Question Video: Determining Conditional Probability Mathematics • 10th Grade

Video Transcript

Suppose 𝐴 and 𝐵 are two events. Given that the probability of 𝐴 intersection 𝐵 is two-thirds and the probability of 𝐴 is nine-thirteenths, find the probability of 𝐵 given 𝐴.

Now you may recall the general formula for conditional probability that the probability of 𝐴 given 𝐵 is equal to the probability of 𝐴 intersection 𝐵 over the probability of 𝐵. But we’ve been asked to find the probability of 𝐵 given 𝐴. So let’s reverse 𝐴 and 𝐵 in our formula. Well, that’s great. We’re looking for the probability of 𝐵 given 𝐴. We’ve been given probability 𝐴 in the question, but we were given the probability of 𝐴 intersection 𝐵, not 𝐵 intersection 𝐴. But let’s consider a Venn diagram.

The region 𝐴 intersection 𝐵 is the same as the region 𝐵 intersection 𝐴. So an equivalent formula is the probability of 𝐵 given 𝐴 is equal to the probability of 𝐴 intersection 𝐵 over the probability of 𝐴. We were told that the probability of 𝐴 intersection 𝐵 is two-thirds and the probability of 𝐴 is nine-thirteenths. So the probability of 𝐵 given 𝐴 is two-thirds divided by nine-thirteenths. And an equivalent calculation to dividing by nine-thirteenths is multiplying by its reciprocal, 13 over nine, which gives us our answer. The probability of 𝐵 given 𝐴 is 26 over 27.