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.

This eighth book of the author on gambling math presents in accessible terms the cold mathematics behind the sparkling slot machines, either physical or virtual. It contains all the mathematical facts grounding the configuration, functionality, outcome, and profits of the slot games. Therefore, it is not a so-called how-to-win book, but a complete, rigorous mathematical guide for the slot player and also for game producers, being unique in this respect. As it is primarily addressed to the slot player, its goal is to present practical applications of the mathematical models of slot games, in order to provide numerical results that a player can use as criteria for gaming decisions or just as information for any slot game and any predicted winning event. These results are focused on probability and expected value, these being the most important parameters for decisional criteria in slots. The book is packed with plenty of figures, tables, and formulas. The content is organized so that readers can skip the theoretical parts and go picking the practical results (numerical, in tables of values where possible, or ready-to-compute formulas) for the desired situation. The practical results are gathered in the last chapter, titled "Practical Applications and Numerical Results," the largest part of the book, for the most popular categories of slot machines, namely with 3, 5, 9, and 16 reels. Any other category of slot games is covered in the theoretical part of the book, where the general formulas apply not only to existing slot games, but also to possible future slot games of any design and configuration. The author does not just throw the slot mathematics to the audience and run away, but offers an ultimate practical contribution with the chapter "How to estimate the number of stops and the symbol distribution on a reel," a surprise for both players and producers, where one can see that mathematics provides players with some statistical methods as well as methods based on physical measurements for retrieving these missing data. Having these data along with the mathematical results of this book, anyone can generate the PAR sheet of any slot machine. In the last decade, mathematics has been taken more and more seriously into account in gaming, as being the essence that governs the games of chance and the only rigorous tool providing information on optimal play, where possible. For the popular game of slots, mathematics already fulfilled its duty by providing all the data that it can provide and that cannot be found on the display of the slot machines - it is all here in this book.

In the flop stage, what are your odds of achieving a straight by river and what are the odds of your opponents achieving something higher? If you will achieve the straight, what are the new odds of your opponents beating it in the river stage? How about the full house? How strong is your hand from mathematical point of view and how can you use this information in your play? This is the kind of questions that this book deals with.In improving poker skills, acquisition of information is vital, whether we talk about data a player collects during a specific game or pre-established mathematical facts behind the card distribution, including prediction. Texas Hold'em Poker is a game highly suitable for probability-based decisions. The power of a single card changing the hand hierarchies at every stage is a strong argument that every objective strategy should be probability based and so the card odds should be part of such a strategy. The book is a complete probability guide of Hold'em Poker, covering all possible gaming situations, an improved edition of the bestseller "Texas Hold'em Odds," published in 2004. In this edition, the author focused on the practical side of the presentation and use of the probabilities involved in Hold'em, while taking into account everywhere the subjective side of the probability-based criteria of each player's strategy.The first part of the book deals with the odds of specific card formations, from one pair to straight flush, at every stage of the game, grouped in three large categories: long-shot odds for own hand, long-shot odds for opponent's hands and immediate odds. These odds are collected in tables at the end of each section and, where generating two-dimensional tables was not possible, the compact formulas returning the odds are provided. All tables and formulas are followed by examples of how to use them for finding the desired odds in a specific gaming situation. All probability results were worked out through compact mathematical formulas and not by the use of any software based on partial simulations. The chapter "Operating with and Weighting the Odds" is the main new material added to the first edition and deals with the rules of estimating and evaluating the probabilities of complex gaming events in order to use them in a strategy, by using the odds provided in the first part of the book. When we are allowed to add together the partial probabilities of a union of events for an overall probability and when we are not, methods of approximation and tips of avoiding hard calculations, all have their dedicated subchapters, packed with suggestive examples of application on concrete hands. The author also introduces the concept of strength matrix of a hand, within the adequate mathematical model of the strength of a Hold'em hand, and argues why the strength of a hand should be defined through mathematical probabilities. At this point, he proves that what most of the so-called odds calculators return are not mathematical probabilities and cannot stand as objective strength indicators for Hold'em hands.The last chapter is a suggestive collection of probability-based hand analyses on concrete Hold'em hands that can be generalized to categories of hands having identical results associated.The author is a recognized authority on casino/gaming mathematics and his books are in the official bibliography of the students of several gambling institutes and organizations around the globe. This book is the most trusted and professional source for the mathematical facts of Hold'em Poker, a must-have completion for any strategy book.

