## Mathematics Of Poker

**Mathematics Of Poker**

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## The Mathematics Of Poker

**Author :**Bill Chen

The Mathematics Of Poker Bill Chen Download at Aboutslots.com are not responsible for any losses from gambling in casinos linked to any of our bonus offers. The player is responsible for how much the person is willing and able to play for. We always urge a use of responsible gambling. Math of Poker - Basics The game of poker is a card game played among two or more players for several rounds.

The Mathematics of Poker. Alvaro Palencias. Download Full PDF Package. A short summary of this paper. 0 Full PDFs related to this paper. The Mathematics of Poker. The Mathematics of Poker. One of the most important features of this new approach is a reliance on quantitative analysis and the application of mathematics to the game. THE MATHEMATICS OF POKER 5 Acknowledgments A book like this is rarely the work of simply the authors; many people have assisted us along the way, both in understanding poker and in the specific task of writing down many of the ideas that wc have developed over the last few years.

**ISBN :**1886070253

**Genre :**Games

**File Size :**27.92 MB

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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 Guerrera

**ISBN :**081840714X

**Genre :**Games & Activities

**File Size :**69.36 MB

**Format :**PDF, Kindle

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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 Patel

**ISBN :**9798686402812

**Genre :**

**File Size :**89.78 MB

## Math Behind Poker

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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 Gaines

**ISBN :**061539745X

**Genre :**

**File Size :**29.52 MB

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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 Barboianu

**ISBN :**9789738752030

**Genre :**Games

**File Size :**

## Mathematics Of Poker Pdf

22.1 MB**Format :**PDF, Docs

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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 Barboianu

**ISBN :**9789738752054

**Genre :**Games

**File Size :**51.1 MB

**Format :**PDF, ePub, Mobi

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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 Hardin

**ISBN :**1516944518

**Genre :**

**File Size :**32.15 MB

**Format :**PDF, Docs

**Download :**931

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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 Barboianu

**ISBN :**9789731991344

**Genre :**Games

**File Size :**71.89 MB

**Format :**PDF, Mobi

**Download :**716

**Read :**850

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 Moshman

**ISBN :**0984619429

**Genre :**Games

**File Size :**36.33 MB

**Format :**PDF, ePub, Mobi

**Download :**926

**Read :**1178

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 Palamourdas

**ISBN :**9781912862221

**Genre :**Games & Activities

**File Size :**80.76 MB

**Format :**PDF, Kindle

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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 Beineke

**ISBN :**9780691183473

**Genre :**Mathematics

**File Size :**28.76 MB

**Format :**PDF, ePub, Mobi

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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 Hardin

**ISBN :**0998294500

**Genre :**

**File Size :**59.36 MB

**Format :**PDF, ePub, Docs

**Download :**612

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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 Schwiethale

**ISBN :**9781257798582

**Genre :**

**File Size :**87.56 MB

**Format :**PDF, Kindle

**Download :**230

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## The Mathematics Of Games And Gambling

**Author :**Edward Packel

**ISBN :**088385628X

**Genre :**Mathematics

**File Size :**40.27 MB

**Format :**PDF, ePub, Docs

**Download :**130

**Read :**693

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 Snyder

**ISBN :**9781580425049

**Genre :**Games & Activities

**File Size :**27.42 MB

**Format :**PDF, Kindle

**Download :**435

**Read :**760

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. Hale

**ISBN :**1884466117

**Genre :**Casinos

**File Size :**58.83 MB

**Format :**PDF, Mobi

**Download :**678

**Read :**877

## Poker

**Author :**David A. Daniel

**ISBN :**1569800936

**Genre :**Games & Activities

**File Size :**64.65 MB

**Format :**PDF, Docs

**Download :**232

**Read :**678

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 Sklansky

**ISBN :**1880685000

**Genre :**Games & Activities

**File Size :**31.50 MB

**Format :**PDF, Kindle

**Download :**143

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'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 Warren

**ISBN :**158042113X

**Genre :**Games & Activities

**File Size :**35.35 MB

**Format :**PDF, ePub, Mobi

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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 Holden

**ISBN :**9780691183312

**Genre :**Computers

**File Size :**74.18 MB

**Format :**PDF, Mobi

**Download :**368

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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:

The one unrelenting fact about poker is that it is truly all about the math. The Mathematics of Poker is very important. Red dog casino payout. There is a probability for every outcome in online poker and a statistic for every hand of the poker game. Some of them are fun, some of them matter and some of them are just for MIT graduates. They are helpful to your online poker game and by having a general knowledge on the mathematics of poker you put yourself in a better position to understand the game of poker over the long run. Here is some more math, with a 100% deposit bonus you can double your starting bankroll, get your Bovada Bonus Codes now.

## Math Odds For Being Dealt Specific Hands In Hold’em

- The odds you will be dealt a pair of aces: 220:1
- Odds you will be dealt any pair: 16:1
- The odds you will be dealt AK suited: 330:1
- Odds you will be dealt AK: 110:1
- The odds you will be dealt either a pair or a hand with an ace: 3.91:1

### Mathematics of Poker For The Flop In Texas Hold’em

- If you hold a pair, the odds against flopping a set are 7.51:1
- You hold AK, the odds against flopping either an ace or a king are: 2.08:1
- If you hold a suited hand, the odds against flopping a four flush are 8:14:1
- When you hold QJ, the odds against flopping an outside straight draw are 15.6:1
- If you hold KQ, the odds against flopping a nut straight (AJT) are 305:1

### Odds For Completing A Hand With Two Cards To Come In Hold’em

- The odds against completing a four flush: 1.86:1
- Odds against completing an outside straight draw: 2.18:1
- The odds against completing an inside straight draw: 5.07:1
- The odds against making a set by the river when you hold a pair: 10.88:1

### Flopping Over Cards With Pocket Pairs In Texas Hold’em

- You hold KK, the odds that no over cards will fall on the flop are: 0.29:1 (an Ace will fall about 22% of the time)
- Odds you hold QQ, the odds that no over cards will fall on the flop are: 0.71:1 ( an over card will fall about 41% of the time)
- You hold JJ, the odds that no over cards will fall on the flop are: 1.32:1 (an over card will fall about 57% of the time)
- You hold TT, the odds that no over cards will fall on the flop are 2.28:1 (an over card will fall about 70% of the time)

## Mathematics of Poker For Five Card Draw Poker

- Odds that you will be dealt a pair of jacks or better: 6.69:1
- The Odds that you will be dealt a pair of tens or lower: 2.42:1
- You will be dealt two pair: 20:1
- Odds that you will be dealt three of a kind: 46.3:1
- The Odds that you will be dealt a straight 254:1
- Odds that you will be dealt a flush: 508:1
- The Odds that you will be dealt a full house: 693:1
- Odds that you will be dealt four of a kind: 4164:1
- The odds that you will be dealt a straight flush: 64973:1
- If you hold one pair and draw three, your odds against making two pair or better are 2.48:1. With the same hand, your odds against making trips or better are 6.86:1
- When you hold three of a kind and draw two, your odds against making a full house or better are 8.65:1
- You hold an open ended straight and draw one, your odds against completing to a straight are 4.88:1
- If you hold an inside straight and draw one, your odds against completing to a straight are 10.75:1

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**Mathematics Of Poker**