Using Hourly Statistics to Estimate Longer Term Win Rates

[harringk]

I started keeping very good records on my Poker sessions starting in January. Since then, I have logged about 100 hours spread over around 60 different sessions (all online $15-$30 limit hold em). During that time I have experienced very good streaks as well as very bad streaks and everything inbetween.

I have calculated my estimated hourly standard deviation based on the formula in this essay:

http://www.twoplustwo.com/mmessay8.html

My question is this: using these statistics, how would I go about calculating some confidence intervals for periods of time longer than one hour?

To put it another way, using the average hourly rate and the estimated hourly standard deviation it is simple enough to calculate confidence intervals for any 1 hour period.

But I am more interested in how likely I would be to make or lose X dollars over a period of say 100 hours of play.

Can anyone point me in the right direction, or just let me know if I'm barking up the wrong tree?

Thanks

[pretender2k]

I am more into BB/100 hands. But I look more at if I generally make x amount per 100 hands, how many hands do I need to play to reach this goal. I will also adjust this as time goes and I get more hands in my records. It may just be me but this made sense because the amount of hands per hour can vary online so I look at playing x hands rather than x hours. This would not work on a tight schedule though. I have a lot of free time outside my job.

[pzhon]

[ QUOTE ]

I have calculated my estimated hourly standard deviation based on the formula in this essay:

http://www.twoplustwo.com/mmessay8.html

My question is this: using these statistics, how would I go about calculating some confidence intervals for periods of time longer than one hour?


[/ QUOTE ]

If you play n times as long, the EV increases by a factor of n, and the standard deviation increases by a factor of sqrt(n). So, if you SD is 15 for an hour, for 100 hours it would be 15*sqrt(100) = 150.

The variance is the square of the standard deviation. One interpretation of the variance per hour is the number of hours you can play without being too surprised to average 2 bets/hour above or below your true average.

[harringk]

[ QUOTE ]
[ QUOTE ]

I have calculated my estimated hourly standard deviation based on the formula in this essay:

http://www.twoplustwo.com/mmessay8.html

My question is this: using these statistics, how would I go about calculating some confidence intervals for periods of time longer than one hour?


[/ QUOTE ]

If you play n times as long, the EV increases by a factor of n, and the standard deviation increases by a factor of sqrt(n). So, if you SD is 15 for an hour, for 100 hours it would be 15*sqrt(100) = 150.

The variance is the square of the standard deviation. One interpretation of the variance per hour is the number of hours you can play without being too surprised to average 2 bets/hour above or below your true average.

[/ QUOTE ]

Thanks for the reply pzhon. Let me put some numbers in here to make sure I understand:

EV = $73 per hour
StDev = $442 per hour

If I plan to play 130 hours next month, my EV for the month would be $73*130= $9,490.

My Standard Deviation would be $442*sqroot(130)=$5,040.

So if I remember correctly +/- 1 Stdev from the EV covers roughly 70% of all outcomes, and +/- 2 Stdev from EV covers roughly 95% of all outcomes.

So for the time period of 130 hours I could estimate that my results would be between $4,450 & $14,530 around 70% of the time and between -$590 & $19,570 around 95% of the time.

Almost makes me want to quit my shitty day job...

[pzhon]

[ QUOTE ]

EV = $73 per hour
StDev = $442 per hour

If I plan to play 130 hours next month, my EV for the month would be $73*130= $9,490.

My Standard Deviation would be $442*sqroot(130)=$5,040.

So if I remember correctly +/- 1 Stdev from the EV covers roughly 70% of all outcomes, and +/- 2 Stdev from EV covers roughly 95% of all outcomes.

So for the time period of 130 hours I could estimate that my results would be between $4,450 & $14,530 around 70% of the time and between -$590 & $19,570 around 95% of the time.

Almost makes me want to quit my shitty day job...

[/ QUOTE ]

The calculations for the EV and SD of a 130-hour session look right, though the confidence intervals depend on the accuracy of the inputs as well as the assumption that the distribution is normal. The most serious possible problem is that you may have misestimated your EV. If your standard deviation is correct, then as a break-even player you would have results this good (+1/6 SD/hour) in a 100 hour session about 5% of the time, assuming a normal distribution.

If you don't assume that the distribution is normal, you can use the Cebyshev inequality to get a cruder confidence interval. You can't be farther than x standard deviations from the mean more than 1/x^2 of the time. Setting 1/x^2 = 5%, x ~ 4.5.

There is another fundamental issue, whether you are predicting the past or the future. It could be that in the past, you played while in peak condition and when you found favorable tables. If you had to play 130 hours/month, would you find better or worse games on average?

Despite these difficulties, it is good to use these methods to try to determine your EV and exposure to risk.