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

[fnord_too]

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If what you're saying is that whether you bust out or not is not certain either way, even over infinity, then I agree. I think there is a chance of busting out, but that it is less than 100% and greater than 0%. Also it seems to follow that the longer this game goes on less chance of you busting out.

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Yeah, that's what I was trying to say. And since you are starting with one trillion BB, I think that your chance of going bust is very close to 0.

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.

[Darryl_P]

If my opponents played in a way that they went all in any more often than, say, once in a million hands, then the assumption of my having a 3 BB/100 advantage would be false...it would be much greater than that.

For me to have only a 3BB/100 advantage we must rule out extremely high variance plays on the part of my opponent since those would lose too much EV.

And when you bring down the variance to "normal" levels, then the random walk model can be used with high precision even in NLH.

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And when you bring down the variance to "normal" levels, then the random walk model can be used with high precision even in NLH.

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Well, it depends on our assumptions about how the game is set up. If we assume that you are playing against opponents with infinitely large bankrolls, then the random walk model no longer holds. On any hand, there is a very small chance that you will lose your entire stack. Your winnings at any rate of BB/100 will not eliminate that chance, and therefore you will bust out eventually.

If you play against opponents who add or subtract from their stacks so they're at 1 trillion BB at the beginning of each new hand, then you're correct, assuming both you and your opponents play to maximize EV, rather than to maximize or minimize the chance you will bust.

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]

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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

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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.

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

[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?

[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.