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The Mathematics of Poker

Bill Chen, Jerrod Ankenman

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Книга посвящена математическим аспектам игры в покер. Она будет интересна прежде всего математикам, заинтересованным в покере, но также и серьезным игрокам в покер, имеющим хорошую математическую подготовку (на уровне ВУЗовского курса теории вероятностей).

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          The Mathematics of Poker Bill Chen Jerrod Ankenman  Other ConJelCo titles:  Cooke's Rules of Real Poker by Roy Cooke and John Bond Hold'em Excellence by Lou Krieger How to Play Like a Poker Pro by Roy Cooke and John Bond How to Think Like a Poker Pro by Roy Cooke and John Bond Internet Poker by Lou Krieger and Kathleen Watterson Mastering No -Limit Hold'em by Russ Fox and Scott T. Harker More Hold' em Excellence by Lou Krieger Real Poker II: The Play of Hands 1992-1999 (2"d ed.) by Roy Cooke and John Bond  Serious Poker by Dan Kimberg Stepping Up by Randy Burgess The Home Poker Handbook by Roy Cooke and John Bond Why You Lose at Poker by Russ Fox and Scott T. Harker Winning Low-Limit Hold 'em by Lee Jones Winning Omahal8 Poker by Mark Tenner and Lou Krieger Winning Strategies for No-Limit Hold' em by Nick Christenson and Russ Fox  Video Poker - Optimum Play by Dan Paymar Software  StatKing for Windows  (  •  The Mathematics of Poker Bill Chen Jerrod Ankenman  ConJelCo LLC Tools for the Intelligent Gambler Pittsburgh, Pennsylvania  tm  The Mathematics of Poker Copyright © 2006 by Bill Chen and Jerrod Ankenman All rights reserved. This book may not be duplicated in any way or stored in an infonnation retrieval system, without the express written consent of the publisher, except in the form of brief excerpts or quotations for the purpose of review. Making copies of this book, or any portion, for any purpose other than your own, is a violation of United States copyright laws .  Publisher's Cataloging-in-Publication Data Chen, Bill Ankenman,j errod The Mathematics of Poker x,382p. ; 29cm. ISB N-13: 978-1-886070-25·7 I SBN-I0: H86070-25-3  I. Ticle. Library of Congress Control Number: 2006924665 First Edition 57986 Cover design by Cat Zaccardi Book design by Daniel Pipitone Digital production by Melissa Neely ConJelCo LLC 1460 Bennington Ave Pittsburgh, PA 152 17 [412] 621-6040  http://www.conje1co.com  Errata, if any. can be found at http://www.conjelco.comlmathofpoker/  able of Contents J.dGlowledgmen; ts .. .. . ... . ... . .  ~o rd  uction .  . V11 . ...... IX  . .... .. ... 1  ~ 1 : Basics  pter 1 pt er 2 ~ p ter 3  Decisions Under Risk: Probability and Expectation. . .... 13 Predicting the Future: Variance and Sample Outcomes. . .............. 22 Using All the Information: Estimating Parameters and Bayes' Theorem .. 32  ?:i:rt II : Exploitive Play  pter 4 pt er 5 Ola pter 6 apter 7 apter 8 apter 9  Playing the Odds: Pot Odds and Implied Odds .. Scientilic Tarot: Reading Hands and Strategies. . ... ... .. . . The Tells are in the Data: Topics in Online Poker .. . . ..... . . Playing Accurately, Part I: Cards Exposed Situations Playing Accurately, Part II: Hand vs. Distribution .. Adaptive Play: Distribution vs. Distribution.  . . . .. 47 . . .. . 59 . .... 70 ... . . 74 . ... 85 94  ;>-..rt II I: Optimal Play  apter 10 Facing The Nemesis: Game Theory . . .... . . . . . . . .lOl apter 11 One Side of the Street: Half-Street Games .. .. . . ....... . . . .... . . ... . . 111 Chapter 12 Headsup With High Blinds: ThcJam-or-Fold Game ...... . . .. . . ..... 123 Chapter 13 Poker Made Simple: The AKQGame .... . .... . ... . ...... . .. . ..... . ... . . . 140 _. .. 148 Chapter 14 You Don't Have To Guess: No-Limit Bet Sizing. Chapter 15 Player X Strikes Back: Full-Street Games . . ..... 158 Appendix to Chapter 15 The No-Limit AKQGame. . . . ... . 171 _.. _. 178 Chapter 16 Small Bets, Big Pots: No-Fold [O,IJ Games .... .. .... . . Appendix to Chapter 16 Solving the Difference Equations . . . ..... . . . . . . 195 Chapter 17 Mixing in BluITs: Fmite Pot [O,IJ Games .. . ..... . ..... . . . . . _. 198 Chapter 18 Lessons and Values: The [O,1J Game Redux . . . . 216 . .. . . ... 234 Chapter 19 The Road to Poker: Static Multi-Street Games . . Chapter 20 Drawing Out: Non-Static Multi-Street Games .. ..... . . . ..249 Chapter 21 A Case Study: Using Game Theory. . .... .. ...... .. _... . . ... 265 Part IV: Risk Chapter 22 Chapter 23 Chapter 24 Chapter 25  Staying in Action: Risk of Ruin . . ... . . 281 Adding Uncertainty: Risk of Ruin with Uncertain Wm Rates.. . ... 295 Growing Bankrolls: The Kelly Criterion and Rational Game Selection ... 304 Poker Fmance: Portfolio Theory and Backing Agreements . . .310  Part V: Other TopiCS Chapter 26 Doubling Up: Tournaments, Part I . .. . ... . ...... . Chapter 27 Chips Aren't Cash: Tournaments, Part II .. Chapter 28 Poker's Still Poker: Tournaments, Part III . Chapter 29 TIrree's a Crowd: Multiplayer Games .. Chapter 30 Putting It All Together: Us;ng Math to Improve Play .. Recommended Reading. About the Authors .. . About the Publisher .. .  THE MATHEMATICS OF POKER  . .. 321 . .... . . . .333 . ..... . . 347 . ... 359 . . ..... 370 . ... . 376 . . .... . . . .381 . ... 382  v  vi  THE MATHEMATICS OF POK ER  Acknowledgments A book like this is rarely the work of simply the authors; lllany people have assisted us along the way, both in understanding poker and in the specific task of writing down many of the ideas  that we have developed over the last few years. A book called The Matlu:maticr  tf IM.er was  conceived by Bill, Chuck Weinstock, and Andrew Latta several years ago, beforeJerrod and Bill had even met. TIlat book was set to be a fairly fOlmal, textbook-like approach to discussing the mathematical aspects of the game. A few years later,Jerrod and Bill had begun [0 collaborate on solving some poker games and the idea resurfaced as a book that was less like a mathematics paper and more accessible to readers with a modest mathematical background.  Our deepest thanks CO those who read the manuscript and provided valuable feedback. Most notable among those were Andrew Bloch , Andl'CW Prock, and Andrew Lacco, w ho scoured sections in detail, providing criticisms and suggestions for improvement. Andrew Prock's PokerStove [001 (lutp:/lww\v.pokerstove.com) was quite v aluable in performing many of the equity calculations. Others who read the manuscript and provided useful feedback were Paul R. Pudaite and Michael Maurer. JcffYass at SusquehaJma International Group (http://www.sig.com) has been such a generous employer in allowing Bill to work on poker, play the World Series, and so on. We thank Doug Costa, Dan Loeb,Jay Siplestien, and Alexei Dvoretskii at SIG for their helpful comments. We have learned a great deal from a myriad of conversations with various people in the poker community. OUf friend and partner Matt Hawrilenko, and former WSOP champion Chris Ferguson are two individuals especially worth si.ngling out for their insight and knowled ge. Both of us participate enthusiastically in the *J\RG community, and discussions with members of that community, too, have been enlightening, particularly round table discussions with players including Saby l Cohen,J ack Mahalingam,JP Massar, Patti Beadles, Steve Landrum, and (1999 Tournament of C hampions winner) Spencer Sun. Sarah Jennings, our editor, improved the book significantly in a short period of time by being willing to slog drrough equations and complain about the altogether too frequent skipped steps. Chuck Weinstock at Conjelco has been amazingly patient with all the delays and extended deadlines that corne from working with authors for whom writing is a secondary occupation. We also appreciate the comments and encouragement from Bill's father Dr. An-Ban C hen who has written his own book and has numerous publications in his field of solid state physics. Jerrod's mother Judy has been a constant source of support for all endeavors, evcn crazy-sounding ones like playing poker for a living. TIrroughout the writing of this book, as in the rest of life, Michelle Lancaster has constantly been wonderful and supportive ofJ errod. This book could not have been completed without her. Patricia Walters has also provided Bill with support and ample encouragement, and it is of no coincidence that the book was completed during the three years she has known Bill.  THE MATHEMATICS OF POK ER  vii  viii  THE MATHEMATICS OF POKER  Foreword ::lex:': believe a word I say.  !: ~  ::lO(  that I'm lying when I tell you that this is an important book. I don't even lie at the not much, anyway -- so why would I lie abom a book I didn't even write?  :ci-c table -  ::"3 just that you can't trust me to be o bjective. I liked this book before I'd even seen a single .?4..... I li."-ed it when it was just a series of cOllversations betvveen Bill, myself, and a handful ::i other math geeks. And if I hadn't made up my mind before I'd read it, I'm pretty sure ~-'d have \von me over with the first sentence. :Jon"t worry, though. You don't have to crust me. Math doesn't lie. And results don 't lie, -neroIn the 2006 WSOP, the authors finished in the money seven times, includingJerrod's ~nd place finish in Limit H old ern, and Bill's two wins in Limit and Short H anded No  :....:rnit Hold·cm. -'lost poker books get people talking. The best books make some people say, "How could nyone publish our carefully guarded secrets?" Other times, you see stuff that looks fishy Clough to make you wonder if the author wasn't deliberately giving out bad advice. I think :his book will get people talking, too, but it won't be the usual sort of speculation. No one is going to argue that Bill andJerrod don't know their math. The argument will be about whether or not the math is important. People like to talk about poker as "any man's game." Accountams and lav"yers , students and housewives can all compete at the same level-- all you need is a buy-in, some basic math and good intuition and you, too, can get to the final table of the World Series of Poker. That norian is especially appealing to lazy people who don't wane to have to spend years working at something to achieve success. It's true in the most literal sense that anyone can win, but with some well-invested effort, you can tip the scales considerably in your favor. The math in here isn't easy_ You don't need a PhD in game theory to understand the concepts in this book, but it's not as simple as memorizing starting hands or calculating the likelihood of making your Hush on the river. There's some work involved. The people who want to believe inruition is enough aren't going to read this book. But the people who make the effort will be playing with a definite edge. In fact, much of my poker success is the result of using some of the most basic concepts addressed in this book. Bill andJerrod have saved you a lot of time. They've saved me a lac of a time, too. I get asked a lot of poker questions, and most are pretty easy to answer. But Pve never had a good response when someone asks me to reconunend a book for understanding game theory as it relates to poker. I usually end up explaining that there are good poker books and good game theory books, but no book addresses the relationship between the two. Now I have an answer. And if I ever find myself teach.ing a poker class for the mathematics department at UCLA, this will be the only book on the syllabus. Chris 'Jesus" Ferguson Champion, 2000 World Series of Poker ::\ovember 2006  THE M ATHEMAT ICS OF POKER  ix  x  THE MATHEMATICS OF POKER  Introduction  (7fyou think the math isn't important) you don't know the right math." Chris "Jesus" Ferguson, 2000 World Series of Poker champion  Introduction  Introduction In the late 1970s and early 1980s, the bond and option markets were dominated by traders who had learned their craft by experience. They believed that their experience and inruition for trading were a renev..able edge; that is, that they could make money just as they always had by continuing to trade as they always had. By the mid-1990s, a revolution in trading had occurred; the old school grizzled traders had been replaced by a ncw breed of quantitative analysts, applying mathematics to the "art" of trading and making of it a science.  