Practical Poker Math: Basic Odds And Probabilities for Hold'Em and Omaha

by Pat Dittmar

Overview

What are the odds of winning at poker if you don’t know the odds in poker?

Practical Poker Math provides a complete and easy-to-understand explanation of the basic odds, probabilities and expectations in Hold’Em and Omaha. All the formulations are completely open and are demonstrated via easy-to-follow, colour-coded calculations.

Pat Dittmar has aimed this book at beginning through semi-pro players who want to improve their results and who know that there is not much chance of improvement without a fundamental understanding of poker odds, probabilities, and expectations.

Pat Dittmar is head of trading and development at TradePointTechnologies.com, an organization that develops and deploys state-of-the-art proprietary trading technologies. Pat feels that successful poker players possess the skills required to play in the one true fast action, around the world, around the clock “Big Game” – the world financials markets, and TradePointTechnologies recruits its traders exclusively from the poker world.

Product Details

ISBN-13:	9781554903269
Publisher:	ECW Press
Publication date:	09/01/2008
Sold by:	Barnes & Noble
Format:	eBook
Pages:	231
Sales rank:	901,518
File size:	8 MB

About the Author

Pat Dittmar has played professional poker for more than twenty years, and has been a consultant to a major online poker site. His math expertise comes from years in the financial community, including partner in a brokerage firm, derivatives trader, big block trader, arbitrageur, and Compliance Registered Options Principal. He lives in Las Vegas.

Read an Excerpt

Practical Poker Math

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By Pat Dittmar

ECW PRESS

Copyright © 2008 Martin M. Dittmar
All rights reserved.
ISBN: 978-1-55490-326-9

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CHAPTER 1

Introduction to Game Theory in Poker

Game Theory — A Historical Perspective

Some believe that the study of Game Theory began with the works of Daniel Bernoulli. A mathematician born in 1700, Bernoulli is probably best known for his work with the properties and relationships of pressure, density, velocity and fluid flow. Known as "Bernoulli's Principle," this work forms the basis of jet engine production and operation today. Pressured by his father to enter the world of commerce, he is also credited with introducing the concepts of expected utility and diminishing returns. This work in particular can be of use when "pricing" bets or bluffs in no-limit poker.

Others believe the first real mathematical tool to become available to game theorists was "Bayes' Theorem," published posthumously in England in the 18th century. Thomas Bayes was born in 1702 and was an ordained minister. His work involved using probabilities as a basis for logical inference. (The author has developed and used artificially intelligent systems based on "Bayes' Theorem" to trade derivatives in today's financial markets.)

Yet still others believe that the study of Game Theory began with the publication of Antoine Augustin Cournot's The Recherches in the early 1800s. The work dealt with the optimization of output as a best dynamic response.

Émile Borel was probably the first to formally define important concepts in the use of strategy in games. Born in 1871 in Saint-Affrique, France, he demonstrated an early penchant for mathematics. In 1909 the Sorbonne created a special chair of "Theory of Functions" which Borel held through 1940. During the years 1921–27 he published several papers on Game Theory and several papers on poker. Important to poker players are his discussions on the concepts of imperfect information, mixed strategies and credibility.

In 1944 Princeton University Press published Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. While not the first work to define certain concepts of strategy in games, it is widely recognized as one that has fostered Game Theory as we know it today.

Also important to poker players is the work of Julia Bowman Robinson. Born in 1919, she discovered her passion for mathematics after a bout with scarlet fever and was the first woman admitted to the Academy of Sciences. For poker players her most important work was An Iterative Method of Solving a Game.

Credited by many as being a primary shepherd of modern Game Theory is John Forbes Nash, Jr. Diagnosed as a paranoid schizophrenic, Nash was long troubled by delusions. This condition, now in remission, became the subject of a popular film. His work earned him 1/3 of the 1994 Nobel Prize in Economic Sciences. Born in the Appalachian town of Bluefield, West Virginia in 1928, Nash became well known for a 28-page work he did at age 21 which defined his "Nash Equilibrium" concerning strategic behavior in non-cooperative games. Poker players are most drawn to the story that while he was in a bar near Princeton and being goaded into approaching an attractive blond-haired lady, he suddenly shouted and then ran off to complete his work on "The Mathematics of Competition" which is one basis of Game Theory today.

