Urn A contains 2 white and 2 black balls while urn B contains 3 white and 2 black balls. One ball is transferred from urn A to urn B and then a ball is drawn out of urn B. What is the probability that the ball is white?

Explanation

Let W_1 and B_1 be the events of transferring a white or black ball from A to B. P(W_1) = 2/4 = 1/2 and P(B_1) = 2/4 = 1/2. If W_1 occurs, B has 4 white and 2 black, so P(W_2|W_1) = 4/6 = 2/3. If B_1 occurs, B has 3 white and 3 black, so P(W_2|B_1) = 3/6 = 1/2. Total probability P(W) = \frac{1}{2}(\frac{2}{3}) + \frac{1}{2}(\frac{1}{2}) = \frac{1}{3} + \frac{1}{4} = \frac{7}{12}.

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