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Math Help - Fun extra credit question

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    Fun extra credit question

    Mary and Paul are playing a game of tossing a coin. Mary says to Paul, "We will toss the coin until the first head appears. If the first head appears after an odd number of tosses, I win the game. If the first head appears after an even number of tosses, you win the game." They each put $60 into a cup and when one of them wins a game, he or she removes a dollar from the cup. They continue to do this until the cup is empty. Mathematically speaking, how much money will each person remove from the cup?
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    Quote Originally Posted by amiv4 View Post
    Mary and Paul are playing a game of tossing a coin. Mary says to Paul, "We will toss the coin until the first head appears. If the first head appears after an odd number of tosses, I win the game. If the first head appears after an even number of tosses, you win the game." They each put $60 into a cup and when one of them wins a game, he or she removes a dollar from the cup. They continue to do this until the cup is empty. Mathematically speaking, how much money will each person remove from the cup?
    Pr(Head on odd toss) = \frac{1}{2} + \left( \frac{1}{2} \right)^3 + \left( \frac{1}{2} \right)^5 + \, .... = x


    Pr(Head on even toss) = \left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^4 + \left( \frac{1}{2} \right)^6  + \, .... = \frac{1}{2} \left[ \frac{1}{2} + \left( \frac{1}{2} \right)^3 + \left( \frac{1}{2} \right)^5 + \, .... \right] = \frac{x}{2}.

    Therefore x + \frac{x}{2} = 1 \Rightarrow x = \frac{2}{3}.

    So out of 60 games, how many is Paul expected to win ....?
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