The Mathematics Of Poker
Posted : admin On 4/10/2022- The Mathematics Of Poker Download
- The Mathematics Of Poker Bill Chen
- Texas Hold Em Math
- The Mathematics Of Poker Epub
- The Mathematics Of Poker Pdf
- The Mathematics Of Poker Review
This book provides an introduction to quantitative techniques as applied to poker and to a branch of mathematics that is particularly applicable to poker, game theory, in a manner that makes seemingly difficult topics accessible to players without a strong mathematical background. Many poker books overlap eachother, reminding pot odds, hand values, tournament phases, etc. This book dives into the fundamental theory. The interesting math of poker is not related with mundane matters of probabilities, pot odds, etc. The interesting math is the math behind bluffing, calling, and value-playing.
Download Book The Mathematics Of Poker in PDF format. You can Read Online The Mathematics Of Poker here in PDF, EPUB, Mobi or Docx formats.
The Mathematics Of Poker
Author :Bill ChenISBN :1886070253
Genre :Games
File Size : 56.71 MB
Format :PDF, Mobi
Download :997
Read :973
For decades, the highest level 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 has risen to prominence within the poker community. Applying the tools of computer science and mathematics to poker and sharing the information across the Internet, these players have challenged many of the assumptions that underlay traditional approaches to the game.
Killer Poker By The Numbers
Author :Tony GuerreraISBN :081840714X
Genre :Games & Activities
File Size : 23.49 MB
Format :PDF, ePub, Mobi
Download :234
Read :1017
The first and only poker book to thoroughly cover the mathematical concepts behind every hand of poker and to do it in an easily accessible manner, Killer Poker By The Numbers is sure to appeal to all poker fans, epsecially fans of the Killer Poker series--from amateurs to advanced players. At its root poker is a numbers game, and this latest installment in John Vorhaus's bestselling Killer Poker series completely demystifies the math used to make winning decisions, including: - How blind structures and table sizes can affect play - How to determine strategy and profitability for various table types - How to work with probability, expected value, pot odds, and variance Killer Poker By The Numbers will not only benefit from John Vorhaus's distinct brand and loyal group of readers, and but also because: - Guerrera is one of the leading experts on the mathematics of poker. His experience studying physics at Cal Tech, along with his experience as a math tutor, make him the perfect author for this book. - Guerrera contributed a chapter to John Vorhaus's recently published Killer Poker Online 2 (Lyle Stuart, September 2006). - John Vorhaus will write the foreword, as well as market this book alongside all his other Killer Poker books. - Killer Poker By The Numbers will be promoted on www.killerpokerbythenumbers.com.
Math Of Poker Basic
Author :Poonam PatelISBN :9798686402812
Genre :
File Size : 38.42 MB
The Mathematics Of Poker Download
Format :PDF, KindleDownload :204
Read :347
Poker math is a vitally important aspect to no-limit hold'em poker, but it's often overlooked or simply not used because many poker players fear it's too difficult to learn. I'm here to tell you it is not. In fact, fundamental poker math is very easy to learn. More importantly, it can yield you a lot more profits at the poker table. In this book, I'll teach you how to use simple math at the poker table to gain a huge skill advantage over your opponents that'll allow you to win more and lose less, you'll be introduced to several fundamental overarching poker concepts that apply to poker math. Then we'll begin our journey into poker math where you'll learn about fundamental poker mathematics, including probabilities and odds, pot equity, pot odds, implied odds, the rule of two and four, expected value (EV), and much more. We'll then embark on a journey of learning about important pre-flop and post-flop poker mathematical concepts, such as pre-flop all-in situations, set-mining, steal attempts, three-bet bluffing, betting with the best hand, semi-bluffing all-in, bluffs and hero calls, you'll learn how to perform basic and intermediate expected value calculations and utilize card combinations, better known as combinatorics.
Poker Math That Matters
Author :Owen GainesISBN :061539745X
Genre :
File Size : 69.32 MB
Format :PDF, Kindle
Download :783
Read :1225
Learning the math behind good poker has never been simpler. Owen's methods and shortcuts will have you applying the math at the poker table overnight.
