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I realized that an earlier question I'd posted was actually different than what I was actually asking.

My question is: say we have a game where you win 2 dollars, or lose 1 dollar, both with probability 1/2. What is the probability that you end up with exactly the same amount of money that you started with?

Is this equivalent to a ruin problem against an infinitely rich adversary?

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    It rather depends on when you decide to end the game. When you run out of money? When you next have what you started with (and what happens if you lose 1 then gain 2)? When you have a million times what you started with? The first of these to happen?2011-05-03
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    say it functions exactly as a random walk on the real number line. It ends when you next have what you started with, and you are allowed to go into the negative.2011-05-03
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    Random walks do not usually jump over integer points, but I will take that as "don't stop".2011-05-03

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