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A: Infinity
B: Chance
Basic, maybe flawed question: Let's say you have a bankroll of 1 trillion BB's, and your winrate is 3bb/100. If you played an infinite amount of time, would you eventually bust out?
Since it is always possible that you could lose X hands, will this eventually happen since the time range is infinity? How does this concept mesh with the fact that you should always be up in the long run since you have an edge?
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If you have an edge, then your probability of going bust is always < 1, no matter how low your win rate, or how small your bankroll. This probability, called the risk of ruin, depends on your win rate, your standard deviation, and your bankroll, and it can be computed by the formula in
this thread for games like blackjack and poker. The derivation can be found
here.
While it is true that you are guaranteed to eventually suffer a downswing of X dollars for X arbitrarily large, these downswings do not cause you to go bust because by the time they occur, your bankroll will have grown large enough to absorb them. Your bankroll grows linearly with the number of hands played, while the likelihood of a downswing of a given size depends on your standard deviation, and this increases as the square root of the number of hands played. Hence the growth of your bankroll out paces the frequency and size of the negative swings.
When considering the risk of ruin, many people have the misconception that it makes a big difference whether you play forever vs. only a few hundred hours. In fact, for a significant winner, risk of ruin is a short-term phenomenon. If he doesn't go broke in the first few hundred hours, chances are he never will. This is because once he doubles his bankroll, his risk of ruin becomes squared, e.g. 1% becomes 0.01% since going bust now requires him to lose the 1% bankroll twice. So his risk of ruin for playing forever is essentially the same as for playing only a few hundred hours. This is for a player who reinvests all his winnings in his bankroll. A player who spends all his winnings will go broke once he hits a big enough losing streak. How long that takes depends on the size of his initial bankroll, but the probability that he goes broke appoaches 1 as he plays to infinity.