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

Suppose 𝐴 and 𝐵 are two events. Given that 𝐵 ⊂ 𝐴 and 𝑃(𝐴) = 5𝑃(𝐵) = 0.8, find 𝑃(𝐴 − 𝐵).

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Video Transcript

Suppose 𝐴 and 𝐵 are two events. Given that 𝐵 is a subset of 𝐴 and the probability of 𝐴 is equal to five multiplied by the probability of 𝐵, which is equal to 0.8, find the probability of 𝐴 minus 𝐵.

In this question, we want to calculate the probability of the difference between two events. We recall that 𝐴 minus 𝐵 is the set of elements 𝑥 that occur in event 𝐴 and do not occur in event 𝐵. And the difference rule of probability states that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐴 intersection 𝐵. This can be represented on a Venn diagram as shown.

In this question, however, we are told that 𝐵 is a subset of 𝐴. This means that every element of 𝐵 is also in 𝐴. And our Venn diagram will look as shown. It is clear in this case that the probability of 𝐴 minus 𝐵 is equal to the probability of 𝐴 minus the probability of 𝐵, as the probability of 𝐴 intersection 𝐵, those elements in both event 𝐴 and event 𝐵, is just equal to the probability of event 𝐵.

We are told that the probability of 𝐴 is 0.8. Five multiplied by the probability of 𝐵 is also equal to 0.8. Dividing both sides of this equation by five, we see that the probability of 𝐵 is 0.16. The probability of 𝐴 minus 𝐵 is therefore equal to 0.8 minus 0.16, and this is equal to 0.64.

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