If the latest backgammon programs, based on neural net technology and mathematical analysis had played in a tournament in the late 1970s, their play would have been mocked as overaggressive and weak by the experts of the time. Today, computer analyses are considered to be the final word on backgammon play by the world's strongest players - and the game is fundamemally changed for it. And for decades, the highest levels of poker have been dominated by players who have learned the game by playing it, "road gamblers" who have cultivated intuition for the game and are adept at reading other players' hands from betting patterns and physical tells. Over the last five to ten years, a whole new breed of player has risen to prominence within the poker community. Applying the tools of computer science and mathematics to poker and sharing information across the Internet, these players have challenged many of the assumptions that underlie rraditional approaches to the game. One of the most important features of this new approach to the game is a reliance on quantitative analysis and the application of mathematics to the game. Our intent in this book is to provide an introduction to quantitative techniques as applied to poker and to the application of game theory, a branch of mathematics, to poker. Any player who plays poker is using some model, no matter what methods he uses to inform it. Even if a player is not consciously using mathematics, a model of the siruation is implicit in his decisions; that is, when he calls, raises, or folds, he is making a statement about the relative values of those actions. By preferring one action over another, he articulates his belief that one action is bettcr than another in a particular situation. Mathematics are a parcicul~rly appropriate tool for making decisions based on information. Rejecting mathematics as a tool for playing poker puts one's decision·making at the mercy of guesswork.  Common Misconceptions We frequently encounter players who dismiss a mathematical approach out of hand, often based on their misconceptions about what this approach is all about. vVe list a few of these here; these are ideas that we have heard spoken, even by fairly knowledgeable players. For each of these, we provide a brief rebuttal here; throughout this book, we vvill attempt to present additional refutation through our analysis. 1) By analyzing what has happened in the past - our opponents, their tendencies, and so on- we can obtain a permanent and recurring edge. TIlls misconception is insidious because it seems very reasonable; in fact, we can gain an edge over our opponents by knowing their strategies and exploiting them. But this edge can be only temporary; our opponents, even some of the ones we think play poorly, adapt and  evolve by reducing the quantity and magnitude of clear errors they make and by attempting to counter-exploit us. We have christened this first misconception the "PlayStation™ theory of THE MATHEMATICS OF POKER  3  Introduction poker" - that the poker world is full of players who play the same fixed strategy, and the goal of playing poker is to simply maximize profit against the fixed srrategies of our opponents. In fact, our opponents' strategies are dynamic, and so we must be dynamic; no edge that we have is necessarily permanent. 2) Mathematical play is predictable and lacks creativity.  In some sense this is true; that is, if a player were to play the optimal strategy to a game, his srrategy would be "predictable" - but there would be nolhing at all that could be done with this information. In th e latter parts of the book, we vvill introduce the concept of balance - this is the idea that each action sequence contains a mixture of hands that prevents the opponent from exploiting the strategy. O ptimal play incorporates a precisely calibrated mixture of bluffs, scmi-bluffs, and value bets that make it appear entirely unpredictable. "Predictable" connotes "exploitable," but this is not necessarily true. If a player has aces every time he raises, this is predictable and exploitable. However, if a player always raises when he holds aces, this is not necessarily exploitable as long as he also raises with some other hands. The opponent is not able to exploit sequences that contain other actions because it is unknown if the player holds aces. 3) Math is not always applicable; sometimes lithe numbers go out the window!' This misconception is related to the idea that for any situation, there is only one mathematically correct play; players assume that even pla)'ing exploitively, there is a correct mathematical play - but that they have a "read" which causes them to prefer a dilferent play. But this is simply a narrow definition of "mathematical play" - incorporating new infonnation into our wlderstanding of our opponent's distribution and utilizing that infonnation to play more accuralely is the major subject of Part II. In fact, math ematics contains tools (notably Bayes' theorem) that allow us to precisely quantify the degree to which new information impacts our thinking; in fact, playing mathematically is more accurate as far as incorporating "reads" than playing by "feel." 4) Optimal play is an intractable problem for real-life poker games; hence, we should simply play expJoitively. TIlls is an important idea. It is true iliat we currently lack the computing power to solve headsup holdem or other games of similar complexity. (\tVc vvill discuss what it means to "solvt:" a game in Part III). We have methods that are known to find the answer, but they will  not run on modem .computers in any reasonable amount of time. "Optimal" play does not even exist for multiplayer games, as we shall see. But this does not prevent us from doing two things : attempting to create strategies which share many of the same properties as optimal strategies and thereby play in a "near-optimal" fashion; and also to evaluate candidate strategies and find om how far away from optimal they are by maximally exploiting them. 5) When playing [online, in a tournament, in high limit games, in low limit games ..•], you have to change your strategy completely to win. This misconception is part of a broader misunderstanding of the idea of a "strategy" - it is in  fact true that in some of these situations, you must take different actions, particularly exploitively, in order to have success. But this is not because the games are fundamentally different; it is because the other players play differently and so your responses to their play take different forms. Consider for a moment a simple example. Suppose you are dealt A9s on the button in a full ring holdem game. In a small-stakes limit holdem game, six players might limp to you, and you should raise. In a high limit game, it might be raised from middle position, and you would fold. In a tournament, it might be folded to you, and you would raise. These are entirely different actions, bur the broader strategy is the same in all - choose the most profitable action. 4  THE MATHEMATICS OF POK ER  Introduction lbroughout this book, we will discuss a wide variety of poker topics, but overall, our ideas could be distilled to one simple piece of play advice: Maximiu average profit. TIlls idea is at the heart of all our strategies, and this is the one thing that doesn't change from game condition to game condition.  Psychological Aspects Poker authors, when faced with a difficult question, are fond of falling back on the old standby, .lIt depends." - on the opponents, on one's 'read', and so on. And it is surely true that the most profitable action in many poker siruations does in fact depend on one's sense, whether intuitive or mathematical, of what the opponent holds (or what he can hold). But one thing that is ofcen missing from the qualitative reasoning thal accompanies "It depends," is a real answer or a methodology for aniving at an action. In reality, the answer does in fact depend on our assumptions, and the tendencies and tells of our opponents are certainly something about which reasonable people can disagree. But once we have characterized their play into assumptions, the methods of mathematics take over and intuition fails as a guide to proper play. Some may take our assertion that quantitative reasoning surpasses intuition as a guide to play as a claim that the psychological aspects of poker are without value. But we do not hold this view. The psychology of poker can b e an absolutely invaluable tool for exploitive play, and the assumptions that drive the answers thar our mathematical models can generate are often strongly psychological in narure. The methods by which we utilize the infonnation that our intuition or people-reading skills give us is our concern here. In addition, we devote time to the question of what we ought to do when we are unable to obtain such infonnation, and also in exposing some of the poor assumptions that often undermine the infonnation-gathering efforts of intuition. 'With that said, we will generally, excepting a few specific sections, ignore physical tells and opponent profiling as being beyond the scope of this book and more adequately covered by other -writers, particularly in the work of Mike Caro.  About This Book We are practical people - we generally do not study poker for the intellecrual challenge, although it rums out that there is a substantial amount of complexity and interest to the game. We study poker: with mathematics because by doing so, we make more money. As a result, we are very focused on the practical application of our work, rather than on generating proofs or covering esoteric, improbable cases. TIlls is not a mathematics textbook, but a primer on the application of mathematical techniques to poker and in how to tum the insights gained into increased profit at the table. Certainly, there are mathematical techniques that can be applied to poker that are difficult and complex. But we believe that most of the mathematics of poker is really not terribly difficult, and we have sought to make some topics that may seem difficult accessible to players without a very strong mathematical background. Btl[ on the other hand, it is math, and we fear that if you are afraid of equ ations and mathematical tenninology, it will be somewhat difficult to follow some sections. But the vast majority of the book should be understandable to anyone who has completed high school algebra. We will occasionally refer to results or conclusions from more advanced math. In these cases, it is not of prime importance that you understand exacdy the mathematical technique that was employed. The important element is the concept - it is very reasonable to just "take our word for it" in some cases.  THE MATHEMATICS OF POKER  5  Introduction To help facilitate this, we have marked off the start and end of some portions r:i.. the text so that our less mathematical readers can skip more complex derivations.. Just look for this icon for guidance, indicating these cases. ~ In addition,  Solution: Solutions to example problems are shown in shaded boxes.  As we said, this book is not a mathematical textbook or a mathematical paper to be submitted to a journal. The material here is not presented in the marmer of formal proof, nor do we intend it to be taken as such. 'Ve justify our conclusions with mathematical arguments where necessary and with intuitive supplemental arguments where possible in order to attempt to make the principles of the mathematics of poker accessible to readers without a formal mathematical background, and we try not to be boring. 1ne primary goal of our work here is not to solve game theory problems for the pure joy of doing so; it is to enhance our ability to win money at poker.  This book is aimed at a wide range of players, from players with only a modest amount of experience to world-class players. If you have never played poker before, the best course of action is to put this book dovvn, read some of the other books in print aimed at beginners. play some poker, learn some more, and then return after gaining additional experience. If you are a computer scientist or options trader who has recendy taken up the game, then welcome. This book is for you. If you are one of a growing class of players who has read a few books, played for some time, and believe you are a solid, winning player, are interested in making the next steps but feel like the existing literatme lacks insight that will help you to raise your game, then welcome. This book is also for you. If you are the holder of multiple World Series of Poker bracelets who plays regularly in the big game at the Bellagio, you too are welcome. There is likely a fair amount of material here that can help you as well.  Organization The book is orgariized as follows:  Part I: Basics, is an introduction to a number of general concepts that apply to all forms of gambling and other situations that include decision making under risk. We begin by introducing probability, a core concept that underlies all of poker. We then introduce the concept of a probability distribution, an important abstraction that allows us to effectively analyze situations with a large number of possible outcomes, each with unique and variable probabilities. Once we have a probability distribution, we can define expected value, which is the metric that we seek to maximize in poker. Additionally, we introduce a number of concepts from statistics that have specific, common, and useful applications in the field of poker, including one of the most powerful concepts in statistics, Bayes' theorem.  