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I gave my friend this problem as a brainteaser; while her attempted solution didn't work, it raised an interesting question.

I flip a fair coin repeatedly and record the results. I stop as soon as the number of heads is equal to twice the number of tails (for example, I will stop after seeing HHT or THTHHH or TTTHHHHHH). What's the probability that I never stop?

I've tried to just compute the answer directly, but the terms got ugly pretty quickly. I'm hoping for a hint towards a slick solution, but I will keep trying to brute force an answer in the meantime.

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    Hopefully whatever method resolves this applies to other ratios than $2$.2011-08-27
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    @anon, good idea... See my answer.2011-08-27
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    @anon: I've generalized my answer up to an infinite series expression as well.2011-08-27
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    See "[a coin toss question](http://math.stackexchange.com/questions/23808/coin-toss-question)" for the similar problem of flipping until achieving the same number of heads as tails.2011-08-31
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    @Elliot I'd like you remind you that this question doesn't currently have an accepted answer.2013-06-16
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    Probability that the game ends after 3 tosses is 0.3752013-11-26

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