Three Cards Solution

The correct answer is 2/3. Many people give the answer 1/2 because they think the chances are 50:50 that they hold the card that is all black or the hybrid card. This is wrong because it ignores a piece of information, namely that you are seeing a black face. If you label the cards as w1w2, w3b1 and b2b3 then it is clear you are holding b1, b2 or b3 face up. So the chances of the other side being black are 2/3 because only in the case of b1 is the other side of your card white.

There is an even easier way to get the right answer. Ask a different question, namely what is the chance that you pick a card that has the same colour on both sides? The answer is clearly 2/3 as only the hybrid card has different colours on its two faces. Since the situation is symmetrical with regard to black and white, the chances of the other side being black if the first side is black are also 2/3.

The moral of the story is that calculating probability is harder than it seems and that you need to be very careful.




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