Question:
You are playing a card game with me. Suppose I shuffle a deck of 52 cards, then I show you the card one by one to you from top to bottom. You can stop me during this whole process, based on your memory of the previous cards you have seen. If you stop me, and there are still cards left in the deck, if the next unshown card is red, you get one dollar from me, otherwise if the next unshown card is black, you give me one dollar; if no cards left in the deck, you get one dollar if the last card is red, and you give one dollar if the last card is black.
Is there a strategy for you to win with probability larger than 50%?
Sol:
There is not strategy for that. Think the game in a different way: when you stop me, suppose instead of deciding whether you win or lose based on the next unshown card, we decide whether you win or lose based on the last card in the deck: if the last card is red you get one dollar, otherwise if it is black I get one dollar. The two games are the same. But for the latter, whatever strategies you choose the probability for you to win is always 50%, and so does the former.
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