The Mathematics Of Poker

Posted on4/14/2022by admin

Over the last five to ten years, a whole new breed has risen to prominence within the poker community. Applying the tools of computer science and mathematics to poker and sharing the information across the Internet, these players have challenged many of the assumptions that underlay traditional approaches to the game. Killer Poker By The Numbers. The Mathematics of Poker Bill Chen, Jerrod Ankenman In the late 1970s and early 1980s, the bond an option markets were dominated by traders who had learned their craft by experience. The mathematics are there to illustrate the game theory that underlies poker. Even with the supplemental explanations and synopses, The Mathematics of Poker is a demanding read. It asks a lot of the reader both in following the arguments. The Mathematics of Poker is very important. There is a probability for every outcome in online poker and a statistic for every hand of the poker game. Some of them are fun, some of them matter and some of them are just for MIT graduates. The Mathematics of Poker by Bill Chen and Jerrod Ankenman. Diligent readers who invest the necessary effort to follow Chen and Ankenman’s arguments and consider their. 'For those who think poker math is only about probability, pot odds, and straightforward, rote play, think again.

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The Mathematics Of Poker Epub
The Mathematics Of Poker Bill Chen

Introduction to the Mathematics of Poker

Let me get you a little bit into the math of poker. First of all, let’s consider its importance on a hand-to-hand basis. First of all, when you’re dealt a hand in poker, and it’s your turn to act, depending on the action that you got from the other players who acted before you, you have to make the decision either to raise, call or fold.

First of all, you consider your hand. Is it a strong hand that I could get all my money in pre-flop with and be profitable? Or do I have a drawing hand, like a suited connector? And what amount of big blinds can I call at most in this spot to be able to play my hand profitably on the river? Can I re-raise as a bluff and will this bring me profit in the long run?

If all the answers here are no, no and no then you’re going to fold. But are you sure you calculated it correctly? We’ll get into depth on these matters in this article.

Also, let’s say you hold a strong top pair and you’ve got to the river with it, and your opponent over-bet shoves. Do we have the right odds to call here? What is he representing here? And can we put him on a range? And after that can we call profitably? These questions will be answered also.

Pot Odds

I always go for percentages everywhere in poker, because I find it a lot easier to understand and also apply. The first thing I learned when I was starting out with poker, was pot odds, and I think they should be the foundation to every poker player’s knowledge.

To explain this as simple as do re mi, for you to be profitable, the breakeven equity that you need to make a call when you get bet into, is the amount that you have to put into the pot divided by the total pot size (including your bet – so the amount that you’re winning when you have the stronger holding).

Let’s first take an example:

You are playing poker against Phil Ivey, and he bets into you on the river. You’re currently holding AJ, and the board is A2742. Some guy told you that he could either have AT, AJ, AQ or AK, and you’re facing a 2/3 pot bet.

Let’s face it, you’re behind. But what is the amount of hands that beats you relative to the size of the bet you have to call? Good question.

So you beat AT and get beat by AQ and AK, and split with AJ, which we shall discount because it’s 0 EV.

Obviously, the same number of combinations exist for AT, AQ and AK, so we win 1 time out of three!

Our estimated equity is 33%, but are the pot odds low enough? If they are, it’s a sure call!

This is how you calculate:

You need to call 2/3 into a pot that will contain your 2/3 bet, the pot size which is 3/3 and the opponent’s bet which is also 2/3. That means that you have to call 2/3 to win a pot of 7/3, so your breakeven equity will be:

(2/3)/(7/3)*100(to display in percentage) = (2/7)*100 = 28%.

We know from the logic above that we have 33% equity, so we have greater equity than the pot odds, so even though we’re behind, we can still call here profitably!

You have 1 dollar and you hate bluffing, but someone offers you the chance to crack his pocket AA’s with 72o and win 100 dollars if you do it. Assuming this is a legit deal, you only need to put in 1 dollar to win 100 so this is
(1/100)*100=1% breakeven frequency.

