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Suppose π΄ and π΅ are two events. Given that π(π΄ β© π΅) = 2/3 and π(π΄) = 9/13, find π(π΅ | π΄).

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.

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