WEBVTT
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Suppose that π΄ and π΅ are events in a random experiment.
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Given that the probability of π΄ is 0.71, the probability of π΅ bar is 0.47, and the probability of π΄ union π΅ is 0.99, determine the probability of π΅ minus π΄.
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In this question, we are trying to calculate the probability of the difference of two events.
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The difference rule of probability states that the probability of π΅ minus π΄ is equal to the probability of π΅ minus the probability of π΄ intersection π΅.
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This can be represented on a Venn diagram as shown.
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Weβre looking for the probability that weβre in event π΅ but not event π΄.
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Weβre not given either the probability of π΅ or the probability of π΄ intersection π΅ in the question.
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However, we are given the probability of π΅ bar.
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This is the complement of event π΅ such that the probability of π΅ bar is equal to one minus the probability of π΅.
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In this question, 0.47 is equal to one minus the probability of π΅.
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Rearranging this equation, we have the probability of π΅ is equal to one minus 0.47, which is equal to 0.53.
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We can calculate the probability of π΄ intersection π΅ using the addition rule of probability.
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This states that the probability of π΄ union π΅ is equal to the probability of π΄ plus the probability of π΅ minus the probability of π΄ intersection π΅.
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Substituting in the values we know, we have 0.99 is equal to 0.71 plus 0.53 minus the probability of π΄ intersection π΅.
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We can rearrange this equation such that the probability of π΄ intersection π΅ is equal to 0.71 plus 0.53 minus 0.99.
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This is equal to 0.25.
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The probability of π΅ minus π΄ is therefore equal to 0.53 minus 0.25, which is equal to 0.28.