Probability Guide To Gambling
Author :Catalin BarboianuISBN :9789738752030
Genre :Games
File Size : 39.4 MB
Format :PDF, Kindle
Download :777
Read :912
Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability (definitions, properties, theorems and calculus formulas), combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one (like many other titles published on this subject), but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.'
Draw Poker Odds
Author :Catalin BarboianuISBN :9789738752054
Genre :Games
File Size : 34.13 MB
Format :PDF, ePub
Download :774
Read :460
Like any variation of poker, draw poker (or classical poker) is also predisposed to probability-based decisions. The author presents the mathematics involved in this card game, with respect to the usage of the numerical results in players' strategies. The whole presentation is focused on the practical aspect of the application of probability theory in draw poker and all the sections are such structured to allow the direct usage of the numerical results. This is why every section is packed with tables, some of them filling dozens of pages. This is not a math book, even if the supporting mathematics is present thorough, but a guide addressed to poker players, who can skip the math parts at any time and pick the needed results from tables. For those interested, the complete methodology, the way probability theory is applied and a part of the calculations are shown, so it teaches the player how to calculate odds for any situation for every stage of the game, even the numerical results are already listed in the book. Want to evaluate the probability of one opponent bluffing? Want to know the probability of at least one opponent holding a card formation higher than yours, at any moment of the game? Want to know the probability of hitting the desired formation if discarding in a certain way? All this information is in the book and is fully mathematically grounded. All probability results from this guide are obtained through compact mathematical formulas and not partial simulations on computer. These formulas are the outcome of one year of study, math work and tests. The author found the right probability model in which to apply the theory and conveniently quantify the card distributions inorder to work out the draw poker probability formulas. They were built with an enough large range of variables to cover all possible situations and were never worked out before. Their numerical returns were gathered in three main categories of odds presented in the book: - Initial probabilities of the first card distribution for your own hand; - Prediction probabilities after first card distribution and before the second for your own hand; - Prediction probabilities for opponents' hands. Every section ends with suggestive examples and there is also a special chapter with a lot of relevant gaming situations presented along with the odds of their associated events. Among author's previously published books on mathematics of gambling, Draw Poker Odds seems to be the most practical one and that is because the author presents the results of applied probability in a gambling-behavioral manner that can influence the balance between the subjective strategies and the real odds in player's favor.
Essential Poker Math
Author :Alton HardinISBN :1516944518
Genre :
File Size : 75.82 MB
Format :PDF, ePub, Docs
Download :376
Read :1045
Poker Math Is Easy to Learn Poker math is a vitally important aspect to No Limit Holdem poker, but it is often overlooked or simply not used because many poker players fear it is too difficult to learn. I'm here to tell you it is not. In fact, fundamental poker math is very easy to learn. More importantly, it can yield you a lot more profits at the poker table. Without using simple math at the poker table, you are simply playing a guessing game. Use Simple Math at the Poker Table and Increase You Winnings In this book I will teach you how to use simple arithmetic at the poker table to gain a huge skill advantage over your opponents that will allow you to win more and lose less. Poker players that don't use math are simply guessing and you'll learn to no longer guess and know the right mathematical move at the poker table. These simple mathematical concepts I will be teaching you will drastically help improve your poker game and allow you to make the most profitable decisions at the poker. Contents and Overview First you will be introduced to some fundamental overarching poker concepts that apply to poker mathematics. Then we will begin our journey into poker mathematics where you will learn about probabilities and odds, pot odds and implied odds, pot equity, and expected value. You will then learn how to quickly estimate your equity at the poker table using the Rule of 2 & 4. Moreover, you'll learn the steps involved in determining if calling with a drawing hand is profitable or not. We will also cover how to size your bets with the best hand and teach you how often bluffs and hero calls need to work to be profitable. Lastly, we will show you how to perform EV calculations and better understand card combinations. Effectively Understand and Utilize Essential Poker Math Develop a keen understanding of Probability and Odds Learn to quickly calculate Pot Odds & Implied Odds at the poker table Effectively use Pot Equity & The Rule of 2 & 4 to Determine the correct poker play Understand how to use Expected Value (EV) both on and off the table to analyze your plays Learn the important math behind Bluffs & Hero Calls to give you a skill advantage over your opponent Learn Card Combinations to further enhance your card reading abilities And Many More Amazing Topics... What You Will Get out of This Book Suitable for both beginning and experienced poker players alike you'll learn many essential fundamental poker mathematical concepts that will help you drastically improve your poker game. After reading this book, you will have mastered fundamental No Limit Holdem mathematics. You will have gained a huge skill advantage over your opponents and you will be able to quickly and effectively use math at the poker table to make are always the most profitable move. Most importantly, you will become a much better and profitable poker player! So what are you waiting for? Purchase this book today to start learning how to advance your poker game with simple poker math!
