Puzzle

Alice repeatedly draws a card randomly, without replacement, from a pack of fifty-two cards. Bob has a one-time privilege to raise his hand just before a card is about to be drawn. Bob must execute his privilege before the last card is drawn. If the card drawn is Red just after Bob raises his hand, Bob wins; otherwise he loses. Is there any way for Bob to be correct more than half the times he plays this game with Alice?

Source

Heard from Prof Huzur Saran in 1994.

Solution

Imagine that whenever Bob raises his hand, Alice draws the *last* card in the pile (instead of the next card). This process is equivalent to the original process as far as the color of the card being drawn is concerned. So the probability that the card drawn is red is half. In other words, Bob can never be correct more than half the times he plays this game with Alice.

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