Sergiu Hart / Puzzles / 100 cards

100 cards

The names of 100 people (all names are different) are written on 100 cards. The 100 cards are taken to a closed room, randomly shuffled and put face down in a row.

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).

Notes

They can coordinate before the game starts; there is no communication (neither explicit nor hidden - such as moving cards, etc.) during the game.
No need to find the precise maximum; just try to make the probability as large as you can.
What happens as the number of people increases?

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