# Always get confused by “probabilities and weighted probabilities” type of questions as cant differentiate between two

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.

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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.

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But wouldnt P(W) still be 1001/2001 OR 1/2000? –  user18601 Jan 12 at 14:12
Of course not, if you have 1000 black balls in the first urn and only one white ball in the second urn, you see that the probability of picking a white ball is equivalent to the probability of picking the second urn ($=\frac{1}{2}$), that is having Tail when the coin is flipped. Try to picture the experiment in your head and count the number of times you pick a white ball as opposed to a black ball. –  ThePawn Jan 12 at 14:19
this clarified everything. thanks a lot!!! –  user18601 Jan 12 at 23:09
Looking at extremes is indeed useful general advice for checking one's thinking--as well as one's calculations. (+1) –  whuber Mar 13 at 20:43