6  THE MATHEMATICS OF POKER  Introduction  Part II: Exploitive Play, is the beginning of our analysis of poker. We introduce the concept of a toy game, which is a smaller, simpler game chat we can solve in order to gain insight about analogous, more complicated games. We then consider examples of toy games in a number of situations. First we look at playing poker with me cards exposed and find that the play in many situations is quite obvious; at the same time, we find interesting situations with some counter-intuitive properties that are helpful in understanding full games. Then we consider what many authors treat as the heart of poker, the situation where we play our single hand against a distribution of the opponent's hands and attempt to exploit his strategy, or maximize our win against his play. This is the subject of the overwhehning majority of the poker literature. But we go further, to the in our view) much more important case, where we are not only playing a single hand against the opponent, but playing an entire distribution of hands against his distribution of hands. It is this view of poker, we claim, that leads to truly strong play. Part III: Optimal Play, is the largest and most important part of this book. In this part, we introduce the branch of mathematics called game theory. Game theory allows us to find optimal strategies for simple games and to infer characteristics of optimal strategies for more complicated games even if we cannot solve them direcdy. We do work on many variations of the AKQ,game, a simple toy game originally introduced to us in Card Player magazine by Mike Caro. We then spend a substantial amount of time introducing and solving [0,1] poker games, of the type introduced by John von Neumann and Oskar ~'Iorganstem in their seminal text on game theory Theory 0/ Games and Economic Behavior 1944), but with substantially more complexity and relevance to real-life poker. We also e:-..-plain and provide the optimal play solution to short-stack headsup no-limit holdem. Part IV: Bankroll and Risk includes material of interest on a very important topic to anyone who approaches poker seriously. We present the risk of ruin model, a method fo r estimating the chance of losing a fixed amount playing in a game with positivc expectation but some variance. We then extend the risk of ruin model in a novel way to include the uncertainty surrounding any observation of win rate. We also address topics such as the Kelly criterion, choosing an appropriate game level, and the application of portfolio theory to the poker metagame. Part V: Other Topics includes material on other important topics. 'lburnamenrs are the fastest-growing and most visible form of poker today; we providc an explanation of concepts and models for calculating equity and making accurate decisions in the tournament envirorunent. We consider the game theory of muItiplayer games, an important and very complex branch of game theory, and show some reasons why me analysis of such games is so difficult. In this section we also articulate and explain our strategic philosophy of play, including our attempts to play optimally or at least pseudooptimally as well as the situations in which we play exploitively.  -:H E MATHEMATICS O F POKER  7  Introduction  How This Book Is Different This book differs from other poker books in a number of ways. One of the most prominent is in its emphasis on quantitative methods and modeling. We believe that intuition is often a valuable tool for understanding what is happening. But at the same time, we eschew its use as a guide to what action to take. We also look for ways to identify situations where Our intuition is often wrong, and attempt to retrain it in such situations in order to improve the quality of our reads and our overall play. For example, psychologists have identified that the human brain is quite poor at estimating probabilities, especially for situations that occur with low frequency. By creating alternate methods for estimating these probabilities, we can gain an advantage over our opponents. It is reasonable to look at each poker decision as a two'part process of gathering information and then synthesizing that information and choosing the right action. It is our contention that inruition has no place in the latter. Once we have a set of assumptions about the situation - how our opponent plays, what our cards are, the pot size, etc., then finding the right action is a simple matter of calculating expectation for the various options and choosing the option that maximizes this. The second major way in which this book differs from other poker books is in its emphasis on strategy, contrasted to an emphasis on decisions. Many poker books divide the hand into sections, such as cCpreflop play," "flop play," "turn play," etc. By doing this, however, they make it difficult to capture the way in which a player's preflop, Hop, rum, and river play are all intimately connected, and ultimately part of the same strategy. We try to look at hands and games in a much more organic fashion, where, as much as possible, the evaluation of expectation occurs not at each decision point but at the begirming of the hand, where a full strategy for the game is chosen. Unfortunately, holdem and other popular poker games are extraordinarily complex in this sense, and so we must sacrifice this sometimes due to computational infeasibility. But the idea of carrying a strategy fonvard through different betting rounds and being constantly aware of the potential hands we could hold at this point, which OUT fellow poker theorists Chris Ferguson and Paul R. Pudaite call "reading your own hand," is essential to our view of poker. A third way in which this book differs from much of the existing literature is that it is not a book about how to play poker. It is a book about how to think about poker. We offer very little in terms of specific recommendations about how to play various games ; instead this book is devoted to examining the issues that are of importance in determining a strategy. Instead of a roadmap to how to play poker optimally, we instead try to offer a roadmap to how to think about optimal poker. Our approach to studying poker, too, diverges from much of the existing literarure. We often work on toy games , small, solvable games from which we hope to gain insight into larger, more complex games. In a sense, we look at toy games to examine dimensions of poker, and how they affect our strategy. How does the game change when we move from cards exposed to cards concealed? From games where players cannot fold to games where they can? From games where the first player always checks to games where both players can bet? From games with one street to games with cwo? We examine these situations by way of toy games - because toy games , unlike real poker, are solvable in practice - and attempt to gain insight into how we should approach the larger game.  8  THE MATHEMATICS OF POKER  •  Introduction  Our Goals It is our hope that our presentation of this material will provide at least two things; that it will aid you to play more strongly in your own poker endeavors and to think about siruanons in poker in a new light, and that it will serve as a jumping-off point toward the incredible amount of serious work that remains to be done in this field. Poker is in a critical stage of growth at this writing; the universe of poker players and the mainstream credibility of the game have never been larger. Yet it is still largely believed that intuition and experience are detennining factors of the quality of play - just as in the bond and options markets in the eady 19805, trading was dominated by old-time veterans who had both qualities in abundance. A decade later, the quantitative analysts had grasped control of the market, and style and intuition were on the decline. In the same way, even those poker players regarded as the strongest in the world make serious errors and deviations from optimal strategies. 1bis is not an indictment of their play, but a reminder that the distance between the play of the best players in the world and the best play possible is still large, and that therefore there is a large amount of profit available to those who can bridge that gap.  THE MATHEMATIC S OF POKER  9  Introduction  10  THE MATHEMATICS OF POKER  Part I: Basics  ((As for as the laws qfmathematics rqer to reality, they are not certain; as for as they are certain, they do not rifer to reality." Albert Einstein  Chapter 1-Decisions Under Risk: Probability and Expectation  Chapter 1 Decisions Under Risk: Probability and Expectation There are as many different reasons for playing poker as there are players who play the game. Some play for social reasons, to feel part of a group or "one of the guys," some play for recreation, just to enjoy themselves. Many play for the enjoyment of competition. Still others play to satisfy gambling addictions or to cover up other pain in their lives. One of the di.fficulties of taking a mathematical approach to these reasons is that it's difficult to quantify  the value of having fun or of a sense of belonging. In addition to some of the more nebulous and difficult to quantify reasons for playing poker, there may also be additional financial incentives not captured within the game itsdf. For e..x.ample, the winner of the championship event of the World Series of Poker is virtually guaranteed to reap a windfall from endorsements , appearances, and so on, over and above the: large first prize. There are other considerations for players at the poker table as well; perhaps losing an additional hand would be a significant psychological blow. "While we may criticize this view as irrational, it must still factor ineo any exhaustive examination of the incentives to play poker. Even if we restrict our inquiry eo monetary rewards, we find that preference for money is non-linear. For most people, winnlng five million dollars is worth much more (or has much more utility) than a 50% chance of winning ten million; five million dollars is life-changing money for most, and the marginal value of the additional five million is much smaller.  In a broader sense, all of these issues are included in the utility theory branch of economics. U tility theorists seek to quantify the preferences of individuals and create a framework under which financial and non-financial incentives can be directly compared. In reality, it is utility that we seek to maximize when playing poker (or in fact, when doing anything). H owever, the use of utility theory as a basis for analysis presents a difficulty; each individual has his own utility curves and so general analysis becomes cxtremdy difficulL In this book, we will therefore refrain from considering utility and instead use money won inside the game as a proxy for utility. In the bankroll theory section in Part IV, we will take an in-depth look at certain meta-game considerations, introduce such concepts as risk ofruin, the Kelry criteri~ and certainty equivalent. All of these are measures of risk that have primarily to do with factors outside the game. Except when expressly stated, however, we will take as a premise that players are adequately bankrolled for the games they are playing in, and that their sole purpose is to maximize the money they will -win by making the best decisions at every point. Maximizing total money won in poker requires that a player maximize the expected value of his decisions. However, before we can reasonably introduce this cornerstone concept, we must first spend some time discussing the concepts of probability that underlie it. The following material owes a great debt to Rlchard Epstein's text The Theory 0/ Gambling and StaJistiud Lo[jc (1967), a valuable primer on probability and gambling.  THE MATHEMATICS OF POKER  13  Part I: Basics  Probability Most of the decisions in poker take place under conditions where the outcome has not yet been determined. When the dealer deals out the hand at the outset, the players' cards are unknown, at least until they are observed. Yet we still have some information about the contents of the other players' hands. The game's rules constrain the coments of their handswhile a player may hold the jack-ten of hearts, he cannot hold the ace-prince of billiard tables, for example. The composition of the deck of cards is set before starting and gives us information about the hands. Consider a holdem hand. "What is the chance that the hand contains two aces? You may know the answer already, but consider what the answer means. "What if we dealt a million hands just like this? How many pairs of aces would there be? What if we dealt ten million? Over time and many trials, the ratio of pairs of aces to total hands dealt v.'ill converge on a particular number. We define probabilii)J as this number. Probability is the key to decision-making in poker as it provides a mathematical framework by which we can evaluate the likelihood of uncertain events.  