This work is a complete mathematical guide to lottery games, covering all of the problems related to probability, combinatorics, and all parameters describing the lottery matrices, as well as the various playing systems. The mathematics sections describe the mathematical model of the lottery, which is in fact the essence of the lotto game. The applications of this model provide players with all the mathematical data regarding the parameters attached to the gaming events and personal playing systems. By applying these data, one can find all the winning probabilities for the play with one line (for each category in part or cumulatively), and how these probabilities change with playing the various types of systems containing several lines, depending on their structure. Also, each playing system has a formula attached that provides the number of possible multiple prizes in various circumstances. Other mathematical parameters of the playing systems and the correlations between them are also presented. The generality of the mathematical model and of the obtained formulas allows their application for any existent lottery (including variations like Keno) and any playing system. Each formula is followed by numerical results covering the most frequent lottery matrices worldwide and by multiple examples predominantly belonging to the 6/49 lottery. The listing of the numerical results in dozens of well-organized tables, along with instructions and examples of using them, makes possible the direct usage of this guide by players without a mathematical background. The author also discusses from a mathematical point of view the strategies of choosing involved in the lotto game. The book does not offer so-called winning strategies (proving that the only strategy is that of choosing), but helps players to better organize their own playing systems and to confront their own convictions (so many times based on false perceptions) with the incontestable reality offered by the direct applications of the mathematical model of the lotto game. As a must-have handbook for any lottery player, this book offers essential information about the game itself and can provide the basis for gaming decisions of any kind.

The problem-solving assumes theoretical and analytical skills, as well as algorithmic skills, coupled with a basic mathematical intuition. The concept of this problem book successfully supports the development of these skills of the solver and meanwhile offers mathematics instructors models for teaching problem-solving as an integral part of the mathematics learning process. The topics of the problems belong to algebra II and algebra of the first two years of college, passing through fields of integer and real numbers, equations, inequalities, powers, logarithms, divisibility, polynomials, and combinatorics. This work is the first of a series that will also operate in other domains and subdomains of mathematics. The problem book is structured in four separate and independent sections, namely Problems, Hints, Algorithms, and Proofs, in this order. The Problems section consists of over 100 problems themselves, which are of medium to advanced difficulty level. The problems can be presented and discussed at mathematics workshops and sessions of preparation for contests and olympiads as well as in the classroom, using problems of different difficulty levels with separate groups of students; for advanced groups, an instructor can use this problem book in its entirety. The Hints offer groups of keywords that suggest to the solver - intuitively, as well as analytically - an initial approach to the problem, important observations upon which the solution is based, categories of theoretical results applied when solving, and specific theoretical results. The hints also suggest indirectly the solving algorithm (found in the next section), but without exposing or synthesizing it. In the Algorithms section, the solving algorithms provide chronological groups of steps necessary for generating the complete solution. The algorithm is presented as a brief list of tasks; it does not reveal the complete solution to the problem, but only points out the partial tasks whose results will finally yield the logical construction of the solution. The Proofs represent the complete integral solutions of the problems, unfolded according to the solving algorithm. This comprehensive presentation includes the detailed steps to be executed, the observations that precede the deductions, and the entire logical motivation. No partial results are left unproved, neither as an exercise nor as being obvious or easily deduced. The sections described previously are separated in this book so that the solver can explore the problem and search for solving paths independently, consulting the next section only when he or she has exhausted, with no success, his or her own approach and individual study methods. As the solver moves progressively from a partial solution to a more complete one, this additional effort itself becomes a useful mathematical exercise. Moreover, the process of moving successively through the indications of the problems together with individual investigation and autocorrection of a wrong approach, stimulates and motivates the solver toward a solution. All these elements give this type of problem book a truly interactive character.

