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Math Help - coin games

  1. #1
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    Cool coin games

    Alice hides either a nickel or a quarter behind her back. Then Bob guesses which it is. If Bob guesses correctly, he wins the coin. If Bob guesses incorrectly, he has to pay Alice 15 cents. Also, Bob has spies and knows what the strategy will be and can guess accordingly.

    What strategy for Alice hs the highest value, and what is its value?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by inneedofhelp View Post
    Alice hides either a nickel or a quarter behind her back. Then Bob guesses which it is. If Bob guesses correctly, he wins the coin. If Bob guesses incorrectly, he has to pay Alice 15 cents. Also, Bob has spies and knows what the strategy will be and can guess accordingly.

    What strategy for Alice hs the highest value, and what is its value?
    To slelect the nickel with probability 5/7 and the quarter with probability 2/7 at random and independently on each play, and the game has a value of -50/7 to Alice.

    That Alice has to have a randomised strategy is obvious as otherwise Bob will win evey play (now it may turn out that the random starategy is to always select the nickel but we will let the sums tell us that).

    So start by assuming that A chooses the nickel with probability p_1, and that B guesses nickel with probbaility p_2. Work out the expected value of the game to A under these assumptions. This is the objective that A will seek to maximise under the assumption that B will simultaneously try to minimise it.

    All of this is subject to the constraints that 0<=p_1<=1, 0<=p_2<=1.

    RonL
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