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Math Help - Expected number of turns in single player coin flipping game

  1. #1
    Newbie Maxim's Avatar
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    Expected number of turns in single player coin flipping game

    Similar to my other topic, again I play a coin flipping game.

    This time I start with zero points. I repeatedly flip a coin. Heads = I get one point, tails = I lose one point. My score can also be negative.

    Question: what's the expected number of turns I have to take, to reach 3 points?
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  2. #2
    Newbie Maxim's Avatar
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    Re: Expected number of turns in single player coin flipping game

    I've been pondering about this for a while now, but I can't seem to get anywhere.

    Defining T_{a\rightarrow b} as the number of turns until reaching b points (starting with a points), I think \mathbb{E}(T_{0\rightarrow 3}) = 3\mathbb{E}(T_{0\rightarrow 1}) and \mathbb{E}(T_{0\rightarrow 1})=\mathbb{E}(T_{n\rightarrow n+1})\ \forall n.

    So basically it boils down to calculating \mathbb{E}(T_{n\rightarrow n+1}), the expected number of turns to reach one point more than I currently have.

    I can write the recurrence relation \mathbb{E}(T_{n\rightarrow n+1}) = \frac{1}{2} + \frac{1}{2}\mathbb{E}(T_{n-1\rightarrow n+1}) but since there's no boundaries to the score, this seems to bring me nowhere?
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