Continuing his series of books on the mathematics of gambling, the author shows how a simple-rule game such as roulette is suited to a complex mathematical model whose applications generate improved betting systems that take into account a player's personal playing criteria. The book is both practical and theoretical, but is mainly devoted to the application of theory. About two-thirds of the content is lists of categories and sub-categories of improved betting systems, along with all the parameters that might stand as the main objective criteria in a personal strategy - odds, profits and losses. The work contains new and original material not published before. The mathematical chapter describes complex bets, the profit function, the equivalence between bets and all their properties. All theoretical results are accompanied by suggestive concrete examples and can be followed by anyone with a minimal mathematical background because they involve only basic algebraic skills and set theory basics. The reader may also choose to skip the math and go directly to the sections containing applications, where he or she can pick desired numerical results from tables. The book offers no new so-called winning strategies, although it discusses them from a mathematical point of view. It does, however, offer improved betting systems and helps to organize a player's choices in roulette betting, according to mathematical facts and personal strategies. It is a must-have roulette handbook to be studied before placing your bets on the turn of either a European or American roulette wheel.

Man's daily life is full of decisional situations. Whether we have math skills or not, we frequently estimate and compare probabilities, sometimes without realizing it, especially when making decisions. But probabilities are not just simple numbers attached objectively or subjectively to events, as they perhaps look, and their calculus and usage is highly predisposed to qualitative or quantitative errors in the absence of proper knowledge. That is why a book explaining the probability concept and its interpretations and applications for non-mathematicians is a necessity. This is an enlightening journey through the world of probability theory. Its multiple goals are to help the reader understand what probability really means, to teach the reader how to rigorously perform and apply the probability calculus, even without a solid mathematical background, and to stimulate the reader to go deeper into the notions involved. In the first part, the author tries to build a clear image of the probability concept by reconstructing its mathematical definition step by step through its constituent notions. It starts with a general presentation of the conceptual ensemble word - definition - notion - model any theory is based on when trying to reproduce reality. Then, the probability notion is defined and explained starting from the classical definition to the definition for the countable case; then probability is presented as a limit and as a measure. This book presents not only the mathematical concept of probability, but also its philosophical aspects, the relativity of probability and its applications and even the psychology of probability. All explanations are made in a comprehensible manner and are supported with suggestive examples from nature and daily life and even with challenging math paradoxes. After these points are laid out the math chapter follows. It contains all the notions and principal theoretical results that ground Probability Theory, starting with fundamental notions like Sets, Functions, Boole algebras, and Sequences, and continuing with Measure Theory Basics - Tribes, Borel sets, Measurable spaces, and Measure, ending with Field of events, Sigma-fields, Probability, Conditional probability, Discrete random variables, Classical probability distributions, and Convergence. And, of course, it includes all important theorems and results dealing with them. A special section is dedicated to Combinatorics and combinatorial calculus. Readers with no minimal mathematical background may choose to skip this chapter because the teaching material is structured for developing probability calculus skills based on algorithmic procedures. This is the subject of the chapter titled Beginners' Calculus Guide, in which the reader is taught to apply the properties of probability and to perform calculations in practical applications. The skills acquired here can be practiced on the more than 200 solved and unsolved problems and exercises in the book. Everyone should find something of interest here: philosophers and mathematicians may focus on the sections on philosophical matters of the probability model and decisional matters, students and non-mathematicians can find solid A to Z teaching material about Probability Theory and the practical person can find all the tools needed to apply and perform probability calculus without a teacher.