IT n trials of an experiment (such as dealing out a holdem hand) produce no occurrences of an event x, we define the probability p of x occurringP(x) as follows:  p(x}  ~  n,  ~ lim n-". n  (1.1 )  Now it happens to be the case that the likelihood of a single holdem hand being a pair of aces is 11 221 • We COUld, of course, determine this by dealin'b out ten billion hands and obscrving,the ratlo 01 palls 01 aces to tota\. banus uea\t. l:"rU.s, 'however, wou\e be a \eng\hy ane Oillicu\t process, and we can do better by breaking the problem up into components. First we consider just one card. What is the probability that a single card is an ace? Even this problem can be broken down further - what is the probability that a single card is the ace of spades? This final question can be answered rather directly. We make the following assumptions: There are fifty-two cards in a standard deck. Each possible card is equally likely. Then the probability of any particular card being the one chosen is 1/ 52 , If the chance of the card being the ace of spades is 1/52 , what is the chance of the card being any ace? This is equivalent to the chance that the card is the ace of spades OR that it is the ace of hearts OR that it is the ace of diamonds OR that it is the ace of clubs. There are four aces in the deck, each with a 1/52 chance of being the card, and summing these probabilities, we have:  PlA)= l4{~2J 1 P(A) = 13 We can sum these probabilities directly because they are mutually exclusive; that is, no card can simultaneously be both the ace of spades and the ace of hearts. Note that the probability 1/ 13 is exactly equal to the ratio (number of aces in the deck)/(number of cards total). This rdationship holds just as well as the summing of the individual probabilities.  14  THE MATHEMATICS OF POKER  Chapter 1-Decision s Under Risk: Probability and Expectation  Independent Events ·et re 1e  Some events, however, are not murually exclusive. Consider for example, these two events:  1. The card is a heart 1. The card is an ace.  :5, lS  w  If we try to figure out the probability that a single card is a heart OR that it is an ace, we find :hat there are thirteen hearts in the deck out of fifty-cards, so the chance that the card is a heart is lt~ . The chance that the card is an ace is: as before, 1/ 13 _ However, we cannot simply add :hese probabilities as before, because it is possible for a card to both be an ace and a heart.  Is :r U-  n  )f  n  There are two types of relationships between events. The first type are events that have no effect on each other. For example, the closing value of the NASDAQ stock index and the value of the dice on a particular roll at the craps table in a casino in Monaco that evening are basically unrelated events; neither one should impact the other in any way that is not ~egligible. If the probability of both events occurring equals the product of the individual probabilities, then the events are said to be 'independent. The probability that both A and B cur is called the joint probability of A and B.  :::. :his case, the joint probability of a card being both a heart and an ace is (1 / 13 )(1/ 4 ), or 1/ 52 . :-- , is because the fact that the card is a heart does not affect the chance that it is an ace - all i:r= suits have the same set of cards.  s e  :::iependent events are not murually exclusive except when one of the events has probability In this example, the total number of hearts in the deck is thirteen, and the total of aces :::J. the deck is four. However, by adding these together, we are double-counting one single card the ace of hearts) . There are acrually thirteen hearts and three other aces, or if you prefer, four aces, and twelve other hearts. It rums out that the general application of this concept is that the probability that at least one of two mutually non·exclusive events A and B will occur is the sum of the probabilities of A and B minus the joint probability of A and B. So the probability of the card being a heart or an ace is equal to the chance of it being a heart (1/4) plus the chance of it being an ace (1/13) minus the chance of it being both (1/52), or 4h3 . This is true for all cyenu, iudcpcnde.nt or dependent. .::::::0.  Dependent Events Some evenrs, by cono-ast, do have impacts on each other. For example, before a baseball game, a certain talented pitcher might have a 3% chance of pitching nine innings and allowing no runs , while his team might have a 60% chance of wllming the game. However, the chance of the pitcher'S team vvinning the game and him also pitching a shutout is obviously not 60% times 3%. Instead, it is very close to 3% itself, because the pitcher's team will virtually always "in the game when he accomplishes this. These events are called dependent We can also consider the ronditUmal probabiliry of A given B, which is the chance that if B happens, A will also happen. The probability of A and B both occurring for dependent events is equal to the probability of A multiplied by the conditional probability ofB given A. Events are independent if the conditional probability of A given B is equal to the probability of A alone. Sununarizing these topics more formally, if we use the following notation: p(A U B) = Probability of A or B occurring. p(A n B) = Probability of A and B occurring.  THE MATHEMATICS OF POKER  15  I  Part I: Basics  peA I B) ~ Conditional probability of A occurring given B has already occurred. The U and n notations are from set theory and fonnally represent "union" and "intersection." We prefer the more mundane terms "or" and "and." Likewise, I is the symbol for "given," so we pronounce these expressions as follows:  p(A U B) ~ "p of A or B" p(A n B) ~ "p of A and B" PtA I B) ~ "p of A given B" Then for mutually exclusive events:  p(A U B) ~ prAY + p(B)  (1.2)  For independent events:  p(A n B)  ~  p(A)p(B)  (1.3)  For all events:  p(A U B) ~ prAY + p(B) - p(A n B)  (1.4)  For dependent events:  p(A n B)  ~ p(A)p(B  I A)  (1.5)  Equation 1.2 is simply a special case of Equation 1.4 for mutually exclusive events,  p(A n B) = O. Likewise, Equation 1.3 is a special case of Equation 1.5, as for independent events, p(B I A) ~ p(B). Additionally, if p(B IA) ~ p(B), then peA IB) ~ peA)· We can now return to the question at hand. How frequently will a single holdem hand dealt from a full deck contain two aces? There are two events here: A: The first card is an ace. B: The second card is an ace. p(A)~ '1 13 , and p(B)~ '113 as well. However, these two events are dependent.lf A occurs (the first  card is an ace), then it is less likely that B will ocrur, as the cards are dealt without replacemem. So P{B IA) is the chance that the second card is an ace given that the first card is an ace. There are three aces remaining, and fifty-one possible cards, so p(B IA) = 3/ 5 1, or 1/17 .  p(A n B) p(A n B) p(A n B)  ~  p(A)p(B I A)  ~  ('I.,) (II,,)  ~  'I",  Tnere are a number of ol.her simple properties that we can mention about probabilities. First, the probability of any event is at least zero and no greater than one. Referring back to the definition of probability, n trials will never result in more than n occurrences of the event, and never less than zero occurrences. The probability of an event that is certain to occur is one. The probability of an event that never occurs is zero. The probability of an evenes complement -that is, the chance that an event does nOt occur, is simply one minus the event's probability. Summarizing, if we use the foUowing notation:  P(A) ~ Probability that A does not occur. 16  THE MATHEMATICS OF POKER  Chapter l-Decisions Under Risk: Probability and Expectation  C = a certain event  1= an impossible event -=nen we have:  o ""P(A )"" 1 for any A  (1 .6)  p(C) ~ 1  (1.7)  prJ)  ~ 0  (1.8)  p(A) + PtA) ~ 1  (1 .9)  Equation 1.9 can also be restated as:  p(A) ~ 1 - PtA)  (1.10)  \'e can solve many probability problems using these rules. Some common questions of probability are simple, such as the chance of rolling double sixes on [wo dice. In tenns of probability, this can be stated using equation 1.3, since the die rolls are independent. Let PtA) be the probability of rolling a six on the first die and P(lJ) be the probability of rolling a six on the second die. Then:  p(A n B) p(A n B) p(A n B)  ~  p(A)p(B)  ~  (110)('10) 1/"  ~  Likewise, using equation 1.2, me chance of a single player holding aces, kings, or queens becomes: P(AA) P(KK)  ~  I/m  ~ 1/221  P(QQJ ~ 1/221 P«AA,KK,QQ,J)  ~ P(AA)  + P(KK) + P(QQJ  ~  J/ 22l  Additionally we can solve more complex questions , such as: How likely is it that a suited hand will flop a Bush? \Ve hold two of the flush suit, leaving eleven in the deck. All three of the cards must be of the Bush suit, meaning that we have A = the first card being a flush card, B = the second card being a Bush card given that the first card is a Bush card, and C~ the third card being a Bush card given than both of the first two are flush cards.  prAY ~ II /S O p(B I A) ~ 10/" p(e I (A n B)) ~ '/48  (two cards removed from the deck in the player's hand) (one flush card and three total cards removed) (two flush cards and four total cards removed)  Applying equation 1.5, we get:  p(A n B) ~ p(A)p(B I A) p(A n B) ~ (11/50)(,0/,,) p(A n B) ~ Il/"s THE MATHEMATICS O F POKER  17  Part I: Basics  Letting D = (A  n B), we can use equation 1.5 again:  p(D n C) ~ p(D)(P(C I D) p(A n B n C) ~ p(A n B)p(C I (A n B)) p(A n B n C) ~ (11/24.;)('/,,) p(A  n B n C) =  33h920,  or a little less than 1%.  vVe can apply these rules to virtually any situation, and throughout the text we will use these properties and rules to calculate probabilities for single events.  Probability Distributions Though single event probabilities are important, it is often the case that they are inadequate to fully analyze a situation. Instead, it is frequendy important to consider many different probabilities at the same time. We can characterize the possible outcomes and their probabilities from an event as a probability distributWn.  =  Consider a fair coin flip. The coin fup has just two possible outcomes - each outcome is mutually exclusive and has a probability of l /2 . We can create a probability distribution for the coin flip by taking each outcome and pairing it with its probability. So we have two pairs: (heads, II,) and (tails, II,) . If C is the probability distribution of the result of a coin flip, then we can write this as: C~  {(heads , II,), (tails, II,)}  Likewise, the probability distribution of the result of a fair six-sided die roll is: D~{ ( 1 , 1 /6),  (2,1/6) , (3 ,IIe), (4,1/6), (5 ,1/6), (6,110))  We can construct a discrete probability disrribution for any event by enumerating an exhaustive and murnally exclusive list of possible outcomes and pairing these outcomes "vith their corresponding probabilities. 