I always get confused by "probabilities and weighted probabilities" type of questions as I can't differentiate between the two of them. Consider following example.
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 $P(W) = P(W|T) P(T) + P(W|T') P(T')\\ = 3/15 \cdot 1/2 + 5/12 \cdot 1/2\\ = 1/10 + 5/24 = 37/120 (= 0.302)$
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.