Random walk on a symmetric interval

[pzhon]

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The analogous statement is false for asymmetric intervals.

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This should be clear from the fact that if there are 2n chips, only the leader can win in n steps.

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Right. It's also false in the continuous case, where you can't make an argument like that quite as easily, but you can still show that a nearly instantaneous vistory is much more likely for the player who starts with the larger stack. Qualitatively, how does the conditional probability of winning vary with time?