Each person in turn enters the room and turns over cards one after another until either he finds the card with his name or he has turned 50 of the 100 cards. After that he leaves the room, without communicating anything to the others, and the cards are all turned back face down before the next person comes in.
How large can they make the probability that everyone will find the card with his name?
(If everyone turns 50 cards at random, the probability will be (1/2)^{100}, which is about 8 * 10^{-31}).
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