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

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃(𝐴) = 5/7 and 𝑃(𝐵) = 4/7. Given that 𝑃(𝐴 ∪ 𝐵) = 6/7, determine 𝑃(𝐴 − 𝐵).

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Video Transcript

Suppose 𝐴 and 𝐵 are two events with probabilities the probability of 𝐴 is five- sevenths and the probability of 𝐵 is four-sevenths. Given that the probability of 𝐴 union 𝐵 is six-sevenths, determine the probability of 𝐴 minus 𝐵.

In this question, we are asked to calculate the probability of the difference of two events. We recall that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. This is known as the difference rule of probability. And in this question, we are given the probability of 𝐴 is five-sevenths. At present, we are not given the probability of 𝐴 intersection 𝐵. Before trying to calculate this, it is worth noting that we can represent this situation on a Venn diagram. We are trying to calculate the probability of being in event 𝐴 but not event 𝐵.

In order to calculate the probability of 𝐴 intersection 𝐵, we recall the addition rule of probability. This states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. We are given three of these values in the question. The probability of 𝐴 union 𝐵 is six-sevenths, the probability of 𝐴 is five-sevenths, and the probability of 𝐵 is four-sevenths. By rearranging and then simplifying this equation, we can calculate the probability of 𝐴 intersection 𝐵. This is equal to five-sevenths plus four-sevenths minus six-sevenths. As the denominators of the three fractions are the same, we simply add and subtract the numerators. Five plus four minus six is equal to three. The probability of 𝐴 intersection 𝐵 is three-sevenths.

We can now substitute this value back into the formula to calculate the probability of 𝐴 minus 𝐵. This is equal to five-sevenths minus three-sevenths, which gives us a final answer of two-sevenths.

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