EV and Variance

[JoeyT]

It has been awhile since I did much math, and I'm sure this has been discussed but I can't find a good answer to it... so if someone can point me to a resource or just answer this I would greatly appreciate it. Basically I want to know that given...

EV of E per some unit time (an hour or 100 hands or whatever)
Variance V for that same unit time
# Units of time T

What are the chances that you make or lose $X over T units of time with said E and V?

So like, if your EV is $10/hr and your variance is $50/hr, what are the chances over a given 10 hours you'll make at least $100, or lose at least $100, or whatever. Also, what is a reasonable variance for a TAG in llhe games?

Thanks.

[BruceZ]

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So like, if your EV is $10/hr and your variance is $50/hr, what are the chances over a given 10 hours you'll make at least $100[/quote

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50%.

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or lose at least $100

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Essentially zero. Being $200 below average would be almost 9 standard deviations below the mean since the standard deviation is sqrt(50*10).

For a general win rate and standard deviation = sqrt(variance), use the method described in this post.


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Also, what is a reasonable variance for a TAG in llhe games?

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Around the beginning of the big poker boom, a reasonable standard deviation for a player winning 3 bb/100 hands in a full ring game was about 16 big bets for 100 hands, or a variance of 256 bb^2/100 hands. Since then the games have generally become first looser and then tighter, so use that to get an idea. For higher or lower win rates, scale the standard deviation proportionately.

[BillC]

I'm not so sure that you can invoke the Central Limit Theorem here. Short-term results can be much more extreme than predicted by the usual random walk model.

[BruceZ]

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I'm not so sure that you can invoke the Central Limit Theorem here. Short-term results can be much more extreme than predicted by the usual random walk model.

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I didn't consider that he was asking about just 10 hours. Still the -$100 result should be quite unlikely, and the 50% would be accurate as long as the short term distribution is at least symmetrical. Even for 10 hours, a normal distribution might not be too far off for convolving most hourly distributions with themselves 10 times. Wasn't it you who once said that it would be surprising if the results for 100 hands were not normal by the central limit theorem being applied to a sum of 100 hand results? 10 hours would be more than 100 hands.

[BillC]

It would be surprising if the distribution for 10 hours is far form normal, but I am just not sure 10 hours suffices. I know that there have been studies about this on this site. I just remember blackjack team members freaking out about -6 sigma drawdowns in a several shoes. I always told them too cool it and use CLT (yet). Of course blackjack is not the same (?) and 10 hours is more like 20 backcounted shoes.