Am I stupid? I can't fit these two concepts into any type of harmony.

These types of questions may seem interesting thought exercises, but really are not. For one, the concept of "infinite time" already makes a supposition that is not reality-based (certainly not for organic poker-playing human lifeforms). So, to project any human activity to infinity is already a flawed analysis. Second, when one says "infinity" what are you really saying? It can't be logically handled because its so abstract that common concepts fail. If some event is truly infinite, then you should think that every possible outcome will eventually be realized simply because it has non-zero probability over an infinite period. Thus, if your bankroll is finite, there is some non-zero probability you bust and an infinite time to realize that non-zero probability sequence. But as you increase in time, that probability goes closer and closer to zero since your finite starting point is generally increasing. So there's an irreconcilible paradox that every sequence should eventually occur, but the sequence becomes more and more improbable as you go towards infinity. It would be an easier question if you had an infinite number of starting bankrolls, then yes, eventually a number of them would bust. The nature of "infinity" discussions will always have these paradoxes when a single trial is considered.

[PairTheBoard]

[ QUOTE ]
kidluckee --

when one says "infinity" what are you really saying? It can't be logically handled because its so abstract that common concepts fail. If some event is truly infinite, then you should think that every possible outcome will eventually be realized simply because it has non-zero probability over an infinite period.

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It can be handled mathematically. We are not looking at a Fixed event with nonzero probablilty and with infinite trials. If that were the case you would be right in saying the event is bound to happen eventually. But in this case we are looking at a sequence of different events, each one with smaller and smaller probability. We Can handle this situation logically. We have developed mathematics to do so and in some cases of this kind - such as this one - it is Not true that one of the events must eventually occur. In this case the sequence of events is "Going broke before doubling". After each trial where the event does Not happen, the chances of the next event in the sequence are smaller.

It is useful to look at the case of infinity for finding a bound to the probabilty. If we compute the probability of not going broke over infinite time to be say 99.9% then that's a lower bound for the probabilties we're interested in for finite times. For whatever finite time you're really interested in the probabilty for not going broke is therefore greater than 99.9%.

We are in the realm of mathematics here. Vague notions don't carry much weight.

PairTheBoard

As long as your winrate is greater than 0 and finite, and your bankroll is greater than 0 and finite, if you played an infinite amount of time, your chances of busting will always be greater than 0% and less than 100%

Even if your bankroll is 1BB, and your winrate is .001bb/100 there is a chance you will never go broke

And no matter how big your bankroll is, you always have a chance of going broke

Almost sounds contradictory, but it's not

[ QUOTE ]

It is useful to look at the case of infinity for finding a bound to the probabilty. If we compute the probability of not going broke over infinite time to be say 99.9% then that's a lower bound for the probabilties we're interested in for finite times. For whatever finite time you're really interested in the probabilty for not going broke is therefore greater than 99.9%.

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Good point.

[Alex/Mugaaz]

Ok since most seem to agree the answer is greater than 0 and less than 100%, what is the actual answer, and how can it be calculated?

[BruceZ]

[ QUOTE ]
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.

[Darryl_P]

Good grief! I already gave you the answer. If you want to hear something more pleasing I recommend visiting a psychic or an astrologist because if you ask any mathematician his answer will be the same as mine.

[FishAndChips]

Let's not forget in our analysis that while our bankroll is finite, so to is the total amount of money that can be won playing poker. Whether that's all the money in the world, or just a fraction of it, it definitely changes some conclusions.

If the amount of money that could be won in the game was infinite, you could eventually hit some astronomical bad luck streak with a trillion (or whatever our current bankroll is) bet down swing etc. However, you must compare the risk of ruin to the chance of winning every sweet penny.

Just imagine what would happen if you won "all the money in the world." [img]/images/graemlins/wink.gif[/img]

[PairTheBoard]

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FishAndChips --

Just imagine what would happen if you won "all the money in the world."

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I think I would keep my 5 year old Mustang because I like it and it only has 40,000 miles on it. But I'd probably get a bigger computer screen.

PairTheBoard

This is a very good answer for a limit game, but if you are playing in a no-limit game against players who have your $1 trillion covered, the probability of you busting out approaches 1 as the number of trials approach infinity.

In this situation, it is possible to bust out in a single hand. You're unlikely to go all in, but it will happen eventually. Most of the times when you go all in, you will be a large favorite but not a lock. If you change your strategy such that you will only go all in on the river with the nuts, you will be beaten very easily. That strategy can very easily be defeated by an opponent who pushes preflop on every hand--even if you had AA, you'd have to fold, because you'd only be a 9:1 favorite.