You build a deck from cards labelled 1 to 5. Each round an opponent who knows the deck names a number and pays you that number, until a drawn card matches. Which deck is best?
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You build a deck out of cards labelled 1 to 5, using as many copies of each label as you like, and your opponent is told exactly what is in it. Each round he names a number and then a random card is drawn. He pays you the number he named whatever the card shows; if the card does not match, the card is returned, the deck reshuffled and another round is played, and if it matches the game ends. He chooses guesses to minimise his total payment. How do you compose the deck to maximise your expected winnings?
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