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

Suppose 𝐴 and 𝐵 are events with probabilities 𝑃(𝐴) = 0.14 and 𝑃(𝐵) = 0.63. Given that 𝐴 ⊂ 𝐵, determine 𝑃(𝐵 − 𝐴).

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Video Transcript

Suppose 𝐴 and 𝐵 are events with probabilities the probability of 𝐴 is 0.14 and the probability of 𝐵 is 0.63. Given that 𝐴 is a subset of 𝐵, determine the probability of 𝐵 minus 𝐴.

In this question, we are asked to determine the probability of 𝐵 minus 𝐴. This is known as the difference of two events, and we can use the difference rule of probability to calculate it. This states that the probability of 𝐵 minus 𝐴 is equal to the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. In this question, we are told the probability of 𝐵 is 0.63. However, we are not explicitly told the probability of 𝐴 intersection 𝐵. We are told that 𝐴 is a subset of 𝐵. This means that all elements of set 𝐴 are also elements of set 𝐵. And this can be represented on a Venn diagram as shown.

This leads us to a general rule when dealing with subsets. If 𝐴 is a subset of 𝐵, then the probability of 𝐴 intersection 𝐵 is simply the probability of 𝐴. The elements that are in both set 𝐴 and set 𝐵 is the entire set of elements in set 𝐴. Since the probability of 𝐴 is 0.14, then the probability of 𝐴 intersection 𝐵 must also be 0.14. We can calculate the probability of 𝐵 minus 𝐴 by subtracting this from 0.63. This is equal to 0.49.

The probability of 𝐵 minus 𝐴, in other words, those elements in set 𝐵 but not set 𝐴, is equal to 0.49.

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