Win Rate Confidence

Published: March 16, 2020

You’ve played for a whole bunch of hours. You’ve kept diligent records of your results. You’ve had your ups and downs, but you think it’s been more ups than downs. You’ve played well and you think you are a big winner in the game.

And it’s true. When you add up all of your results the total is a nice big number.

And when you divide your total winnings by the number of hours that you played that number looks good too. It compares favorably with your job.

If you can make this much money playing poker you can tell your boss what you really think and never have to deal with them again.

Then you snap back into reality.

How much can you really expect to win per hour of poker? Your observed win rate is the best possible estimate of your actual win rate, but how much confidence can you have in it?

Mathematical theory

Because of the central limit theorem we know that your actual results for each hour of poker that you play will be normally distributed.

To put it another way, your results will fall on a bell curve. A lot of your hours of poker will have results that are close to your average. A few of your hours of poker will have results that are far from your average.

The actual mean (aka average or middle) of that bell curve is your expected value for one hour of poker.

The maximum likelihood estimate of the mean is the arithmetic mean of the samples.

In simple language, the best estimate of your expected value for an hour of poker is the average of your observed results for each hour of poker.

The more difficult question is how good is that estimate?

Confidence Intervals

When you play poker and record your results you are performing an experiment with the intent of finding out what your average hourly win rate is.

A confidence interval can tell you how confident you can be that your actual average win rate lies within the interval.

For a given level of confidence, the more experiments you do, the smaller your confidence interval will get.

After only a few hours it is not uncommon for a 95% confidence interval to be huge and completely useless. But it only takes a few dozen hours of play for me to identify the games where I am a big winner.

How to

The Wikipedia article on confidence intervals describes how to calculate it if you have an estimate of your standard deviation.

The Definitive Guide to Poker Bankroll Requirements has a section on standard deviation and how to estimate it from your poker results by using the Weitzman Variance Estimator.

But the easiest way to calculate your win rate confidence intervals is to sign up for a free account on Neverbust and enter your results there.

Neverbust calculates your win rate with 95% confidence.

And of course, your win rate means nothing if you run out of money and can’t play anymore.

Neverbust will also calculate your risk of going bust and your minimum bankroll requirements.