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  • Alspach's Mathematics and Poker Page
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Mathematics, Probability and Poker

I'll be posting some comments and links to basic ideas about probability and mathematics and their relationship to poker. Any questions or comments, please post to rgp for discussion.

Successful gambling is about making bets with the best of it. Making bets where the money odds you're getting are better than the card odds against you. Probability is the branch of mathematics that's most relevant to the study of making winning bets. Most books on poker or gambling tend to not really get into that past combinatorial counting methods, which doesn't really give you the concepts of probability you need to lay a solid foundation of understanding the road to success in gambling.

I'm going to take a conceptual approach, so I'm not going to assume any mathematical training beyond a good high school education. So we need to start by introducing a few basic concepts.

Section I. Mathematics of Poker

When most people think of the topic of poker math they tend to think in terms of calculation of simple probabilities and pot odds. What is the probability of flopping a set in Hold’em if you start with a pocket pair? What is the probability of flopping top pair if you start with AK? What is the probability of completing your hand by the river if you flop a flush draw and a gut shot straight draw? What money odds do you need from the pot to call a bet on 6th street in 7-card stud when you need to complete a 4-flush to win?

All these questions can be answered by using a collection of mathematical procedures called counting methods. Counting methods involve using some mathematics to count various combinations of possible outcomes, and you can calculate probabilities by just taking the ratio of all those combinations that fulfill the criteria and all possible combinations.

But, there is more to mathematics than counting and calculating simple probabilities. A lot more. Other branches of mathematics that are applicable to poker are game theory, fixed point theory, optimazation theory, stochastic processes, hypothesis testing.

Section II. Some introductory conceptsMost recent update 2/18/04

Section III. The Long Run

Section IV. The Gambler's Fallacy

There appears to be a lot of fuzzy thinking about poker earn. Years ago Sklansky made the idea popular that you earn your hourly win rate by just sitting at the table. Other's, like Kreiger and Danny N, seem to just accept that idea without any critical thought.

It's fuzzy thinking in it's classic form, an application of the Gambler's Fallacy.

The Gambler's Fallacy is the idea that the future will even out the past results by offsetting any deviations from the average that the past has. If you have a run of red at the wheel then sometime in the future that will be counter balenced by a run of black. The Gambler's Fallacy is based on a misunderstanding of the Law of Large Numbers which says that the future will last long enough that any past results become irrelavent to your long run average.

If your average win rate is $10 an hour, and you play 5 hours and lose $100, you didn't earn $50. You lost $100. It's a real loss, and it won't be made up in the futre. It's gone.

The idea that you earn your win rate every hour applies to the future, but not to the past. If you lost money in the past then you lost money and it's gone forever.

If you think that losing isn't really losing because on average you're a winning player then you're simply deluding yourself.

That doesn't mean you should worry about loses. It's inevitable that you'll have some losing sessions, and trying to avoid them isn't going to lead to playing good poker.

But, if you have an average win of $40 an hour and play five hours and lose $1,000 then you didn't win $200. I'm not saying that you're only as good as your last session. Having a losing session does not mean you're a losing player. But if you lost $100 your last session then that's $100 that's gone for good.

If you play long enough it may be that that $100 becomes unimportant to you -- It certinaly won't have much of an impact on your lifetime hourly average -- that's what the law of large numbers says. But, you didn't "earn" 1 big bet an hour by sitting at the table last night, you lost $100.

Some players think this way, "If your Aces get cracked because you raised and got called by T8 and they flopped trips, you should be happy they called because you'll win it back next time."

That's the gambler's fallacy, the idea that you'll "win it back". You might win in the future because they make bad calls, but that future is independent of whether or not you won the last pot. That money is gone. I'm glad they called because they made a bad call. But, they still won and that money is now their money. If they keep making bad calls, then you'll probably win in the future. But, that money is still gone.

The law of large numbers says that it doesn't matter that it's gone because you'll keep playing long enough so that your total wins in the future will just overwhelm today's results, today doesn't really matter at all.

It does not say you'll "get it back".

Section V. Game Theory

Right now I'm just putting up some links to past rgp threads on game theory. It's not really my area of expertise so I'll probably just stick to providing some links.

Math and Poker Blog

Kelly betting

These are all somewhat technical articles.

Secretary Problem