Easy Math Question?

I posted this question on RGP, but nobody responded. Maybe it's a stupid question? Anyhow...


Two players sit down to play a heads-up match of limit holdem. Assume that both players play at a somewhat constant skill level (In other word's, they don't play well one week and awful the next). At what point (measured in big bets) can we conclude with 95% certainty that the player with the lead is the better player?

Not a stupid question, just very difficult to answer. Perhaps nobody responds because nobody knows?


Also, the answer in terms of big bets can vary significantly depending on how loose and aggressive the two players play.


Jim

You haven't given enough information; just reaching a given lead is not enough.


It should be easy to recognize that the std. deviation is important here, because if the total win of a player is small in comparison to std. deviation, then the win could be due to a swing.


Secondly, you cannot have an experiment, and continually check the results until a statistical condition holds. You must define beforehand a given number of iterations (in this case, hands) and do your analysis afterwards.


IIRC, there is a theorem (a portion or lemma of central limit theorme, I think): Give me an arbitrary number of std. deviations X. As the number of iterations of an experiment increases, the probability that the outcome will at some point fall outside X std deviations approaches 1. Meaning if you do enough iterations, you can get results as far away from the mean as you'd like.


So, I am not certain that you can wait until a player is up some number of bets at all.

If you had the players play enough beforehand that you could make a reasonable estimate of the standard deviation, then you could determine how many hours they would have to play so that the results were 95% accurate.


***Give me an arbitrary number of std. deviations X. As the number of iterations of an experiment increases, the probability that the outcome will at some point fall outside X std deviations approaches 1***


It is also true, however, that if you give me an arbitrary value e, then as the number of iterations increases, the probability that the final result will lie in the range [X-e,X+e] (where X is the mean) goes to 1. Therefore, you can, given a predetermined std. devitation, determine the number of hours required to have at least 95% certainty that the real value is within some preset error bar around your result.

Thank you Lenny, this was sort of what I was getting at. So aren't there known common ranges in deviation for heads up play? Sort of like the 1 BB/hr thing you always seen thrown around for average profit? If you applied those ranges, could you extrapolate and come up with range of bets or hours? Can you give me a ballpark?