Texas Hold Em Poker Odds For Your Strategy With Probability Based Hand Analyses
Author :Catalin BarboianuISBN :9789731991344
Genre :Games
File Size : 60.48 MB
Format :PDF, Docs
Download :565
Read :825
A complete probability guide of Hold'em Poker, this guide covers all possible gaming situations. The author focuses on the practical side of the presentation and use of the probabilities involved in Hold'em, while taking into account the subjective side of the probability-based criteria of each player's strategy.
The Math Of Hold Em
Author :Collin MoshmanISBN :0984619429
Genre :Games
File Size : 60.26 MB
Format :PDF
Download :795
Read :1026
Winning big at hold'em requires practical math knowledge, which is precisely what this book will teach you. Both of the authors of this book graduated with degrees in Mathematics from Caltech. Collin Moshman is the best-selling author of Sit 'n Go Strategy and Heads-Up No-Limit Hold'em. Douglas Zare is a mathematician and poker coach famous for his in-depth analyses. The authors use their extensive math, poker, and coaching backgrounds to present the most important hold'em math concepts designed to immediately increase your winrate at the tables. You will learn to: Understand the fundamental probabilities and statistics underlying Hold'em Count hand combinations while playing Master expected value calculations Play optimally with game theory Seize full advantage of the latest software Utilize deductive logic to put your opponent on a specific hand range And more
Why Alex Beats Bobbie At Poker
Author :Duncan PalamourdasISBN :9781912862221
Genre :Games & Activities
File Size : 65.83 MB
Format :PDF, ePub, Mobi
Download :140
Read :906
Meet Alex and Bobbie, who both like to play poker. Alex is a professional poker player who plays for a living and is a solid long-term winner. Bobbie is a recreational player who plays a decent enough game but mainly wants to have a good time. If you play poker regularly you will meet thousands of players like Bobbie in your games and very few like Alex. Of course one would expect that, in the long run, Alex will perform better than Bobbie. But have you ever wondered EXACTLY what it is that Alex understands better and does differently to Bobbie? This is a rather complex question that does not have a simple answer. In this book, UCLA maths professor Duncan Palamourdas addresses this question via a journey through human psychology, game theory, easy-to-understand mathematics and even philosophy. Topics include: * Understanding the instinctive but unprofitable tendencies of inexperienced players. * How to identify what a mistake actually is in poker – and how to exploit it. * Why poker does not revolve around bluffing. * The great impact of variance in poker and how to account for it. * How to develop a consistent approach that allows you to play like Alex and not Bobbie.
