Suppose you only get one flip. If you flip coin A and it comes up heads, you will of course guess that it is the biased coin; if it comes up tails, you will guess that it's the fair coin. Now if you flip coin B and get the opposite face, you will be even happier with your previous decision. If you get the same face, you will be reduced to no information and may as well stick with the same choice. Thus, flipping the second coin is worthless.
Could flipping the same coin twice change your mind? Yes. If you get heads the first time, you are inclined to guess that that coin is biased, but seeing tails the next time will change your best guess to "unbiased." We conclude that flipping one coin twice is strictly better than flipping each coin once.
As it turns out, it's a theorem that in trying to determine which is which of two known probability distributions, it's better to draw twice from one than once from each. However, unlike in the puzzle above, it may happen that after seeing the result of your first draw you prefer to draw from the other distribution. We'll leave it to the reader to concoct an example.
[The remarkable theorem mentioned above was proved by Romanian mathematician Gheorghe Zbaganu.]
