How to solve this "probabilities and weighted probabilities" type of question? 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.
 A: Suppose that you are in a probability course in which the professor begins every class by doing the experiment you described. However, you are late to class every day and come in just as the ball is being withdrawn. In the rush to get to your seat, you fail to observe that on some days Urn 1 is being used while on other days Urn 2 is being used.  You keep track of whether the ball is white.  Over 120 days (what can I say, it is a long course), on 60 days it was Urn 1 that was used from which 25 white and 35 black balls white drawn while on the other 60 days it was Urn 2 from which 12 white and 48 black balls were drawn.  So, as far as you can tell, the probability of drawing a white ball is (25+12)/120 = 37/120. Your
more punctual classmates, on the other hand, have more data and know that the probability of drawing a white ball depends on which urn is chosen, but over 120 days, and ignoring
the fine details of the data that they have gathered, they too work out the relative 
frequency of white balls as 37/120, same as you, and this is indeed the maximum-likelihood estimate for the probability that a white ball is drawn at the beginning of class
in your course.
A: Now, imagine the same experiment with the first urn containing 1000 white balls plus 1000 black balls and the second urn containing only one white ball. 
Another case would consist in having only 2000 black balls in the first urn and only one white in the second. You see that in those two examples, your first reasoning is obviously wrong.
Bottom line: always check the extreme cases to make sure your solution is correct. 
