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11 years ago
We are given two drawers, one contains only black balls, the other one 50% black balls, and 50% white balls. I pick up a ball at random, it turns out that the ball is black, what is the probability that this black ball comes from the first drawer?
P(A|B) = P(B|A) * P(A) / P(B), where:
* P(A) is the prior probability of A
* P(A|B) is the conditional probability of A given B - or posterior probability
* P(B|A) is the conditional probability of B given A
* P(B) is the marginal probability of B
What is the probability that I picked up the first drawer (the one with only black balls), given that I picked up a black ball.
* P(A) is the prior probability that I picked up drawer #1 without any other info, that is 1/2
* P(B) is the prior probability that I picked up a black ball without any other info, that is 3/4, because there are 1 black balls in #1, 0.5 black balls in #2, and 0.5 white balls in #2
* P(B|A) is the probability of getting a black ball given I chose drawer #1, because there are only black balls, it is 1
So we have P(A|B) = (1 * 1/2) / (3/4) = 2/3
So the answer is 2/3.