Game Theory and Poker

More than the study and application of a set of principles, Game Theory in poker is primarily about two goals:

1. The study and understanding of the opposition

2. The development of an efficient strategy to dominate the competition.

To be of maximum value, this study and understanding must be translated into effective action, and these actions must be the antithesis of everything a poker opponent thinks or does.

The difference between an antithesis and a correct response defines the utility of Game Theory in poker. An opponent makes an obvious bluff. You are certain that your hand will not even beat the bluff. A correct response in poker is to fold your hand when you know you are beat. The antithesis is to raise or re-raise and make your opponent fold his.

This is an example of a move. In an environment of ever-increasing odds and stakes such as a poker tournament, good hands just don't come along often enough for a player to make it on the strength of his hand alone. A winning player must make moves and the study and application of the principles of Game Theory can help him to know

1. When and where to make the move

2. How likely the move is to succeed.

Expressed in the specific terminology of advanced theorists, poker can be defined as an asynchronous, non-cooperative, constant-sum (zero-sum), dynamic game of mixed strategies. While the game is played in an atmosphere of common knowledge and no player possesses complete knowledge, some players are better able to process this common knowledge into a more complete knowledge than are their opponents. A player is most able to make the best-reply dynamic (sometimes referred to as the Cournot adjustment) and earn a cardinal payoff after using the process of backward induction to construct and deploy a dominant strategy.

In poker, beyond a certain set of rules, players do not cooperate with each other because each is trying to win at the expense of all others (zero-sum). Moreover, they will repeatedly change their strategies at different intervals and for different reasons (therefore, asynchronously).

While information about stakes, pot-size, board-cards and players' actions and reactions is available as common knowledge to all players at the table, no player has complete knowledge of such factors as the other players' cards or intentions.

The one characteristic common to most outstanding players is their ability to better process and better use — that is, they get more value from — the information that is commonly available to everyone else at the table.

Two of the most basic assumptions of Game Theory are that all players

1. Have equal common knowledge

2. Will act in a rational manner.

But in poker, while all players at any given table have access to the same common information, not all of them are smart enough to do something with it. Players who know more about odds and probabilities, and whose instincts and keen observation enable them to better process the common knowledge around them, will take far better advantage of this information and will have correspondingly higher positive expectations.

So while all the players at the table have access to the same common knowledge, some players are able to base their actions on knowledge that is more complete. That all players in all games will always act rationally is never a safe assumption in poker.

Equilibrium versus Evolution

According to the "Nash Equilibrium," a game is said to be in a state of equilibrium when no player can earn more by a change in strategy. It has been argued that, using the process of backward induction, players will evolve their strategies to the point of equilibrium.

In poker, the astute player's strategy will always be in a state of evolution so that his opponents, in order not to be dominated, will also be compelled to modify their strategies. In a real game of poker there is constant evolution and therefore hardly ever a point of absolute equilibrium.

Efficiency and Diminishing Returns

The concept of efficiency and the notion of diminishing returns relate to the conservation and the most effective use of power/momentum and resources. In poker this is about a player's chips, cards and table image. These two concepts are particularly helpful to the no-limit player when it comes to "pricing" bets and raises.

In any game, and most especially in no-limit poker tournaments, a player's "Mo" (power) comes from an amalgamation of

1. His image at the table

2. The size of his stack

3. The strength of his hand.

If a player is rich in one, he can lean on the other two. If a player has a very strong hand, not only will he win chips, he will add to his image at the table. If he is possessed of a strong image, he can steal with smaller bets. If his stack is huge, players with smaller stacks will often avoid conflict and he will be able to play weaker hands.

In tournament poker, a player's stack goes down in value with the passage of time. At any point, a player's stack size is finite and, therefore, he must carefully consider pricing when he makes his bet. He must consider

1. At what price will his opponent be "priced out" of the hand

2. At what price will his opponent be "priced into" calling

3. Given a careful study of the opposition, what is the best price to achieve the best result without risking more chips than absolutely necessary.