La vida cotidiana est llena de situaciones que exigen tomar decisiones. Y en estos casos comparamos y hacemos estimaciones de probabilidades, a veces casi sin darnos cuenta, especialmente en el momento de decidir. Pero las probabilidades no son nmeros simples asociados objetiva o subjetivamente a los eventos, como nos podra parecer, y el clculo y el uso que le damos estn especialmente proclives a errores cualitativos y cuantitativos, si no se maneja un conocimiento apropiado. Por ello es una necesidad que exista un libro que explique el concepto de probabilidad junto con sus interpretaciones y aplicaciones, dirigido a gente sin conocimientos profundos de matemtica. Este libro es un viaje iluminador por el mundo de la teora de la probabilidad. Su objetivo mltiple es afianzar en el lector una comprensin de lo que realmente significa la probabilidad, ensearle el manejo y la aplicacin rigurosa del clculo probabilstico, an cuando carezca de una preparacin matemtica slida. Adems se le estimula a profundizar en las nociones que la fundamentan. En la primera parte del libro, el autor intenta crear una imagen del concepto de probabilidad mediante la reconstruccin de su definicin recorriendo punto por punto las nociones que all se encuentran. Comenzando por una presentacin general del conjunto conceptual palabra - definicin - nocin - modelo, en el que se sustenta toda teora cuando se trata de reproducir la realidad. Por eso la nocin de probabilidad se define y explica partiendo de la definicin clsica hasta la definicin del caso numerable; la probabilidad se presenta como un lmite y como una medida. Se presenta no solamente el concepto matemtico de probabilidad, sino tambin sus aspectos filosficos, la relatividad de la probabilidad y sus aplicaciones, y tambin la psicologa de la probabilidad. Todas las explicaciones estn hechas de una manera comprensible y se afianzan con ejemplos sugerentes tomados de la naturaleza y de la vida diaria, y hasta tambin con desafiantes paradojas matemticas. Luego de dejar en claro estos puntos, se contina con el captulo de matemtica. Contiene all todas las nociones y resultados tericos que son el basamento de la teora de la probabilidad, partiendo de las nociones fundamentales como conjuntos, funciones, lgebra de Boole, sucesiones y continuando con los fundamentos de la teora de la medida - tribus - conjuntos de Borel, espacios mensurables y medida, y finalizando con campo de eventos, campos -sigma, probabilidad, probabilidad condicional, variables aleatorias discretas, distribuciones clsicas de la probabilidad, y convergencia. Y por supuesto, se incluyen todos los teoremas importantes y los resultados relevantes. Una seccin especial est dedicada a la combinatoria y al clculo combinatorio. Los lectores sin preparacin matemtica previa pueden evitar este captulo porque el material didctico a lo largo de todo el libro est estructurado para desarrollar la habilidad de hacer clculos probabilsticos basados en procedimientos algortmicos. Este es el enfoque del captulo titulado Gua de Clculo para el Principiante, en el que se ensea al lector a aplicar las propiedades de la probabilidad y a realizar clculos para las aplicaciones prcticas. Los conocimientos que se adquieren pueden practicarse en ms de 200 problemas resueltos y sin resolver presentados en el libro. Todos tendrn aqu su cuota de inters: los matemticos y filsofos se concentrarn en los aspectos filosficos del modelo de la probabilidad y en la toma de decisiones; los estudiantes y los no matemticos podrn encontrar un material didctico completo sobre la teora de la probabilidad y la gente prctica hallar todas las herramientas que se requieren para aplicar y resolver clculos probabilsticos sin necesidad de un profesor.

Continuando con su serie de libros en las matemticas de juegos de azar, el autor muestra como un juego de reglas simples como la ruleta es idneo para un modelo de matemtica compleja cuyas aplicaciones generan sistemas mejorados de apuestas que toma en cuenta el criterio de juego personal de un jugador. El libro es prctico y terico, pero es principalmente dedicado a la aplicacin de la teora. Cerca de dos tercios del contenido son listas de categoras y subcategoras de sistemas mejorados de apuestas, junto con todos los parmetros que pudieran representar el criterio primordial en una estrategia personal de probabilidades, utilidades y prdidas. La obra contiene material nuevo y original nunca antes publicado. El captulo de matemticas describe las apuestas complejas, la funcin de utilidad, la equivalencia entre las apuestas y todas sus propiedades. Todos los resultados tericos estn acompaados de ejemplos concretos, sugerentes y pueden ser entendidos por cualquier persona con un mnimo de conocimiento matemtico porque solamente involucran habilidades bsicas de algebra y teora bsica de conjuntos . El lector puede tambin elegir saltarse las matemticas e ir directamente a las secciones que contienen las aplicaciones, donde l o ella pueden elegir los resultados numricos deseados de las tablas. El libro no ofrece las tan llamadas estrategias ganadoras, aunque habla de ellas desde un punto de vista matemtico. Lo que si hace, sin embargo, es ofrecer sistemas mejorados de apuestas y ayuda a organizar las elecciones del jugador cuando apuesta a la ruleta, de acuerdo a factores matemticos y estrategias personales. Es un manual de ruleta imprescindible para estudiar antes de colocar sus apuestas en el giro ya sea de la rueda de la ruleta europea o americana.