Urn 1 contains 5 white balls and 7 black balls. Urn 2 contains 3 whites and 12 black. A fair coin is flipped; if it is Heads, a ball is drawn from Urn 1, and if it is Tails, a ball is drawn from Urn 2. Suppose that this experiment is done and you learn that a white ball was selected. What is the probability of choosing a white ball?
I thought P(W) = 8/27 or 0.29 since P(H or T) = 0.5
BUT P(W) = P(W|T) P(T) + P(W|T') P(T') = 3/15 x 1/2 + 5/12 x 1/2 = 1/10 + 5/24 = 37/120 (= 0.302)
I understand the even though getting H or T is same, the frequency of white balls in each urn is different. But I still think probability is 8/27 (of course I am wrong but dont know how to change my opinion). Could someone better explain whats happening and/or point me to other examples? I always get such questions wrong and want to train myself instinctively for such weighted probabilities. Need to train myself to spot them.