Conventional poker wisdom has long held that a primary obligation of the big stack is to knock out the small stacks. A game theorist would say that the primary obligation of the big stack is to not pump up the small stacks. The increasing stakes and other, more desperate players will eliminate most of the short stacks — which will leave the big stacks to carefully pick their spots and eliminate anyone left.

Strategy

The value of applying Game Theory principles in any arena is primarily to help a player develop an efficient strategy that dominates the competition.

From a practical perspective, Game Theory is about strategic development. As no effective strategy is likely to be developed in a vacuum, its formation must be the result of a close study of not only the opposition but also past interactions.

As more players play more poker, many of them increase their skills, and the result is that the game becomes ever tougher to beat.

The players who have dominated the game in the past and the players who will do so in the future are the few who can convert the common knowledge available to every player at the table into more complete knowledge.

Assuming that all greatly successful players are possessed of advanced knowledge of or instinct for odds and strategy, theprimary property of their game that differentiates them from the rest of the field is an uncanny ability to discern an opponent's strength and likely action or reaction.

Today the bulk of poker is about one game — No-Limit Texas Hold'em. In a game where you can either greatly increase or lose all your chips in a single move, knowledge of opponents and their tendencies in certain situations becomes much more important than either stack size or the power of a hand.

With complete information on odds and strategy available to a growing and increasingly more able and competitive pool, the one area of opportunity open to aspiring poker kings is the study and application of Game Theory in poker.

The most difficult calculation in all of poker, especially in No-Limit Texas Hold'em, is that of a player's Total Odds of winning the pot. This calculation includes far more than the odds of making a certain hand — it includes the odds of making the hand, the likelihood that your opponent's hand is stronger or weaker, and (most difficult of all) the probability of an opponent's action or reaction to your action or reaction.

The only hope of coming to a reasonably accurate calculation of this matrix of probabilities is via the very essence of Game Theory. You must make a detailed and almost instantaneous analysis of your opponent and his strengths, weaknesses and other propensities.

CHAPTER 2

The Basic Calculations

Odds & Probabilities

Odds and Probabilities are simply two methods of expressing the relationship of Positives, Negatives and Total Possibilities. They represent the same basic information and each can be easily converted into the other.

Odds

Odds in poker are expressed as a ratio of the Negatives to the Positives:

WILLNOTs : WILLs

With a well-shuffled deck of cards, what are the Odds that the first card from the deck will be one of the 4 Aces?

Total Possibilities = 52 Cards in the Deck

Positives/WILLs = 4 Aces in the Deck

Negatives/WILLNOTs = 48 Cards not Aces

To calculate odds or probability, you only need to know any 2 of 3 values: WILLs, WILLNOTs and Total Possibilities. There are 3 easy ways to find the third value:

Total Possible (52) – WILLs (4) = WILLNOTs (48)

Total Possible (52) – WILLNOTs (48) = WILLs (4)

WILLNOTs (48) + WILLs (4) = Total Possible (52)

In a deck of 52 cards (Total Possibilities) there are 4 cards (WILLs) that are Aces and 48 cards (WILLNOTs) that are not Aces.

The odds of drawing an Ace is the ratio of the cards that WILLNOT be Aces to the cards that WILL be Aces:

WILLNOTs: WILLs
48 : 4
Reduce
48 / 4 : 4 / 4
12 : 1

The odds against drawing an Ace are 12 to 1.

Probabilities

Probabilities are expressed as either a decimal ratio to 1 or as a percentage. Here is an easy-to-use formula to calculate either:

WILLs/Total Possibilities = Probability as decimal ration to 1

WILLS/Total Possibilities*100 = Probability as %

Total Possibilities is the number of cards in the deck, 52. The number of cards that WILL, is the number of Aces in the deck, 4.

The probability that the first card off a well-shuffled deck will be an Ace is

WILLS/Total Possibilities = Probability as decimal ration to 1

4/52 = .0769

WILLS/Total Possibilities*100 = Probability as %

4/52*100 = 7.69%

Another way to calculate probabilities is to multiply the probability of the components. For example, the probability of finding a pair of Aces in the hole in Hold'em can be found by

WILLS/Total Possibilities = Probability as decimal ration to 1

With 6 possible pairs of Aces and 1,326 possible starting hands, the probability of Aces is

6/1,326 = .0045249

Another way to get the same answer is to multiply the probability that your first card in the hole will be an Ace by the probability that your second card in the hole will be an Ace:

4/52 * 3/51 = .0045249

Conversions

Because ALL odds and probability calculations begin with two of three basic values (Total Possibilities, WILLs and WILLNOTs), the conversion of odds to probability or vice versa is straightforward.

