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For 100 turns you either have $1 placed in one of two boxes at random or empty a random box and keep the money. With optimal play, what is the expected payoff?

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A one-player game with two opaque boxes lasts 100 turns. Each turn you either 'place' (a third party puts $1 into a random box) or 'take' (you empty a random box and keep the money). You do not learn how much you have taken until the end. With optimal play, what is the expected payoff?

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