The Mathematics Of Various Entertaining Subjects
Author :Jennifer BeinekeISBN :9780691183473
Genre :Mathematics
File Size : 78.45 MB
Format :PDF, ePub, Docs
Download :740
Read :747
The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, and surreal numbers were inspired by the game of Go. Yet even with such groundbreaking findings and a wealth of popular-level books exploring puzzles and brainteasers, research in recreational mathematics has often been neglected. The Mathematics of Various Entertaining Subjects brings together authors from a variety of specialties to present fascinating problems and solutions in recreational mathematics. Contributors to the book show how sophisticated mathematics can help construct mazes that look like famous people, how the analysis of crossword puzzles has much in common with understanding epidemics, and how the theory of electrical circuits is useful in understanding the classic Towers of Hanoi puzzle. The card game SET is related to the theory of error-correcting codes, and simple tic-tac-toe takes on a new life when played on an affine plane. Inspirations for the book's wealth of problems include board games, card tricks, fake coins, flexagons, pencil puzzles, poker, and so much more. Looking at a plethora of eclectic games and puzzles, The Mathematics of Various Entertaining Subjects is sure to entertain, challenge, and inspire academic mathematicians and avid math enthusiasts alike.
Essential Poker Math Expanded Edition
Author :Alton HardinISBN :0998294500
Genre :
File Size : 39.98 MB
Format :PDF, Mobi
Download :664
Read :1112
Expanded Edition Just Released! Expanded, Updated & Greatly Improved New Expanded Edition Includes: More Than 150 Pages of New Material with Greatly Expanded Content and 4 New Chapters Over 75 Carefully Devised Practice and Example Poker Hands That Are Analyzed in Great Detail Free Enrollment into My Essential Poker Math eLearning Online Video Training Course Poker Math Is Easy to Learn Poker math is a vitally important aspect to No Limit Holdem poker, but it's often overlooked or simply not used because many poker players fear it is too difficult to learn. I'm here to tell you it is not. In fact, fundamental poker math is very easy to learn. More importantly, it can yield you a lot more profits at the poker table. Without using simple math at the poker table, you are simply playing a guessing game. Use Simple Math at the Poker Table & Increase You Winnings In this book I'll teach you how to use simple arithmetic at the poker table to gain a huge skill advantage over your opponents that will allow you to win more and lose less. Poker players that don't use math are simply guessing and you'll learn to no longer guess and know the correct mathematical move at the poker table. These simple mathematical concepts I'll be teaching you will drastically help improve your poker game and allow you to make the most profitable decisions at the poker. Contents & Overview First you'll be introduced to several fundamental overarching poker concepts that apply to poker mathematics. Then we'll begin our journey into poker mathematics where you'll learn about fundamental poker mathematics, including probabilities and odds, pot equity, pot odds, implied odds, the Rule of 2 and 4, expected value (EV) and much more. We'll then embark on a journey of learning about important pre-flop and post-flop poker mathematical concepts, such as pre-flop all-in situations, set-mining, steal attempts, 3-bet bluffing, betting with the best hand, semi-bluffing all-in, bluffs and hero calls. Lastly, you'll learn how to perform basic and intermediate expected value calculations and utilize card combinations, better known as combinatorics. Effectively Understand & Utilize Essential Poker Math Develop a keen understanding of Probability & Odds Learn to quickly calculate Pot Odds & Implied Odds at the poker table Effectively use Pot Equity & The Rule of 2 & 4 to Determine the correct poker play Understand how to use Expected Value (EV) both on & off the table to analyze your plays Understand how to properly Set-Mine, Steal & 3-Bet Bluff with basic mathematics Learn the important math behind Bluffs & Hero Calls to give you a skill advantage over your opponent Learn to utilize EV Calculations to analyze your previous play off the table Learn to leverage Card Combinations to further enhance your card reading abilities & develop balanced ranges What You'll Get out of This Book Suitable for both beginning and experienced poker players alike you'll learn many essential fundamental poker mathematical concepts that'll help you drastically improve your poker game. After reading this book, you'll have mastered fundamental No Limit Holdem mathematics. You'll have gained a huge skill advantage over your opponents and will be able to effectively use math at the poker table to make profitable moves. Most importantly, you'll become a much better and profitable poker player! What are you waiting for? Purchase this book today to start advancing your poker game with simple poker math!
