability of 𝐴. Simplifying the right-hand side, we have five multiplied by the probability of 𝐴.

We can then divide through by five to calculate the probability of 𝐴. This is equal to 0.19. The probability of 𝐵, which is what we are trying to calculate, is therefore equal to four multiplied by 0.19. This is equal to 0.76. We can represent this information on a Venn diagram. The probability of 𝐴 is 0.19 and the probability of 𝐵 0.76. As these sum to give us the union, which is equal to 0.95, the probability that an event is not in event 𝐴 and not in event 𝐵 is equal to 0.05.

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