Video Transcript
Suppose π΄ and π΅ are two
events. Given that the probability of π΄
intersection π΅ is two-thirds and the probability of π΄ is nine-thirteenths, find
the probability of π΅ given π΄.
Now you may recall the general
formula for conditional probability that the probability of π΄ given π΅ is equal to
the probability of π΄ intersection π΅ over the probability of π΅. But weβve been asked to find the
probability of π΅ given π΄. So letβs reverse π΄ and π΅ in our
formula. Well, thatβs great. Weβre looking for the probability
of π΅ given π΄. Weβve been given probability π΄ in
the question, but we were given the probability of π΄ intersection π΅, not π΅
intersection π΄. But letβs consider a Venn
diagram.
The region π΄ intersection π΅ is
the same as the region π΅ intersection π΄. So an equivalent formula is the
probability of π΅ given π΄ is equal to the probability of π΄ intersection π΅ over
the probability of π΄. We were told that the probability
of π΄ intersection π΅ is two-thirds and the probability of π΄ is
nine-thirteenths. So the probability of π΅ given π΄
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 π΅ given π΄ is
26 over 27.