Ace High Mastering Low Stakes Poker Cash Games
Author :Nathan SchwiethaleISBN :9781257798582
Genre :
File Size : 81.46 MB
Format :PDF
Download :347
Read :728
The Mathematics Of Games And Gambling
Author :Edward PackelISBN :088385628X
Genre :Mathematics
File Size : 48.19 MB
Format :PDF, ePub, Mobi
Download :431
Read :212
This book shows you what determines the best bet in a game of chance or the optimal strategy in a strategic game.
Poker Tournament Formula 2 Advanced Strategies
Author :Arnold SnyderISBN :9781580425049
Genre :Games & Activities
File Size : 45.45 MB
Format :PDF, ePub, Docs
Download :868
Read :312
Snyder adapts the loose aggressive fast tournament strategies of his groundbreaking first book to the big buy-in events where the real money is made. Players learn never-before-revealed concepts and secrets that shows players why cards don't matter as much as the dynamics of a tournament. Readers learn how to alter their strategy for any tournament structure and opponent, why hands must be played differently from cash games, and why players can't figure out what winners are doing just from watching them play. The book also covers optimal satellite strategy, sit'n'go strategy, methods for estimating tournament win rate and edge
The Life And Times Of A Gentleman Gambler
Author :H. D. HaleISBN :1884466117
Genre :Casinos
File Size : 61.10 MB
Format :PDF, Kindle
Download :523
Read :276
Poker
Author :David A. DanielISBN :1569800936
Genre :Games & Activities
File Size : 22.25 MB
Format :PDF, Kindle
Download :270
Read :293
Describes the varieties of poker games, offers advice for assessing an opponent's skills, explains when to bluff and how to tell when someone else is bluffing, and provides tips for betting money
The Theory Of Poker
Author :David SklanskyISBN :1880685000
Genre :Games & Activities
File Size : 86.97 MB
Format :PDF, Mobi
Download :447
Read :532
'Now, for the millions of poker players who know the basics, but do not fully understand the logic and principles of skillful play, here is a serious, comprehensive guide that shows how to think like a professional poker player'--Page 4 of cover.
The Big Book Of Poker
Author :Ken WarrenISBN :158042113X
Genre :Games & Activities
File Size : 55.59 MB
Format :PDF, Docs
Download :179
Read :1095
Best-selling author and poker expert Ken Warren takes you from the basics of poker to the ins and outs of playing and winning at every type of game -- cardrooms, casinos, home games, and even on the internet! Learn how to compete in local tournaments and multi-million dollar televised events, including the legendary World Series of Poker. You'll also learn how to play hold'em in all its forms, including the exciting no-limit games. In these pages you'll find valuable information on how to bet, raise, checkraise, bluff, semi-bluff, and how to take advantage of position and pot odds. Warren teaches you how to read a tell and manage your bankroll. He also teaches you the lingo: this book is packed with poker slang -- common and obscure -- and its definitions. Whether you're a pro, a newbie, or somewhere in between, The Big Book of Poker will help you cash in on pots and hold your own against the toughest players around. Filled with charts, detailed diagrams, easy-to-read examples, and all sorts of color about the game, this book shows you everything you need to know to play and win at poker. Book jacket.
The Mathematics Of Secrets
Author :Joshua HoldenThe Mathematics Of Poker Bill Chen
ISBN :9780691183312Genre :Computers
File Size : 84.64 MB
Format :PDF, ePub
Download :766
Read :155
The Mathematics of Secrets takes readers on a fascinating tour of the mathematics behind cryptography—the science of sending secret messages. Using a wide range of historical anecdotes and real-world examples, Joshua Holden shows how mathematical principles underpin the ways that different codes and ciphers work. He focuses on both code making and code breaking and discusses most of the ancient and modern ciphers that are currently known. He begins by looking at substitution ciphers, and then discusses how to introduce flexibility and additional notation. Holden goes on to explore polyalphabetic substitution ciphers, transposition ciphers, connections between ciphers and computer encryption, stream ciphers, public-key ciphers, and ciphers involving exponentiation. He concludes by looking at the future of ciphers and where cryptography might be headed. The Mathematics of Secrets reveals the mathematics working stealthily in the science of coded messages. A blog describing new developments and historical discoveries in cryptography related to the material in this book is accessible at http://press.princeton.edu/titles/10826.html.
