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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.
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.
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.
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