Random walk question

[Izverg04]

What average number of fair coin flips does it take to lose N bets?

[SumZero]

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What average number of fair coin flips does it take to lose N bets?

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The expected number of flips of a fair coin to lose N bets is 2N.

But I suspect based on the subject you meant what is the expected number of flips to be -N bets in a running total of equally sized bets about flips of a fair coin. For large M number of flips the expected value of the absolute value of the running total is around sqrt(2M/pi). For smaller M the expected value of the absolute value is ((M-1)!!)/((M-2)!!) when M is even and ((M!!)/((M-1)!!)) for M being odd. Of course half the time the distance is positive, so this may not be quite what you are looking for.

[BruceZ]

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What average number of fair coin flips does it take to lose N bets?

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If you play until you lose N bets or bust your opponent, the average duration is N*(opponent's bankroll). If your opponent is infinitely wealthy so he can't be busted, then the average duration is infinite.