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.
