[fnord_too]

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this one seems easy. your bankroll is finite, $1 trillion you said. so of course probability dictates that you will eventually go bust.

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I disagree. This is a random walk about a positively sloped line. If he were even money, then I would say he will go bust, since eventually he will hit a bad spot long enough to depleat any arbitrary number, but here the length of the bad streak needed is increasing, since he has positive expectation.

I have a picture in my head that I am not expressing well. Let me try it this way: Say this were flipping a coin where hero is getting 1.1:1 on his money. As time wears on, he will need a bad streak considerably longer than his longest good streak to go bust. To frame it slightly differently, and possibly sophisticly, over any set of trials, cancel the wins and losses. In this case, you cancel 11 losses with 10 wins. Then you are left with pure loss or gain. For him to go bust in this coin example, his longest loosing streak needs to be very much longer than his longest winning streak to date. (I know he can have multiple losing streaks, but you can just extend the period where you cancel wins and losses there.)

Is this making any sense? Basically, if you take a random walk whos expectation has a non zero slope, you do not have to eventually get to zero if you start sufficiently away from zero on the side that your slope is pointing.