In the example above the odds are 12 : 1. Odds are an expression of WILLNOTs : WILLs.

To convert these odds to a probability expressed as a percentage:

WILLS/Total Possibilities*100 = Probability as %

4/52*100=7.6923%

To re-convert this 7.6923% probability back to odds, you know that this probability, 7.6923%, represents the WILLs and that in any probability situation the Total Possibilities are 100%. Thus you have the WILLs and only need the WILLNOTs to express this probability as odds:

Total Possibilities = 100%

– WILLs = 7.6923%. WILLNOTs = 92.3077%

WILLNOTs (92.3077) : WILLs (7.6923)

92.3077 : 7.6923

Reduce92.3077 / 7.6923 : 7.6923 / 7.6923

12 : 1

To complete this circle of conversions, we will reconvert the 12 : 1 odds back to a probability expressed as a decimal ratio to 1.

To calculate the probability expressed as a decimal ratio to 1 you need to know the WILLs and the Total Possibilities. From odds of 12 : 1 we know that the WILLs are 1 and the WILLNOTs are 12. WILLs (1) + WILLNOTs (12) = Total Possibilities (13). You now have the all numbers needed to calculate 12 : 1 as a decimal probability to 1.

WILLS/Total Possibilities = Probability as decimal ration to 1

1/13 = .076923

Quickie conversion finale: the conversion of the probability of .076923 back into odds is

Total Possibilities = 1

– WILLs = .076923

WILLNOTs = .923077

WILLNOTs (.923077) : WILLs (.076923)

.923077 : .076923

Reduce

.923077 / .076923 : .076923 / .076923

12 : 1

Combinations, Permutations & Factorials

Whether it is called combinatorics or combinatorial math, it is a branch of discrete mathematics that for our purposes deals with the determination of the size of certain sets.

Most odds and probability calculations can be reduced to 2 basic steps:

1. Define the Sets — The calculation of any set of odds or probabilities involves the comparison of 2 of 3 sets:Total Possibilities, WILLs and WILLNOTs.

2. Compare the Sets — Odds in poker are an expression of the comparison of the number of WILLNOTs :WILLs. Probabilities are an expression of WILLs/Total Possibilities.

Formulae that calculate the populations of combinations, permutations or factorials are used to calculate the population of the WILL and WILLNOT sets.

Mathematicians express combinations as "C(52, 2)," in Practical Poker Math, we have used "Comb(52, 2)." Permutations are expressed as "P(52, 2)" and we have used "Perm(52, 2)." For factorials we have used the standard 52!.

Combinations

The mechanics of the math used to determine the enumeration of combinations is more easily demonstrated than explained.

Of the 3 (combinations, permutations and factorials) the most used calculation in poker is the determination of the size of certain combinations.

The calculation of a combination in the format of Comb (52, 2), will produce the number of different 2-card combinations — starting hands — that are possible from a 52-card deck:

(52*51)/(1*2)=1,326

To find the number of 3-card Flops possible from the 50 remaining unseen cards after you have been dealt your 2 hole cards in Hold'em, use Comb(50, 3):

(50*49*48)/(1*2*3)=19,600

There are 6 different pairs or 2-card combinations of Aces that can be made from the 4 Aces in a 52-card deck:

This can be calculated using Comb(4, 2):

(4*3)/(1*2)=6

With 10 great picks for a weekend of sports betting, to find the number of 4-bet parlays —Comb(10, 4):

(10*9*8*7)/(1*2*3*4)=210

There are basically two ways to calculate probabilities:

1. The comparison of sets

2. The mathematical comparison of probabilities.

The comparison of sets is more commonly used and is helpful when trying to determine the size of multiple sets.

(Continues...)

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Excerpted from Practical Poker Math by Pat Dittmar. Copyright © 2008 Martin M. Dittmar. Excerpted by permission of ECW PRESS.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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