01  We can therefore create different probability distributions from the same physical event. From our die roll we could also create a second probability distribution, this one the distribution of the odd-or-evenness of the roll: D'~  {(odd, 1/,), (even, II,)}  b  In poker, we are almost always very concerned with the contents of our opponents' hands. But it is seldom possible to narrow down our estimate of these contents to a single pair of cards. Instead, we use a probability distribution to represent the hands he could possibly hold and the corresponding probabilities that he holds them. At the beginning of the hand, before anyone has looked at their cards, each player's probability distribution of hands is identical. As the hand progresses, however, we can incorporate new information we gain through the play of the hand, the cards in our own hand, the cards on the board, and so on, to continually refine the probability estimates we have for each possible hand. Sometimes we can associate a numerical value with each element of a probability distribution. For example suppose that a friend offers to flip a fair coin with you. The winner will collect $10 from the loser. Now the results of the coin flip follow the probability distribution we identified earlier:  18  "n  THE MATHEMATICS OF POKER  Chapter 1-Decisions Under Risk: Probability and Expectation  c = [(head s, If,) , (tails, If,)) 5:'nce we know the coin is fair, it doesn't matter who calls the coin or what they call, so we can Xienrify a second probability distribution that is the result of the bet:  C' = [(win, If,), (lose, If,)) '\'(: can then go further, and associate a numerical value -...vith each result. If we win the flip, our friend pays us $10. If we lose the flip, then we pay him $10. So we have the following:  B = [(+$10, If,) , (-$1 0, I/,)} te nt os  \ , n en a probability distribution has numerical values associated with each of the possible outcomes, we can find the expected value (EV) of that distribution, which is the value of each outcome multiplied by its probability, all sununed together. lbroughout the text, we will use :he notation <X> to denote "the expected value of X " For this example, we have:  <B> = (1/,)(+$10) + <B> = $5 + (-$5) <B> =O  (1/,)(-510)  Hopefully this is intuitively obvious - if you fup a fair coin for some amount, half the time you win and half the time you lose. The amounts are the same, so you break even on average. _-\lso, the EV of declining your friend 's offer by not Hipping at all is also zero, because no money changes hands. For a probability distribution P, where each of the n outcomes has a value Xi and a probability  Pi. then P'J expected value <P> is : (1.11 ) At the core of winning at poker or at any type of gambling is the idea of maximizing expected value. In this example, your mend has offered you a fair bet. On average, you are no better or worse offby flipping with him than you are by declining to flip. )low suppose your friend offers you a different, better deal. He'll fup with you again, but when you win, he'll pay you $11, while if he wins, you'll only pay him $10. Again, the EV of not flipping is 0, but the EV of flipping is not zero any more. You'll win $11 when you win but lose $10 when you lose. Your expected value of this new bet Bn is:  <Bn> = <Bn> =  ('/,)(+$11) + (1/, )(-$10) $0.50  On average here, then, you will win fifty cents per flip. Of course, this is not a guarameed win ; in fact, it's impossible for you to win 50 cents on any particular flip. It's o nly in the aggregate that this expected value number exists. H owever, by doing this , you will average fifty cents better than declining. As another example, let's say your same mend o ffers you the following deal. You 'll roll a pair of dice once, and if the dice come up double sixes, he'll pay you $30, while if they come up any other number, you'll pay him $1. Again, we can calculate the EV ofthis proposition.  THE MATHEM ATICS OF POKER  19  Part I: Basics  <Bd> = (+$30)(1/36) + <Bd> = $30/" ~$ 35/36  (~$1 )(35"6)  <Bd> = _$5/36 or about 14 cents. The value of this bet to you is about negative 14 cents. The EV of not playing is zero, so this is a bad bet and you shouldn't take it. Tell your friend to go back to offering you 11-10 on coin fups. Notice that this exact bet is offered on craps layouts around the world. A very important property of expected value is that it is additive. That is, the EV of six different bets in a row is the sum of the individual EVs of each bet individually. Most gambling games - most things in life, in fact, are just like this. We are continually offered little coin fups or dice rolls - some with positive expected value, others with negative expected value. Sometimes the event in question isn't a die roll or a coin ffip, but an insurance policy or a bond fund. The free drinks and neon lights of Las Vegas are financed by the summation of millions of little coin flips, on each of which the house has a tiny edge. A skillful poker player takes advantage of this additive property of expected value by constantly taking advantage of favorable EV situations. In using probability distributions to discuss poker, we often omit specific probabilities for each hand. When we do this, it means that the relative probabilities of those hands are unchanged from their probabilities at the beginning of the hand. Supposing that we have observed a very tight player raise and we know from our experience that he raises if and only if he holds aces, kings, queens, or ace-king, we might represent his distribution of hands as:  H = (AA, KK,QQ, AKs, AKo) The omission of probabilities here simply implies that the relative probabilities of these hands are as they were when the cards were dealt. We can also use the <X> notation for situations where we have more than one distribution under examination. Suppose we are discussing a poker situation where two players A and B have hands taken from the following distributions:  = (AA, KK, QQ, lJ, AKu, AKs) B= (AA,KK,QQ}  A  We have the following, then:  <A,B> <A,AAIB>  :the expectation for playing the distribution A against the distribution B. :the expectation for playing the distribution A against the hand AA taken from the distribution B.  <AAIA, AAIB >:the expectation for playing AA from A against AA from B. <A, B>=P(AA)<A, AAIB> + p(KK)<A, KKIB> + P(Q9J <A,QQIB> ... and so on. Additionally, we can perfonn some basic arithmetic operations on the elements of a distribution. For example, if we multiply all the values of the outcomes of a distribution by a real constant, the expected value of the resulting distribution is equal to the expected value of the original distribution multiplied by the constant. Likewise, if we add a constant to each of the values of the outcomes of a distribution, the expected value of the resulting distribution is equal to the expected value of the original distribution plus the constant.  20  THE MATHEMATICS OF POKER  Chapter l-Decisions Under Risk: Probability and Expectation '.\~  :lis Dn  ax :>g  ps ~e .  should also take a moment to describe a common method of expressing probabilities,  edds. Odds are defined as the ratio of me probability of the event not happening to the :zobability of the event happening. These odds may be scaled CO any convenient base and are ..:ommonly expressed as " 7 to 5," "3 to 2," etc. Shorter odds are those where the event is more ...1.dy : longer odds are those where the event is less likely. Often, relative hand values might .x e.....'Pressed this way: "That hand is a 7 to 3 favorite over the other one," meaning that it has !  j"00!0 of winning, and so on.  Odds are usually more awkward to use than probabilities in mathematical calculations because ~- cannot be easily multiplied by outcomes to yield expectation. True "gamblers" often use .::.dds. because odds correspond to the ways in which they are paid out on their bets. Probability ~ :nare of a mathematical concept. Gamblers who utilize mathematics may use either, but often  ;:rder probabilities because of the ease of converting probabilities  [0  expected value.  a  Df Key Concepts The probability of an outcome of an event is the ratio of that outcome's occurrence over an arbitrarily large number of trials of that event.  h d Y ;,  A probability distribution is a pairing of a list of complete and mutually exclusive outcomes of an event with their corresponding probabilities. The expected value of a valued probability distribution is the sum of the probabilities of the outcomes times their probabilities. Expected value is additive.  s  If each outcome of a probability distribution is mapped to numerical values, the expected  s  value of the distribution is the summation of the products of probabilities and outcomes.  ,,  A mathematical approach to poker is concerned primarily with the maximization of expected value.  THE MATHEMATICS OF POKER  21  Part I: Basics  Chapter 2 Predicting the Future: Variance and Sample Outcomes Probability distributions that have values associated with the clements have two characteristics which, taken together, describe mOSt of the behavior of the distribution for repeated trials. The first, described in the previous chapter, is expected value. The second is variance, a measure of the dispersion of the outcomes from the expectation. To characterize these two tcrtIl51oosely, expected value measures how much you will win on average; variance measures how far your speciEc results may be from the expected value.  When we consider variance, we are attempting to capture the range of outcomes that can be expected from a number of trials. In many fields, the range of Outcomes is of particular concern on both sides of the mean. For example, in many manufacturing environments there is a band of acceptability and outcomes on either side of this band are undesirable. In poker, there is a tendency to characterize variance as a one-sided phenomenon, because most players are unconcerned with outcomes that result in winning much more than expectation. In fact, "variance" is often used as shorthand for negative swings. This view is somewhat practical, especially for professional players, but creates a tendency to ignore positive results and to therefore assume that these positive outco mes are more representative of the underlying distribution than they really are. One of the important go als of statistics is to find the probability of a certain measured outcome given a set of initial conditions, and also the inverse of this - inferring the initial conditions from the measured outcome. In poker, both of these are of considerable use. We refer to the underlying distribution of uutCOInes frUIn a set of initial conditions as the population and the observed outcomes as the sample. In poker, we often cannot measure all the clements of the popula tion, but must content ourselves with observing samples. Most statistics courses and texts provide material o n probability as well as a whole slew of sampling methodologies, hypothesis testS, correlation coefficients, and so on. In analyzing poker we make heavy use of probability concepts and occasional use of o ther statistical methods. What follows is a quick-and-dirty overview of some statistical concepts that are useful in analyzing poker, especially in analyzing observed results. Much information deemed to be irrelevant is omitted from the following and we encourage you to consult statistics textbooks for more information on these topics. A commonly asked question in poker is "How often should I expect to have a winning session?" Rephrased, this question is "what is the chance that a particular sample taken from a population that eonsists of my sessions in a game will have an outcome> O?" TIl.e most straightforward method of answering this question would be to examine the probability distribution of your sessions in that game and sum the probabilities of all those outcomes that are greater than zero. U nfortunately, we do not have access to that distribution - no m atter how much data you have collected about your play in that game from the past, all you have is a sample. However, suppose that we know somehow your per-hand expectation and variance in the game, and we know how long the session you are concemed with is. Then we can use statistical m ethods to estimate rhe probability that you will have a winning session. The first of these items, expected value (which we can also call the mean of the distribution) is familiar by now; we discussed it in Chapter 1.  22  THE MATHEMATICS OF POKER  Chapter 2-Pred iding the Future: Variance and Sample Outcomes  Variance  5  The second of these measures, variance, is a measure of the deviation of outcomes from the expectation of a distribution. Consider two bets, one where you are paid even money on a coin flip, and one where you are paid 5 to 1 on a die roll, winning only when the die comes up 6. Both of these distributions have an EV of 0, but the die roll has significantly higher \w ance. lh of the time, you get a payout that is 5 units away from the expectation, while 5/6 of the time you get a payout that is only 1 unit away from the expectation. To calculate variance, we first square the distances from the expected value, multiply them by the probability they occur, and sum the values. For a probability distribution P, where each of the n outcomes has a value  p,. then the variance of P, Vp Vp ~  Xi  and a probability  is:  I,p,(x,- < P »'  (2.1)  ; =1  );"otice that because each teIm is squared and therefore positive, variance is always positive. Reconsidering our examples, the variance of the coinHip is:  Vc = (1/,)(1 - 0)' + (1/,)(- 1 - 0)' Vc= 1 ' Vhile the variance of the die roll is:  VD = ('10)(· 1 - 0)' + (1/6)(5 - 0)'  VD = 5  In poker, a loose·wild game will have much higher variance than a tight-passive game, because me outcomes will be further from the mean (pots you win will be larger, but the money lost in pots you lose will be greater). Style of play will also affect your variance; thin value bets and semi·bluff raises are examples of higher-variance plays that might increase variance, expectation, or both. On me orner hand, loose-maniacal players may make plays that increase their variance while decreasing expectation. And playing too tightly may reduce both quantities. In Part IV, we will examine bankroll considerations and risk considerations and consider a framework by which variance can affect our utility value of money. Except for that part of the book, we will ignore variance as a decision-making aiterion for poker decisions. In this way variance is for us only a descriptive statistic, not a prescriptive one (as expected value is). Variance gives us information about the expected distance from the mean of a distribution. The most important property of variance is that it is directly additive across trials, just as e:\'}Jectation is. So if you take the preceding dice bet twice, the variance of the two b ets combined is twice as large, or 10. Expected value is measured in units of expectation per event; by contrast, variance is measured in units of expectation squared per event squared . Because of this , it is not easy to compare variance to expected value directly. If we are to compare these two quantities, we must take the square root of the variance, which is called the .standard deviation. For our dice example, the standard deviation of one roll is {5::::: 2.23. We often use the Greek letter 0 (sigma) to represent standard deviation, and by extension 0 2 is often used to represent variance in addition to the previously utilized V.  