Top Download:
This is a very important lesson and can also be quite intimidating to a lot of people as we are going to discuss Poker Math!
But there is no need for you to be intimidated, Poker Maths is very simple and we will show you a very simple method in this lesson.
You won’t need to carry a calculator around with you or perform any complex mathematical calculations.
What is Poker Math?
As daunting as it sounds, it is simply a tool that we use during the decision making process to calculate the Pot Odds in Poker and the chances of us winning the pot.
Remember, Poker is not based on pure luck, it is a game of probabilities, there are a certain number of cards in the deck and a certain probability that outcomes will occur. So we can use this in our decision making process.
Every time we make a decision in Poker it is a mathematical gamble, what we have to make sure is that we only take the gamble when the odds are on in our favour. As long as we do this, in the long term we will always come out on top.
When to Use Poker Maths
Poker Maths is mainly used when we need to hit a card in order to make our hand into a winning hand, and we have to decide whether it is worth carrying on and chasing that card.
To make this decision we consider two elements:
- How many “Outs” we have (Cards that will make us a winning hand) and how likely it is that an Out will be dealt.
- What are our “Pot Odds” – How much money will we win in return for us taking the gamble that our Out will be dealt
We then compare the likelihood of us hitting one of our Outs against the Pot Odds we are getting for our bet and see if mathematically it is a good bet.
The best way to understand and explain this is by using a hand walk through, looking at each element individually first, then we’ll bring it all together in order to make a decision on whether we should call the bet.
Texas Hold Em Math
Consider the following situation where you hold A 8 in the big blind. Before the flop everyone folds round to the small blind who calls the extra 5c, to make the Total pot before the Flop 20c (2 players x 10c). The flop comes down K 9 4 and your opponent bets 10c. Let’s use Poker Math to make the decision on whether to call or not.
Poker Outs
When we are counting the number of “Outs” we have, we are looking at how many cards still remain in the deck that could come on the turn or river which we think will make our hand into the winning hand.
In our example hand you have a flush draw needing only one more Club to make the Nut Flush (highest possible). You also hold an overcard, meaning that if you pair your Ace then you would beat anyone who has already hit a single pair on the flop.
From the looks of that flop we can confidently assume that if you complete your Flush or Pair your Ace then you will hold the leading hand. So how many cards are left in the deck that can turn our hand into the leading hand?
- Flush – There are a total of 13 clubs in the deck, of which we can see 4 clubs already (2 in our hand and 2 on the flop) that means there are a further 9 club cards that we cannot see, so we have 9 Outs here.
- Ace Pair – There are 4 Ace’s in the deck of which we are holding one in our hand, so that leaves a further 3 Aces that we haven’t seen yet, so this creates a further 3 Outs.
So we have 9 outs that will give us a flush and a further 3 outs that will give us Top Pair, so we have a total of 12 outs that we think will give us the winning hand.
So what is the likelihood of one of those 12 outs coming on the Turn or River?
Professor’s Rule of 4 and 2
An easy and quick way to calculate this is by using the Professor’s rule of 4 and 2. This way we can forget about complex calculations and quickly calculate the probability of hitting one of our outs.
The Professor’s Rule of 4 and 2
- After the Flop (2 cards still to come… Turn + River)
Probability we will hit our Outs = Number of Outs x 4 - After the Turn (1 card to come.. River)
Probability we will hit our Outs – Number of Outs x 2
So after the flop we have 12 outs which using the Rule of 4 and 2 we can calculate very quickly that the probability of hitting one of our outs is 12 x 4 = 48%. The exact % actually works out to 46.7%, but the rule of 4 and 2 gives us a close enough answer for the purposes we need it for.