Last time I checked Equilab, 72o has around 12% chance to crack aces, so it’s a sure call. I would take this deal every time I get the occasion. Here are some default pot odds that you should know by heart, because they will prove very useful when thinking about calling:

1/3 Pot – 20%
1/2 Pot – 25%
2/3 Pot – 28%
3/4 Pot – 30%
4/5 Pot – 31%
Full Pot – 33%
1.5x Pot – 37%
2x Pot – 40%

$Mathematics$

Fold equity

If you’re the one who’s betting, there’s always a combination of your hand’s equity and your total fold equity involved. Let’s say you’re bluffing the river this time, and you’re wondering how to determine the amount of times he has to fold to make your bet profitable, look no further!

Let’s say that we’re betting with a hand that, if we get called, we can never win, like a busted flush draw on the river. If we think that the opponent will fold enough times, we can make this bet.

The formula is: Breakeven Fold Equity = (your bet) divided by (the sum of your bet and the pot size).

Thinking about this, it becomes quite logical that if you bet full pot, you need him to fold 50% of the time, because 1/(1+1) = 1/2 = 50%. Some other frequencies that are good to remember are:

1/3 Pot = 25%
1/2 Pot = 33%
2/3 Pot = 40%
3/4 Pot = 42%
4/5 Pot = 44%
1.5x Pot = 60%
2x Pot = 66%
3x Pot = 75%.

It’s rumored that a certain player named Isildur1 has used the latter numbers to his complete advantage! What if you thought about that first? You’d be probably playing the higher stakes. So now, having showed you how cool this math stuff is, let me show you how to apply it.

Pre-flop actions

First of all, let’s get into notice some flop hitting probabilities:

A pair – 29%
Two Pairs – 2%
A Set (when holding a pocket pair) – 12%
Trips – 1.35%
A Full house – 0.09%
Four of a Kind – 0.01%
A pair or better - 32%
A flush holding 2 suited cards – 0.84%
A flush draw holding 2 suited cards – 11%
A straight with suited non-gapped connectors – 1.31%
An open-ended straight draw with non-gapped – 10.5%
A straight with one-gapers – 0.98%
An OESD with one-gapers – 8.08%
A straight with two-gapers – 0.65%
An OESD with two-gapers – 5.2%
A gutshot (suited connectors) – 16.6%
Any unsuited hand flopping 2p+ - 3.45%
Any suited hand flopping 2p+/flush – 4.29%
Suited connectors flopping 2p+/straight/flush – 5.59%

Having known all these, we are now inclined to play a more calculated game pre-flop. Although the reason for calling a raise pre-flop should not be pure mathematical, sometimes it does help to know how often you’ll flop an OESD + FD + pair or better with suited connectors, which is close to 50%.

I don’t like playing small suited connectors in multi-way pots, because usually you get dominated by higher flushes and you get the dummy end of a straight. They also aren’t that great to 3-bet against a raiser because they don’t have blockers, so what are we supposed to do with them when we’re not stealing blinds?

Well, let’s say that the EP raiser is opening a wide array of hands, like 14% and there’s also a caller, so we’re thinking with 56s in the BTN. This is a perfect spot to do the squeeze play, because 1) you’re getting a massive amount of folds, and 2) When you get called you will flop a lot of strong draws that you can represent along with your pairs and 2p+.

The Mathematics of Poker
by Bill Chen and Jerrod Ankenman

Diligent readers who invest the necessary effort to follow Chen and Ankenman’s arguments and consider their implications, however, will not be disappointed. Careful study of this text will reveal a world of insight into poker concepts such as value betting, range balancing, and optimal strategy. Although more could have been done to elucidate its practical applications, The Mathematics of Poker is nevertheless an extremely valuable text for any poker player willing to give it the thoughtfulness it deserves.

It is not an easy read, but it should not be beyond the grasp of anyone with a high school education. Game theory is serious mathematics, and nearly every page of this book is packed with equations, charts, and graphs. This looks intimidating, but in fact, the authors do all the heavy lifting and help a diverse audience follow along in a variety of ways. For the real mathematicians, they show their work and occasionally suggest follow-up problems that readers might consider attacking on their own. However, they bracket these sections so that the mathematically challenged can skip past them to the (relatively) plain-language explanations of the process and results that follow. Chen and Ankenman do a remarkably good job of elucidating the conceptual meaning of equations and solutions, and every chapter concludes with a summary of the “Key Concepts.” Still, a passing acquaintance with statistical notation, graphical representation, and high school algebra is all but required to make sense of the text.