THE MATHEMATIC S O F POKER  23  Part I: Basics  (2.2) 0- 2  = V  (2.3)  The Normal Distribution When we take a single random result from a distribution, it has some value that is one of the possible outcomes of the underlying distribution. We call this a random variable. Suppose we flip a coin. The flip itself is a random variable. Suppose that we label the two outcomes 1 (heads) and 0 (tails). The result of the flip will then either be 1 (half the time) or 0 (half the time). If we take multiple coin flips and sum them up, we get a value thac is the summation of the outcomes of the random variable (for example, heads), which we call a sample. The sample value, then, vvill be the number of heads we flip in whatever the size of the sample. For example, suppose we recorded the results of 100 coinHips as a single number - the total number of heads . The expected value of the sample "",ill be 50, as each flip has an expected value of 0.5. The variance and standard deviation of a single fup are:  (1 /2)(1 - ' /2)2 + (1/2)(0 - ' /2)' a 2 = 1h a = 1/ 2 0 2 ~  From the previous section, we know also that the variance of the flips is adclitive. So the variance of 100 flips is 25. Just as an individual flip has a standard deviation, a sample has a standard deviation as welL However, unlike variance, standard deviations are not additive. But there is a relationship between the twu. For N trials, the variance will be:  a 2N = Na 2 (2.4) The square root relationship of trials to standard deviation is an important result, because it shows us how standard deviation scales over multiple trials. If we flip a coin once, it has a standard deviation of 1/ 2 . If we flip it 100 times , the standard deviation of a sample containing 100 trials is not 50, but 5, the square root of 100 times the standard deviation of one Hip. We can see, of course, that since the variance of 100 flips is 25, the standard deviation of 100 flips is simply the square root,S. The distribution of outcomes of a sample is itself a probability clistribution, and is called the sampling distribuiWn. An important result from statistics, the Central Limit 7heorem, describes the relationship between the sampling distribution and the underlying distribution. 'What the Central Limit Theorem says is that as the size of the sample increases, the distribution of the aggregated values of the samples converges on a special distribution called the normal  distribution.  -  24  THE MATHEMATICS OF POKER  Chapter 2-Prediding the Future: Variance and Sample Outcomes  The normal distribution is a bell-shaped curve where the peak of the curve is at the population mean, and the tails asymptotically approach zero as the x-values go to negative or positive infinity. The curve is also scaled by the standard deviation of the population. The total area under the curve of the normal distribution (as with all probability distributions) is equal to 1, and the area under the curve on the interval [xl> x2] is equal to the probabili[}' that a particular result will fall between XI and x2' 1bis area is marked region A in Figure 2.1 . .4  he Ne  1 he of he  tal  ,d  0.35 .3 0.25 0.2 .1 5 0.1 0.05 X2  0 -1  -2  ·3  0  Figure 2.1. Std. Normal Dist, A = p(event between  ll.  Ip  3  2 XI  and  ~)  ...\ little less formally, the Central Limit Theorem says that if you have some population and take a lot of big enough samples (how big depends on the type of data you're looking at), the outcomes of the samples will follow a bell-shaped curve around the mean of the population \\ith a variance that's related to the variance of the underlying population. The equation of the normal distribution function of a distribution with mean Ji and standard deviation 0" is: N (x,)1.,rr) =  j,  a Ig {e JS  Ie  os Ie Ie  21  :R  1  (x-)1. )'  rrv2n  2rr  ~ exp( - - -,-)  (2.5)  Finding the area between two points under the normal distribution curve gives us the probability that the value of a sample with the corresponding mean and variance will fall between those two points. The normal distribution is symmetric about the mean, so '/2 of the total area is to the right of the mean, and 1/2 is to the left. A usual method of calculating areas under the normal curve involves creating for each point something called a 7--score, where z = 6· - Ji)/a. lbis z·score represents the number of standard deviations that the outcome x is away from the mean.  , ~(x  - p}la  TH E MATHEMATI CS OF POKER  (2.6)  25  Part I: Basics  me  We can then find something called cumulative normal distribution for a z-score Z, which is the area to the left of z under the curve (where me mean is zero and the standard deviation is 1). We call this function <1>(z). See Figure 2.2  If z is a normalized z-score value, then the cumulative normal distribution function for z is:  <1>(,) = r;:1  f' ,xp( - -2)  (2.7)  .,2" -  0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05  a ·3  ·2  ·1  01  x  3  Figure 2.2. Cumulative normal distribution  Fmding the area between two values xl and x2 is done by calculating the z-scores zi and 1.2 for Xl and X2, finding the cumulative normal disttibution values tD (Z \) and <I>{z2) and subo-acting them.  If <1> (z) is the cumulative nonnal distribution function for a z-score of Z, then the probability that a sample taken from a normal distribution function ,"lith mean p and standard deviation 0" will fall between two z-scores Xl and X2 is: (2.8)  Statisticians have created tables of <b(z) values and spreadsheet programs can derive these values as well. Some key values are well known to students of statistics. The area between Z= -1 and Z= +1 is approximately 0.68; the area between Z= -2 and Z= +2 is approximately 0.955, and the area between z = -3 and Z = +3 is approximately 0.997.  26  THE MATHEMATICS OF POKER  Chapter 2-Predicting the Future: Variance and Sample Outcomes Nruch iation  ~  values mean that the value of a sample from a nonnal distribution will fall:  Between (I' . a) and (" + a) of the mean 68% of the time. Between (I" 2a) and (" + 2a) of the mean 95.5% of the time. Between (". 3a) and (I' + 3a) of the mean 99.7% of the time.  us: _~  e.xample may help to make this clear. Let us modify the parameters of the die roll game discussed before. A die is thrown, and the player loses 1 unit on 1-5, but wins 6 units on :oil of 6. We'll call tllis new game D].  ~ ~  ~  e.'\.-pectation of the game is:  < D]> = ('1.)(· 1)  + (tlo)(6)  < D2> = Ih unitsltrial ""ben the player wins, he gains six units. Subtracting the mean value of 1/6 from this outcome, obtain:  ~"C'  Vw;n = (6 -  'I.)'  ~I.jn = (35/ 6)2  :.ike\\ise, when he loses he loses one unit. Subtracting the mean value of Ih from this, we have:  Via," = (·1 - 'I,)' Vlose = ("1.)' The variance of the game is:  VD2 = p (win)(Vwin) + p(los e)( flo,e) VDl = (' /,)( 35 /,)' + ('1.)(" 1.)' VD2 z 6.806 units 2ltria12 Suppose we toss the die 200 times. What is the chance that the player will "vin overall in 200 tosses? Will 40 units or more? Will 100 units or more?  \\e c;m solve this problem using the techniques we have just summarized. We first calculate the standard deviation, sometimes called the standard error, of a sample of 200 trials. TIlis will be: a  = ff= --16.806 = 2.61 unitsltrial  oility  oon  .\pplying equation 2.4 we get: aN = a-.JN a200 = 2.61 --1200 = 36.89 units /200 trial,.  For 200 trials, the expected value, or mean (P), of this distribution is 1/6 units/trial times 200 trials or 33.33 units. Using Equation 2.6, we find the z-score of the point 0 as: lese 'een  tely  <ER  Ix  =  {x - p)/cr  where x = 0  Zo = (0 . 33 .33) 13 6.89 Zo = ·33.33 / 36.89 Zo = ·0.9035  THE MATHEMATICS OF POKER  27  Part I: Basics  Consulting a probability table in a statistics textbook, we find that the probability that an observed outcome will lie to the left of chis z-score, <1>(-0.9035) is 0.1831, Hence, there is an 18.31% chance that the player will be behind after 200 trials. To find the chance of being 40 units ahead, we find that point's z-score: '40 ~ (40 - 33.33)/ (36.89) ~ 0.1807  <1>(0.1807)  ~  0.5717  But Cl> (O.l807) is the probability that the observed outcome lies to the left of 40 units, or that we lose at least 40 units. To find the probability that we are to the right of this value, or are ahead 40 units, we must acrually find 1 - <1> (0.1807).  1 - <1> (0.1807)  ~  1 - 0.5717  ~ 0.4283  So there is a 42.830/0 chance of being ahead at least 40 units after 200 tosses.  And similarly for 100 units ahead: '100 ~ (100 - 33.33)/(36.89) ~ 1.8070  From a probability table we find that <1>(1.8070) ~ 0.9646. Thus, for:  p ~ 1 - <1> (1.8070) p~  0.0354  TIle probability of being 100 units ahead after 200 tosses is 3.54%. These values , however, are only approximations; the distribution of 200-roll samples is not quite normal. vVe can actually calrulate these values directly with the aid of a computer. Doing this yidds:  Chance of being ahead after 200 trials:  i Direct Calculation  !Nonnal Approx.  i 81.96%  • 81.69%  40.460/0  i 42.83°/0  Chance of being ahead at least 40 units :  :  Chance of being ahead at least 100 units :  : 4.44%  : 3.54%  As you can see, these values have slight differences. Much of this is caused by the fact that the direct calculations have only a discrete amount of values. 200 trials of this game can result in outcomes of +38 and +45, but not +39 or +42, because there is only one outcome possible from winning1 a +6. The normal approximation assumes that all values are possible.  Using chis method, we rerum to the question posed at the beginning of the chapter: "How often 'will I have a wirming session?" To solve this problem, we need to know the player's CA-p<!ctation per hand , his variance per hand and the size of his sessions. Consider a player whose ...Yin rate in some particular game is approximately 0.015 bets per hand, and whose standard deviation over the same interval is 2 bets per hand. Assume this player plans to play sessiol15 of 300 hands (these would be roughly nine-hour sessions in live poker; perhaps only tv\'o to three hours online). How often should this player expect to win (have a result greater than 0) in a session?  28  THE MATH EMATICS OF POKER  Chapter 2-Predicting the Future: Variance and Sampl e Outcom es  FIrst we can calculate the expected value of a sample, IW: )i;< ~ N)i )i;< ~  (300)(0.015)  )i;< ~  4.5  Second: we calculate his standard deviation for a 300-hand session. · that rare  cr N ~ cr{Ji cr300 ~ (2 )(~ 300) cr300 ~  34.6  :\"ext, we find the z-score and lD(z) for this value: Zx ~ {x - I'N}lcr  Zo Zo  ~ ~  (0 - 4.5 )/34.6 ·0.1 299  :::om a probability table we find:  <1>(·0.1299) ~ 44.83%  P~ 1 . <1> (·0.1299) p~  1- 0.4483  P ~ 0.5517  no<  nero  1b.is indicates that the player has a result of 0 or less 44.83% of the time - or correspondingly has a wirming session 55.17% of the time. In reality, players may change their playing habits in order to produce more winning sessions for psychological reasons, and in reality "win rate" flu ctuates significantly based on game conditions and for other reasons. But a player who simply plays 300 hands at a time with the above perfonnance memes should expect to be above zero just over 55% of the time. A more exn-eme example: A player once told one of the authors that he had kept a spreadsheet where he marked down every AQvs. AK all-in preflop confrontation that he saw online over a period of months and that after 2,000 instances, they were running about 50-50.  