If we don’t hit one of our Outs on the Turn then with only the River left to come the probability that we will hit one of our 12 Outs drops to 12 x 2 = 24% (again the exact % works out at 27.3%)
To compare this to the exact percentages lets take a look at our poker outs chart:
| After the Flop (2 Cards to Come) | After the Turn (1 Card to Come) | ||||
|---|---|---|---|---|---|
| Outs | Rule of 4 | Exact % | Outs | Rule of 2 | Exact % |
| 1 | 4 % | 4.5 % | 1 | 2 % | 2.3 % |
| 2 | 8 % | 8.8 % | 2 | 4 % | 4.5 % |
| 3 | 12 % | 13.0 % | 3 | 6 % | 6.8 % |
| 4 | 16 % | 17.2 % | 4 | 8 % | 9.1 % |
| 5 | 20 % | 21.2 % | 5 | 10 % | 11.4 % |
| 6 | 24 % | 25.2 % | 6 | 12 % | 13.6 % |
| 7 | 28 % | 29.0 % | 7 | 14 % | 15.9 % |
| 8 | 32 % | 32.7 % | 8 | 16 % | 18.2 % |
| 9 | 36 % | 36.4 % | 9 | 18 % | 20.5 % |
| 10 | 40 % | 39.9 % | 10 | 20 % | 22.7 % |
| 11 | 44 % | 43.3 % | 11 | 22 % | 25.0 % |
| 12 | 48 % | 46.7 % | 12 | 24 % | 27.3 % |
| 13 | 52 % | 49.9 % | 13 | 26 % | 29.5 % |
| 14 | 56 % | 53.0 % | 14 | 28 % | 31.8 % |
| 15 | 60 % | 56.1 % | 15 | 30 % | 34.1 % |
| 16 | 64 % | 59.0 % | 16 | 32 % | 36.4 % |
| 17 | 68 % | 61.8 % | 17 | 34 % | 38.6 % |
As you can see the Rule of 4 and 2 does not give us the exact %, but it is pretty close and a nice quick and easy way to do the math in your head.
Now lets summarise what we have calculated so far:
- We estimate that to win the hand you have 12 Outs
- We have calculated that after the flop with 2 cards still to come there is approximately a 48% chance you will hit one of your outs.
Now we know the Odds of us winning, we need to look at the return we will get for our gamble, or in other words the Pot Odds.
Pot Odds
When we calculate the Pot Odds we are simply looking to see how much money we will win in return for our bet. Again it’s a very simple calculation…
Pot Odds Formula
Pot Odds = Total Pot divided by the Bet I would have to call
What are the pot odds after the flop with our opponent having bet 10c?
- Total Pot = 20c + 10c bet = 30 cents
- Total Bet I would have to make = 10 cents
- Therefore the pot odds are 30 cents divided by 10 cents or 3 to 1.
What does this mean? It means that in order to break even we would need to win once for every 3 times we lose. The amount we would win would be the Total Pot + the bet we make = 30 cents + 10 cents = 40 cents.
| Bet number | Outcome | Stake | Winnings |
|---|---|---|---|
| 1 | LOSE | 10 cents | Nil |
| 2 | LOSE | 10 cents | Nil |
| 3 | LOSE | 10 cents | Nil |
| 4 | WIN | 10 cents | 40 cents |
| TOTAL | BREAKEVEN | 40 cents | 40 cents |
Break Even Percentage
Now that we have worked out the Pot Odds we need to convert this into a Break Even Percentage so that we can use it to make our decision. Again it’s another simple calculation that you can do in your head.
Break Even Percentage
Break Even Percentage = 100% divided by (Pot odds added together)
Let me explain a bit further. Pot Odds added together means replace the “to” with a plus sign eg: 3 to 1 becomes 3+1 = 4. So in the example above our pot odds are 3 to 1 so our Break Even Percentage = 100% divided by 4 = 25%
Note – This only works if you express your pot odds against a factor of 1 eg: “3 to 1” or “5 to 1” etc. It will not work if you express the pot odds as any other factor eg: 3 to 2 etc.
So… Should You call?
So lets bring the two elements together in our example hand and see how we can use the new poker math techniques you have learned to arrive at a decision of whether to continue in the hand or whether to fold.
