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Math Help - Expected amount a person wins/looses after m games at a casino

  1. #1
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    Smile Expected amount a person wins/looses after m games at a casino

    Hi everyone,

    Assume that Jack goes to a casino. He plays n games. Every time he wins a game he gets 100$ and every time he looses a game he looses 50$. The probability that he wins i-th game is 1/i. In other words, the probabilities of winning first five games are 1,1/2,1/3,1/4,1/5, respectively.

    Can we compute the expected amount of money he has won or lost after he
    has played n games? An easier version will be how can we compute the expected amount after he has played 5 games.

    Thanks in advance

    Best Regards,
    mac
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  2. #2
    Grand Panjandrum
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    Re: Expected amount a person wins/looses after m games at a casino

    Quote Originally Posted by mac154 View Post
    Hi everyone,

    Assume that Jack goes to a casino. He plays n games. Every time he wins a game he gets 100$ and every time he looses a game he looses 50$. The probability that he wins i-th game is 1/i. In other words, the probabilities of winning first five games are 1,1/2,1/3,1/4,1/5, respectively.

    Can we compute the expected amount of money he has won or lost after he
    has played n games? An easier version will be how can we compute the expected amount after he has played 5 games.

    Thanks in advance

    Best Regards,
    mac
    The expected return on the i th game is

    E(r_i)= -50\left(1-\frac{1}{i}\right)+100\left( \frac{1}{i}\right)

    Therefore the expected total return after n games is:

    E(R_n)=E\left(\sum_{i=1}^n r_i\right)=\sum_{i=1}^nE(r_i)

    CB
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  3. #3
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    Re: Expected amount a person wins/looses after m games at a casino

    Many thanks
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