How likely is this, assuming the site's cards were fair and we have no additional information based on the hands that were folded? E rst, let's consider the variance or standard deviation of a single confrontation.  the tin ble  .AK vS . AQis about 73.5% all-in preflop (including all suitedness combinations). Let's assign a result of 1 to AK winning and 0 to AQwinning. Then our mean is 0.735. Calculating the variance of a single confrontation:  DW  v~  T'S  V~  ier  cr  ~  (0.735)(1· 0.735)' + (0.265 )(0· 0.735)' 0.1948 0.44 13  )S e  lay liy ter  ER  The mean of 2,000 hands is: )i" ~ N)i )i" ~  (2000)(0.735)  )i" ~  1470  THE MATHEMATICS OF POKER  29  Part I: Basics  For a sample of 2,000 hands, using Equation 2.4, the standard deviation is:  fJN~& fJ2 000 ~ (0.4413)('1 2000) fJ2000 ~ 19.737  The result reported here was approximately 50% of 2000, or 1000, while the sample mean would be about 1470 out of 2000. We can find the z-score using Equation 2.6:  ,  ~  (x  ~ ?N)/fJ  '1000 ~  (1000-1470)/(19.737)  '1000 ~ -23.8 15  The result reported in this case was 1000 out of 2000, while the expected population mean would be 1470 out of 2000. This result is then about 23.8 15 standard deviations away from the mean. Values this small are not very easy to evaluate because they are so small - in fact, my spreadsheet program calculates <D(-23.815) to equal exactly zero. Suffice it to say that this is a staggeringly low probability. What's likely is that in fact, this player was either exaggerating, outright lying, or perhaps made the common mistake of forgetting to notice when AK beat A(1 because that is the "expected" result. This is related to a psychological effect known as "perception bias" - we tend to notice things that are out of the ordinary while failing to notice things that are expected. Or perhaps the online site was in fact ':rigged." When this example was posed to a mailing list, the reaction of some participants was to point to the high variance of poker and assert that a wide range of outcomes is possible in 2000 hands. However, this confuses the altogether different issue of win/loss outcomes in 2000 poker hands (which docs have quite high variance) with the outcome of a single poker hand (which has far less). The variance of a single hand in terms of big bets in most forms of poker is much higher than the variance of the winner of an all-in pre80p confrontation. One lesson of this example is not to confuse the high variance of poker hand dollar outcomes with the comparatively low variance of other types of distributions. When we play poker, many random events happen. We and the other players at the cable are dealt random cards taken from a distribution that includes all two-card combinations. There is some betting, and often some opportunity to either change our cards, or to have the value of our hand change by the dealing of additional cards, and so on. Each hand results in some outcome for us, whether it is winning a big pot, stealing the blinds, losing a bet or two, or losing a large pot. This outcome is subject to all the smaller random events that occur within the hand, as well as events that are not so random - perhaps we get a tell on an opponent that enables us to win a pot we would not othcIWise have, or save a bet when we are beaten. Nevertheless, the power of the Central Limit Theorem is that outcomes of individual hands function approximately as though they were random variables selected from our "hand outcomes" distribution. And likewise, outcomes of sessions, and weeks, and months, and our whole poker career, behave as though they were appropriately-sized samples taken from this distribution.  The square root relationship of trials to standard deviation makes iliis particularly useful, because as the number of trials increases, the probability that our results will be far away from our Ch-pected value in relative terms decreases.  30  THE MATHEMATICS OF POKER  Chapter 2-Predicting the Future: Variance and Sample Outcomes  .ean  .-\ssume we have a player whose distribution of hand outcomes at a particular limit has a :nean of $75 per 100 hands, with a variance of $6,400 per hand. If we sample different numbers of hands from this player's distribution, we can see how the size of the sample impacts the dispersion of the results. We know that the probability that this player's results for 3. given sample yvill be between the mean minus two standard deviations and the mean plus nvo standard deviations is 95.5%. We will identify for each sample size:  The meanJiN The standard deviation cr The two endpoints of the 95.5% probability interval.  ean "Om  act, this  aps  me we ed.  mg  Hands  . ?N  ! cr  ~  Lower endpoint  100  $75 .00  500  $375 .00  $1 ,788.85  ($3,202.71 )  $3,952.71  1,000  $750.00  : $2,529.82  . ($4,309.64)  . $5,809 .64  5,000  $3,750.00  : $5,656.85  25,000  $18,750.00  50,000 100,000 1,000,000  $800.00  , ($1,525.00)  : Higher endpoint , $1,675.00  , ($7,563 .71)  $15,063.71  $12,649.11  , ($6,548.22)  $44,048.22  $37,500.00  $17,888.54  ' $1,722.91  $75,000.00  , $25,298 .22  $24,403.56  $750,000.00  : $80,000 .00  • $590,000.00  , $73,277.09 $125 ,596.44 : $910,000.00  hat  l er igh  fa of he .er  As you can see, for smaller numbers of hands, outcomes vary widely between losses and wins. However, as the size of the sample grows, the relative closeness of the sample result becomes larger and larger ~ although its absolute magnitude continues to grow. Comparing the standard deviation of one million hands to the standard deviation for one hundred hands, the size of a standard deviation is a hundred times as large in absolute terms, but more than a hundred times smaller relative to the number of hands. This is the law of large numbers at work; the larger the sample, the closer on a relative basis the outcomes of the sample will be.  lYe  :rc  ue ne or  Key Concepts Variance, the weighted sum of the squares of the distance from the mean of the outcomes of a distribution, is a valuable measurement of dispersion from the mean.  ill  at n.  To compare variance to expected value, we often use its square root, standard deviation. Variance is additive across trials. This gives rise to the square root relationship between the standard deviation of a sample of multiple trials and the standard deviation of one trial.  .IT  15  The Central limit Theorem tells us that as the size of a sample increases, the distribution of the samples behaves more and more like a normal distribution. This allows us to approximate complex distributions by using the normal distribution and to understand the  ~,  behavior of multiple-trial samples taken from a known distribution_  n  THE MATHEMATICS OF POKER  31  Part I: Basics  Chapter 3 Using All the Information: Estimating Parameters and Bayes' Theorem In the last chapter, we described some statistical properties of valued probability distributions, as well as the relationship of samples taken from those distributions to the nonnal distribution. However, throughout the chapter, we simply assumed in examining the sampling distribution that we knew the parameters of the underlying distribution. But in real life, we often don't know these things. Even for simple things such as coin flips, the "true" distribution of outcomes is that the coin is very slightly biased in one direction or the other. A die has tiny imperfections in its surface or composition that make it more likely to land on one side or another. However, these effects are usually small, and so using the "theoretical" coin which is truly 50-50 is a reasonable approximation for our purposes. Likewise, we generally assume for the purposes of poker analysis that the deck is fair and each card is random until it is observed.  We can mitigate real-world difficulties with distributions that we can reasonably approximate (such as coin flips or die rolls). Other types of distributions, however, pose much more difficult problems. In analyzing poker results, we are often interested in a distribution we discussed in the last chapter - per hand wonfloss amounts. When playing in a casino, it would be quite difficult and time-consuming to record the results of every hand - not to mention it might draw unwanted attention. The process is somewhat easier online, as downloadable hand histories and software tools can automate the process. But even if we have all this data, it's just a sample. For poker hands, the probabilities of the underlying distribution won't be reHected in the observed data unless you have a really large amount of data. We can get around this to some extent by using the subject matter of the last chapter. Suppose that we could get the mean and variance of our hand outcome distribution. Then we could find the sampling distribution and predict the aggregate outcomes from playing differem numbers of hands. We can't predict the actual outcome of a particular future sample, but we can predict the distribution of outcomes that will occur. Now the problem is to try to infer the population mean and variance from the sample mean and variance. We examine two approaches to this process. The first is the approach of classical statistics, and the second is an approach that utilizes the primary topic of this chapter, Bayes' theorem. The first approach takes as one of its assumptions that we have no infonnation other than the sample about the likelihood of any particular win rate. The second approach postulates a distribution of win rates that exists outside of our particular sample that can be used to refine our estimates of mean and variance for the population distribution.  ""ill  Estimating Parameters: Classical Statistics Suppose that we have a player who has played a total of 16,900 hands of limit poker. Nonnalizing his results (Q big bets (BB) in order to accoum for different limits he has played, he has achieved a win rate of x = 1.15 BB/100 hands with a standard deviation of s = 2.1 BB/hand. Here instead of l' and a, which represent population parameters, we use x and $, which are sample parameters. Assuming that he plans to continue (Q play in a similar mix of games with similar lineups of players, what can we say about his "true" win rate fJ in the games he has played? We assume in this section that we have no other information about the 32  THE MATHEMATICS OF POKER  Chapter a-Using All the Information: Estimating Parameters and Bayes' Theorem ~elihood of various win rates that might be possible; all win rates from -1 BB/hand to - l BB/hand are deemed to be equally probable.  m-ibutions, i.m ibution. listribution Jft en don'r ibution of cre has tiny me side or in which is fly assume t unci! it is  proximate difficult scussed in d be quite D it might able hand la. it's jusr e reflected  !l'e  Fm c of all, it's important to note that we only have a sample to work v.rith. As a result, there ;,'ill be uncertainty surrounding the value of his win rate. However, we know that the sampling disrribution of 16,900 hands of limit poker will be approximately nonnal because of the Central Limit Theorem. The observed standard deviation 5 = 2.1 BB/h is a reasonable ~rima te of the standard deviation of the population, particularly because the sample is :datively large. We can use these facts to obtain the maximum likelihood estimate of the ?Opulation mean.  Consider alI possible win rates. For each of these win rates, dlere will be a corresponding ~pling distribution, a nonnal curve v.rith a mean of Jl and a standard deviation crN. The ;ca-"'" of each of these normal curves will be at x = p, and alI the other points will be lower. ~ow suppose that we assume for a moment that the population mean is in fact some particular -. The height of the curve at x = X ",111 be associated with the probability that the observed s::.:nple mean would have been the result of the sample. We can find this value for alI possible -;-alues of p. Since all these nonnal curves have the same standard deviation crN, they will all x identical, but shifted along the X -axis, as in Figure 3.1. :35 r----,----,-----,----,----,-----,----,----, ';_3  2 25 : .2  ~.  