To do this we compare the percentage probability that we are going to hit one of our Outs and win the hand, with the Break Even Percentage.
Should I Call?
- Call if…… Probability of Hitting an Out is greater than Pot Odds Break Even Percentage
- Fold if…… Probability of Hitting an Out is less than Pot Odds Break Even Percentage
Our calculations above were as follows:
- Probability of Hitting an Out = 48%
- Break Even Percentage = 25%
If our Probability of hitting an out is higher than the Break Even percentage then this represents a good bet – the odds are in our favour. Why? Because what we are saying above is that we are going to get the winning hand 48% of the time, yet in order to break even we only need to hit the winning hand 25% of the time, so over the long run making this bet will be profitable because we will win the hand more times that we need to in order to just break even.
Hand Walk Through #2
Lets look at another hand example to see poker mathematics in action again.
Before the Flop:
- Blinds: 5 cents / 10 cents
- Your Position: Big Blind
- Your Hand: K 10
- Before Flop Action: Everyone folds to the dealer who calls and the small blind calls, you check.
Two people have called and per the Starting hand chart you should just check here, so the Total Pot before the flop = 30 cents.
Flop comes down Q J 6 and the Dealer bets 10c, the small blind folds.
Do we call? Lets go through the thought process:
The Mathematics Of Poker Epub
How has the Flop helped my hand?
It hasn’t but we do have some draws as we have an open ended straight draw (any Ace or 9 will give us a straight) We also have an overcard with the King.
How has the Flop helped my opponent?
The Dealer did not raise before the flop so it is unlikely he is holding a really strong hand. He may have limped in with high cards or suited connectors. At this stage our best guess is to assume that he has hit top pair and holds a pair of Queens. It’s possible that he hit 2 pair with Q J or he holds a small pair like 6’s and now has a set, but we come to the conclusion that this is unlikely.
How many Outs do we have?
So we conclude that we are facing top pair, in which case we need to hit our straight or a King to make top pair to hold the winning hand.
- Open Ended Straight Draw = 8 Outs (4 Aces and 4 Nines)
- King Top Pair = 3 Outs (4 Kings less the King in our hand)
- Total Outs = 11 Probability of Winning = 11 x 4 = 44%
What are the Pot Odds?
Total Pot is now 40 cents and we are asked to call 10 cents so our Pot odds are 4 to 1 and our break even % = 100% divided by 5 = 20%.
Decision
So now we have quickly run the numbers it is clear that this is a good bet for us (44% vs 20%), and we make the call – Total Pot now equals 50 cents.
Turn Card
Turn Card = 3 and our opponent makes a bet of 25 cents.
After the Turn Card
This card has not helped us and it is unlikely that it has helped our opponent, so at this point we still estimate that our opponent is still in the lead with top pair.
Outs
We still need to hit one of our 11 Outs and now with only the River card to come our Probability of Winning has reduced and is now = 11 x 2 = 22%
Pot Odds
The Total Pot is now 75 cents and our Pot odds are 75 divided by 25 = 3 to 1. This makes our Break Even percentage = 100% divided by 4 = 25%
Decision
So now we have the situation where our probability of winning is less than the break even percentage and so at this point we would fold, even though it is a close call.
The Mathematics Of Poker Pdf
Summary
Well that was a very heavy lesson, but I hope you can see how Poker Maths doesn’t have to be intimidating, and really they are just some simple calculations that you can do in your head. The numbers never lie, and you can use them to make decisions very easy in Poker.
The Mathematics Of Poker Review
You’ve learnt some important new skills and it’s time to practise them and get back to the tables with the next stage of the Poker Bankroll Challenge.
Poker Bankroll Challenge: Stage 3
- Stakes: $0.02/$0.04
- Buy In: $3 (75 x BB)
- Starting Bankroll: $34
- Target: $9 (3 x Buy In)
- Finishing Bankroll: $43
- Estimated Sessions: 3
Use this exercise to start to consider your Outs and Pot Odds in your decision making process, and add this tool to the other tools you have already put into practice such as the starting hands chart.