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

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that 𝑃(𝐴) = 0.71, 𝑃(𝐵 bar) = 0.47, and 𝑃(𝐴 ∪ 𝐵) = 0.99, determine 𝑃(𝐵 − 𝐴).

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### Video Transcript

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that the probability of 𝐴 is 0.71, the probability of 𝐵 bar is 0.47, and the probability of 𝐴 union 𝐵 is 0.99, determine the probability of 𝐵 minus 𝐴.

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 𝐵 minus 𝐴 is equal to the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. This can be represented on a Venn diagram as shown. We’re looking for the probability that we’re in event 𝐵 but not event 𝐴. We’re not given either the probability of 𝐵 or the probability of 𝐴 intersection 𝐵 in the question. However, we are given the probability of 𝐵 bar. This is the complement of event 𝐵 such that the probability of 𝐵 bar is equal to one minus the probability of 𝐵.

In this question, 0.47 is equal to one minus the probability of 𝐵. Rearranging this equation, we have the probability of 𝐵 is equal to one minus 0.47, which is equal to 0.53. We can calculate the probability of 𝐴 intersection 𝐵 using 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. Substituting in the values we know, we have 0.99 is equal to 0.71 plus 0.53 minus the probability of 𝐴 intersection 𝐵. We can rearrange this equation such that the probability of 𝐴 intersection 𝐵 is equal to 0.71 plus 0.53 minus 0.99. This is equal to 0.25.

The probability of 𝐵 minus 𝐴 is therefore equal to 0.53 minus 0.25, which is equal to 0.28.