Suppose we could r different Ie. bur we  .15  J.1  !pIe mean proach of s chapter, o nnation approach a t can be  it poker. s played, r J ~ 2.1 x and s, cr mix of u in the bout the ·PO KER  .0 5  J  ·3  -2  -1  o  2  3  4  5  Figure 3.1. Shifted normal distributions (labeled points at x = 1.15)  Since the peak of the (1ll\Te is the highest point, and the observed value x is the peak when ;J =x , this means that x = 1.15 BBIlDOh is the maximum likelihood estimate of the mean of the distribution. This may seem intuitive, but we VI'ill see when we consider a different approach to dUs problem that the maximum likelihood estimate does not always have to equal the sample :nean, if we incorporate additional information. Knowing that the single win rate that is most likely given the sample is the sample mean is a useful piece of data, but it doesn't help much with the uncertainty. After alI, our hypothetical player might have gotten lucky or unlucky. We can calculate the standard deviation of a sample of 16,900 hands and we can do some what-if analysis about possible win rates. TH E MATHEMATICS OF POKER  33  Part I: Basics Suppose we have a sample N that consists of 16,900 hands taken from an underlying distribution with a mean, or win rate, of 1.15 BB/100h and a standard deviation of 2.1 BBIh. Then, using equation 2.4:  aN = a-.fN a16,900  = (2.1 BB /h) N 16,900 hands)  a16,900 = 273 BB, so  aNI I00h = 2731169" 1. 61 The standard deviation of a sample of this size is greater than the win rate itself. Suppose that we knew that the parameters of the underlying distribution were the same as the observed ones. If we took another sample of 16,900 hands, 32% of the time, the observed Outcome of the 16,900 hand sample would be lower than ·0.46 BB/lOO or higher than 2.76 BB/I00. This is a litde troubling. How can we be confident in the idea that the sample represents the true population mean, when even if that were the case, another sample would be outside of even those fairl y wide bounds 32% of the time? And what if the true population mean were actually, say, zero? Then 1.15 would fall nicely into the one-sigma interval. In fact, it seems like we can't (ell the difference very clearly based on this sample between a win rate of zero and a win rate of 1.15 BB/lOO. What we can do to help to capture this uncertainty is create a conjidence interval To create a confidence imerval, we must first decide on a level of tolerance. Because we're dealing vvith statistical processes, we can't simply say that the probability that the population mean has some certain value is zero-we might have gotten extremely lucky or extremely unlucky. However, we can choose what is called a significance level. This is a probability value mat represents our tolerance for error. Then the confidence interval is the answer to the question, "'\tVhat al'e all the population mean values such that the probability of the observed outcome occurring is less than the chosen significance level?" Suppose that for our observed player, we choose a significance level of95%. Then we can find a confidence level for our player. If our population mean is Ji, then a sample of this size taken from this population will be between (p - 2a) and ( J1 + 2a) 95% of the time. So we can find all the values of Ji such that the observed value x = 1.15 is benveen these two boundaries. As we calculated above, the standard deviation of a sample of 16,900 hands is 1.61 units/100 hands:  aN = a-.fN a = (2.1 BB /h) (..J 16,900 ) a = 273 BB per 16,900 hands 01100h = 273 BB /169 = 1.61 So as long as the population mean satisfies the following confidence interval:  (p - 2<» (p  twO  equations, it will be within the  < 1.15  + 20) > 1.15  (1'- 20) < 1.15 p - (2)(1.61) < 1.15 p < 4.37 (p + 2a) > 1.15 p + (2)(1.61) > 1.15 J1 > -2.07 34  THE MATHEMATICS OF POKER  Part I: BasIcs Suppose we have a sample N that consists of 16,900 hands taken from an underlying distribution with a mean, or ",,>in rate, of 1.15 BB/ IOOh and a standard deviation of2.1 BBIh. Then, using equation 2.4:  aN ~ a..fJi a16,900 ~ (2,1 BB/h) (~ 16,900 hands) a16,900 ~ 273 BB, so  aN / l00h  ~  273/169  ~  1.61  The standard d eviation of a sample of this size is greater than the win rate itself. Suppose that  we knew that the parameters of the underlying distribution were the same as the observed ones. If we took another sample of 16,900 hands , 32% of the time, the observed outcome of the 16,900 hand sample would be lower than -0,46 BB/ l00 or higher than 2,76 BBIlOO, This is a little troubling. How can we be confident in the idea that the sample represents the true population mean, when even if that were the case, another sample would be outside of even those fairly wide bounds 32% of the time? And what if the true population mean were actually, say, zero? Then 1.15 would fall nicely into the one-sigma interval. In fact, it seems like we can't tell the difference very clearly based on this sample between a win rate of zero and a win rate of 1.15 BB/ IOO, ' -\That we can do to help to capture this uncertainty is create a confidence interval. To create a confidence interval, we must first decide on a level of tolerance. Because we're dealing with statistical processes, we can't simply say that the probability that the population mean has some certain value is zero- we might have gotten extremely lucky or extremely unlucky. However, we can choose what is called a significance level. Tills is a probability value that represents our tolerance for error. TIlen the confidence intelVal is the answer to the question, "What are all me population mean values such that the probability of the observed outcome occurring is less than the chosen significance level?" Suppose mat for our obselVed player, we choose a significance level of95%. Then we can find a confidence level for our player. If our population mean is 1', then a sample of this size taken from this population will be between ()I - 2u) and ()I + 2a) 95% of the rime, So we can find all the values of I' such that the obselVed value x = 1.15 is becween these two boundaries.  As we calculated above, the standard deviation of a sample of 16,900 hands is 1.61 units/100 hands:  a N ~ a..fJi a ~ (2, 1 BE /h ) (~ 1 6,900) (J ~ 273 BB per 16,900 hands a / IOOh ~ 273 BB/169 ~ 1.61 So as long as the population mean satisfies the follovving two equations, it will be within the confidence interval: ()I - 2a) < U5 ()I + 2a) > 1.15 ()I- 2a) < U5 )1 - (2)(1.61) 1.15 )I < 4,37 ()I + 2a) > 1.15 I' + (2)(1.61) > 1.15 I' > -2,07  <  34  THE MATHEMATICS OF POKER  Chapter 3-Usin g All the Information: Estimating Parameters and Bayes' Theorem  ng  So a 95% confidence interval for this player's win rate (based on tbe16,900 hand sample he  Ih.  has collected) is [-2.07 BBIJOO, 4.37 BB/IOO].  lat  ed of  he of Te  os ro  'a  as  y. at  n, Ie  .d n  d  10  This does not mean that his true rate is 95% likely to lie on this interval. TIlls is a conunon misunderstanding of the definition of confidence intervals. The confidence interval is all values that, if they were the true rate, then the observed rate would be inside the range of values that would occur 95% of the time. C lassical statistics doesn't make probability estimates of parameter values - in fact, the classical view is that the true win rate is either in the interval or it isn't, because it is not the result of a random event. No amount of sampling can make us sure or unsure as to what the parameter value is. Instead, we can only make claims abom the likelihood or unlikelihood thal we would have observed particulaJ outcomes if a parameter had a particular value. The maximum likelihood estimate and a confidence interval are useful tools for evaluating what information we can gain from a sample. In this case, even though a rate of 1.15 BBI100 might look reasonable, concluding that this rate is close to the true one is a bit premature. The confidence interval can give us an idea of how wide the range of possible population rates might be. However, if pressed, the best single estimate of the overall win rate is the maximum likelihood estimate, which is the sample mean of 1.15 BBI100 in this case. To this point, we have assumed that we had no information about the likelihood of different win rates - that is, that our eh-perimental evidence was the only source of data about what win rate a player might have. But in truth, some win rates are likelier than others, even before we take a measurement. Suppose that you played, say, 5,000 hands of casino poker and in those hands you won 500 big bets, a rate of 10 big bets per 100 hands . In this case, the maximum likelihood estimate from the last section would be that your \vID rate was exactly that - 10 big bets pcr 100 hands. But we do have other information. We have a fairly large amount of evidence, both anecdotal and culled from hand history databases and the like that indicates that among players who playa statistically significant number of hands, the highest win rates are near 3-4 BB/IOO. Even the few outliers who have win rates higher than this do not approach a rate of 10 BB/lOO. Since this is information that we have hefore we even start measuring anything, we call it a priori information.  In fact, if we just consider any particular player we measure to be randomly selected from the universe of all poker players, there is a probability distribution associated with his win rate. We don't know the precise shape of this distribution, of course, because we lack. observed evidence of the entire universe of poker players. However, if we can make correct assumptions about the underlying a priori distribution of win rates, we can make better estimates of the parameters based on the observed evidence by combining the two sets of evidence.  e  Bayes' theorem In Chapter 2, we stated the basic principle of probability (equation 1.5). P (A  n B ) ~ P(A )P(B I A )  In this form, this equation allows us to calculate the joint probability of A and B from the probability of A and the conditional probability of B given A. However, in poker, we are often most concerned with calculating the conditional probability of B given that A has already occurred - for example, we know the cards in OUT Q\o'l'n hand (A ), and now we want to know THE MATHEMATICS OF POKER  35  Part I: Basics  how this infonnation affecrs the cards in our opponents hand (B). What we are looking for is the conditional probability of B given A. So we can reorganize equation 1.5 to create a formula for conditional probability. This is the equation we will refer to as Bayes ' theorem:  (BIA) = P(A n B) P(A) p  (3.1)  Recall that we defined B as the complement of B in Chapter 1; that is: P(B) = 1- P(B) P(B)t P(B) = 1 We already have the definitions: P(A  n B) =  P(A)P(B I A)  Since we know that B and B sum to 1, P(A) can be expressed as the probability of A given B when B occurs, plus me probability of A given B when B occurs. So we can restate equation 3.1 as: (BIA )= p(AIB)P(B) P(A IB)P(B) + P(A IB)P(B) P  (3.2)  In poker, Bayes' theorem allows us to refine our j udgments about probabilities based on new information that we observe. In fact, strong players use Bayes' theorem constandy as new infonnation appears to continually refine their probability assessments; the process of Bayesian inference is at the heare of reading hands and exploitive play, as we shall see in Part II.  A classic example of Bayes' theorem comes from the medical profession. Suppose that we have a screening process for a particular condition. If an iIl{liviuual wilh the condition is screened, the screening process properly identifies the condition 80% of the time. If an individual without the condition is screened, the screening process improperly identifies him as having the condition 10% of the time. 5% of the population (on average) has the condition. Suppose, then, that a person selected randomly from the population is screened, and the screening returns a positive result. What is the probability that this person has the condition (absent further screening or diagnosis)? If you answered "about or a little less than 800/0," you were wrong, but not alone. Studies of doctors' responses to questions such as these have shown a perhaps frightening lack of understanding of Bayesian concepts. We can use Bayes' theorem to find the answer to this problem as follows: A = the screening process reUlms a positive result. B = the patient has the condition.  36  TH E MATHEMATICS OF POKER  Chapter 3-Using All the Information: Estimating Parameters and Bayes' Theorem  ;for  Then we are looking for the probability P{B IA) and the following are UUe: P(A I B) ~ 0.8 (if the patient has the condition, the screening will be positive 80% of the time) P(A IE) ~ 0.1 (if the patient doesn't have the condition, the result will be positive 10% of the time) p(B) ~ 0.05 (5% of all people have the condition) pre) ~ 0.95 (95% of all people don't have the condition)  .-\nd using Equation 3.2, we have:  P(A IB)p(B) I )~ p(AIB)P(B)+P(A IB )p(B)  B A ~ P(  (B I A) ~  nB  (0.8)(0.05)  ~ (0.8)(0.05)+(0.1)(0.95)  P  P(B IA) ~ 29.63% .As you can see, the probability mac a patiem with a positive screening result acrually has the condition is much lower than the 80% "accuracy" the test has in identifying the condition in someone who has it. Testing.vith this screening process would        

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