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Motivation: A friend asked me this question.

The Problem: Suppose you start off with a dollar. You flip a fair coin, if it lands on heads you win $50$ cents otherwise you lose $50$ cents. If after $n$ flips you have a nonzero amount of money, you win. What's the probability you win? What about the limiting case as $n$ tends to infinity?

edit: In this game you are not allowed to have negative money. Thanks, Jonathan Fischoff, the linked helped greatly.

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    You might find this interesting http://math.stackexchange.com/questions/607/given-enough-time-what-are-the-chances-i-can-come-out-ahead-in-a-coin-toss-conte2010-08-07
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    Are you allowed to have a negative amount of money or does the game end when you lose all your money? In either case I think the keyword you want is "one-dimensional random walk."2010-08-07
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    Can the 'house' (the person betting against the player) ever run out of money?2010-08-07
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    I want Qiaochu's question answered a little more explicitly. If I get 60 flips and I get 2 tails on my first 2 flips, is that considered a loss automatically? If my first 59 flips come up tails, but I then flip again and get heads, do I finish the game with 50 cents and thus a nonzero amount of money? Or have I already lost?2015-07-27

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