### Video Transcript

Suppose that π΄ and π΅ are events in a random experiment. 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΄.

In this question, we are trying to calculate the probability of the difference of two events. 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΅. This can be represented on a Venn diagram as shown. Weβre looking for the probability that weβre in event π΅ but not event π΄. Weβre not given either the probability of π΅ or the probability of π΄ intersection π΅ in the question. However, we are given the probability of π΅ bar. This is the complement of event π΅ such that the probability of π΅ bar is equal to one minus the probability of π΅.

In this question, 0.47 is equal to one minus the probability of π΅. Rearranging this equation, we have the probability of π΅ is equal to one minus 0.47, which is equal to 0.53. We can calculate the probability of π΄ intersection π΅ using the addition rule of probability. 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΅. 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΅. We can rearrange this equation such that the probability of π΄ intersection π΅ is equal to 0.71 plus 0.53 minus 0.99. This is equal to 0.25.

The probability of π΅ minus π΄ is therefore equal to 0.53 minus 0.25, which is equal to 0.28.