After some opening chapters that cover concepts like variance, sample sizes, hand reading, and pot odds, Chen and Ankenman introduce the concept of optimal strategy, a style of play that cannot be exploited even if your opponents knew ahead of time exactly how you would be playing. In other words, they are interested in finding solutions such as the exact ratio of bluffs to value bets that would make your opponent indifferent to calling or folding with a weak made hand on the river.

All commonly played versions of poker are far too complex to solve with the tools of game theory, however. Instead, poker must be attacked indirectly, through a series of “toy games” that represent highly simplified poker situations. One oft-revisited example involves two players each dealt one card from a three-card deck containing exactly one A, one K, and one Q. Throughout the book, the authors consider situations where the second player to act knows what card his opponent holds, situations where he does not have this information, situations where the first player is forced to check dark, and finally a full-street game where neither player knows the other’s card but may bet, check, or raise as he sees fit.

The text is very helpful in explaining the optimal strategy for each player in each game and how the addition of new strategic options affects these results. Still, the ultimate solution is nothing but the optimal strategy for a game that no one will ever play. Undoubtedly, understanding what optimal play entails and how it is derived can be enormously valuable at the poker table. But these games are accompanied by a few sentences, at best, explaining their relevance to actual poker situations.

Part of my frustration stems from the fact that the tidbits of practical advice that Chen and Ankenman do include are tantalizingly thought-provoking: “Bluffing in optimal poker play is often not a profitable play in and of itself. Instead, the combination of bluffing and value betting is designed to ensure the optimal strategy gains value no matter how the opponent responds”; “Betting preemptively is a perfect example of a play that has negative expectation, but the expectation of the play is less negative than the alternative of checking”; “it’s frankly terrible to find oneself in a situation with a marked open draw.” Any of these insights is worthy of several pages of extrapolation, but this work is largely left to the reader.

This is more than an error of omission. When Chen and Ankenman do directly address the question of how their material translates into poker strategy, they concern themselves overly much with pursuing optimal versus exploitive strategies, which is to say strategies that deviate from unexploitable, or optimal, play in order to capitalize on perceived weaknesses in an opponent’s strategy.

Their stated reasons for this preference are that it is difficult to determine another player’s strategy with certainty and that opponents will eventually change their play to counter your exploitation. While these are both reasonable concerns, neither is prohibitive. Although certainty is impossible, the ability to make quick and reasonably accurate assessments of a player’s strengths and weaknesses and the ability to adapt and re-adapt to him more quickly than he can do the same are skills from which a successful poker player derives his edge.

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Whereas Chen and Ankenman advocate playing optimally against unknown opponents, I would argue that you often can and should assume and default to exploiting certain weaknesses until you see some evidence to the contrary. Balancing one’s river betting range in order to avoid exploitation by a check-raise bluff, for instance, is a poor default strategy because very few poker players are capable of such a tactic.

The Mathematics Of Poker Epub

The real strength of The Mathematics of Poker, in my opinion, is not that it will help you to play a near-optimal strategy. Rather, it will help you to understand optimal strategy so that you can better recognize and exploit your opponents’ inevitable deviations from it. For example, one toy game illustrates how a player in position must value bet and bluff fewer hands when his opponent is allowed to check-raise than in a game where he is not. The lesson I take from this is that against an opponent who rarely check-raises the river, I should value bet and bluff more often than optimal strategy would suggest.

But ultimately, these are shortcomings, not flaws. The mathematics are there to illustrate the game theory that underlies poker. Even with the supplemental explanations and synopses, The Mathematics of Poker is a demanding read. It asks a lot of the reader both in following the arguments and in making the jump from toy games to real life poker. Those who invest the requisite time and energy, however, will be rewarded with a deeper understanding of how to exploit their opponents and how to avoid such exploitation themselves.

The Mathematics Of Poker Bill Chen

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