Event A Event B
There are 5 apartments. One has 2 men, one has a woman and 2 men, one has 2 women and 3 men, one has 6 women and 1 man, and one has a married couple (one man, one woman). If I knock on the door and a woman answers, what is the probability that I've reached the one with the married couple?
The answer is a/b, enter your answer as a + b.
The answer is 544.
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If the apartments are A1, A2, A3, A4, and A5, we're looking for P(A5|W), where W is woman. This should be equal to P(A5 and W)/P(W). Meaning what is the probability that we are at the married couple's apartment given that a woman has answered the door.
(1) P(A5 and W) = (1/5)(1/2) = 1/10.
P(W) = P(A1 and W) + P(A2 and W) + P(A3 and W) + P(A4 and W) + P(A5 and W).
Since there are no women in A1,
P(A1 and W) = 0.
P(A2 and W) = (1/5)*(1/3)
P(A3 and W) = (1/5)*(2/5)
P(A4 and W) = (1/5)*(6/7)
P(A5 and W) = (1/5)*(1/2)
This gives P(W) = 0 + (1/5)(1/3) + (1/5)(2/5) + (1/5)(6/7) + (1/5)(1/2) = 439/1050
Combining this last result with (1) gives
P(A5|W) = (1/10)/(439/1050) = 105/439.