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Question Video: Determining the Probability of Difference of Two Events Mathematics

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that P(𝐴) = 0.71, P(𝐵 bar) = 0.47, and P(𝐴 ∪ 𝐵) = 0.99, determine P(𝐵 − 𝐴).

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Video Transcript

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that the probability of 𝐴 is 0.71, the probability of 𝐵 bar is 0.47, and the probability of 𝐴 union 𝐵 is 0.99, determine the probability of 𝐵 minus 𝐴.

Before starting this question, we recall that 𝐵 bar means the complement of event 𝐵. The probability of the complement is the same as the probability of the event not occurring. As probabilities sum to one, we know the probability of 𝐵 bar is equal to one minus the probability of 𝐵. Rearranging this formula, the probability of event 𝐵 is therefore equal to one minus the probability of the complement of 𝐵. As this is equal to 0.47, we can subtract this from one to calculate the probability of event 𝐵. The probability of event 𝐵 is therefore equal to 0.53. The reason we need this value is we are asked to calculate the probability of 𝐵 minus 𝐴. Recalling the difference rule for probability, we know this is equal to the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵.

We now know that the probability of 𝐵 is 0.53, and we can calculate the probability of 𝐴 intersection 𝐵. We can do this using the addition rule of probability, one form of which states that the probability of 𝐴 intersection 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 union 𝐵. We are told in the question that the probability of 𝐴 is 0.71. We have calculated that the probability of 𝐵 is 0.53, and we are also given that the probability of 𝐴 union 𝐵 is 0.99. The probability of 𝐴 intersection 𝐵 is therefore equal to 0.71 plus 0.53 minus 0.99. This is equal to 0.25. Substituting this value into the difference rule for probability, we see that the probability of 𝐵 minus 𝐴 is equal to 0.53 minus 0.25. This gives us a final answer of 0.28.

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