Guessing each other’s coins

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You and a friend each has a fair coin.  You can decide on a strategy and then play the following game, without any further communication with each other.  You flip your coin and then write down a guess as to what your friend’s coin will say.  Meanwhile, your friend flips her coin and writes down a guess as to what your coin says.  There’s a third person involved: The third person collects your guesses and inspects your coins.  If both you and your friend correctly guessed each other’s coins, then your team (you and your friend) receive 2 Euros from the third person.  But if either you or your friend (or both) gets the guess wrong, then your team has to pay 1 Euro to the third person.  This procedure is repeated all day.  Assuming your object is to win money, are you happy to be on your team or would you rather trade places with the third person?

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    In this game, the outcomes of the coin flips are independent, which means that the result of one flip does not affect the outcome of the other flip. Therefore, each player has a 50-50 chance of correctly guessing the other player’s coin.

    If both players guess randomly, the probability of both players guessing correctly is (1/2) x (1/2) = 1/4, and the probability of at least one player guessing incorrectly is 1 – 1/4 = 3/4.

    Therefore, on average, for every 4 rounds played, the team will win 2 Euros once and lose 1 Euro three times, resulting in a net loss of 1 Euro for every 4 rounds. This means that over the long run, the team will lose money and the third person will make a profit.

    So, if your objective is to win money, you would be better off trading places with the third person.

    Moshe Expert Answered on 23rd March 2023.
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