The Theory of Gambling and Statistical Logic

Richard A. Epstein

• Publisher: Academic Press
• Year: 2009
• ISBN:
  9780080958613
• Edition: 2
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Early in his rise to enlightenment, man invented a concept that has since been variously viewed as a vice, a crime, a business, a pleasure, a type of magic, a disease, a folly, a weakness, a form of sexual substitution, an expression of the human instinct. He invented gambling.

Recent advances in the field, particularly Parrondo's paradox, have triggered a surge of interest in the statistical and mathematical theory behind gambling. This interest was acknowledge in the motion picture, "21, " inspired by the true story of the MIT students who mastered the art of card counting to reap millions from the Vegas casinos. Richard Epstein's classic book on gambling and its mathematical analysis covers the full range of games from penny matching to blackjack, from Tic-Tac-Toe to the stock market (including Edward Thorp's warrant-hedging analysis). He even considers whether statistical inference can shed light on the study of paranormal phenomena. Epstein is witty and insightful, a pleasure to dip into and read and rewarding to study. The book is written at a fairly sophisticated mathematical level; this is not "Gambling for Dummies" or "How To Beat The Odds Without Really Trying." A background in upper-level undergraduate mathematics is helpful for understanding this work.

• Comprehensive and exciting analysis of all major casino games and variants
• Covers a wide range of interesting topics not covered in other books on the subject
• Depth and breadth of its material is unique compared to other books of this nature
• Richard Epstein's website: www.gamblingtheory.net

Chapter One

Kubeiagenesis

Shortly after pithecanthropus erectus gained the ascendency, he turned his attention to the higher-order abstractions. He invented a concept that has since been variously viewed as a vice, a crime, a business, a pleasure, a type of magic, a disease, a folly, a weakness, a form of sexual substitution, an expression of the human instinct. He invented gambling.

Archaeologists rooting in prehistoric sites have uncovered large numbers of cube-shaped bones, called astragalia, that were apparently used in games some thousands of years ago. Whether our Stone Age ancestors cast these objects for prophecy or amusement or simply to win their neighbor’s stone axe, they began a custom that has survived evolution and revolution.

Although virtually every culture has engaged in some form of dice play, centuries elapsed before thought was directed to the “fairness” of throwing dice or to the equal probability with which each face falls or should fall. The link between mathematics and gambling long remained unsuspected.

Most early civilizations were entrapped by the deep-rooted emotional appeal of absolute truth; they demanded Olympian certitude and could neither envision nor accept the inductive reliability sought by modern physics. “Arguments from probabilities are impostors,” was the doctrine expressed in Plato’s Phaedo. Carneades, in the second century B.C., was the first to shift from the traditional Greek rationalist position by developing an embryonic probability theory that distinguished three types of probability, or degrees of certainty. However, this considerable accomplishment (against the native grain) advanced the position of empiricist philosophy more than the understanding of events of chance.

Throughout the entire history of man preceding the Renaissance, all efforts aimed at explaining the phenomena of chance were characterized by comprehensive ignorance of the nature of probability. Yet gambling has flourished in various forms almost continuously from the time Paleolithic hominids cast polished knucklebones and painted pebbles. Lack of knowledge has rarely inhibited anyone from taking a chance.

Reasoned considerations relating games of chance to a rudimentary theory of probability first emerged in the 16th century. Gerolamo Cardano (1501–1576), physician, philosopher, scientist, astrologer, religionist, gambler, murderer, was responsible for the initial attempt to organize the concept of chance events into a cohesive discipline. In Liber de Ludo Alea (The Book on Games of Chance), published posthumously in 1663, he expressed a rough concept of mathematical expectation, derived power laws for the repetition of events, and conceived the definition of probability as a frequency ratio. Cardano designated the “circuit” as the totality of permissible outcomes of an event. The circuit was 6 for one die, 36 for two dice, and 216 for three dice. He then defined the probability of a particular outcome as the sum of possible ways of achieving that outcome divided by the circuit.

Cardano investigated the probabilities of casting astragalia and undertook to explain the relative occurrence of card combinations, notably for the game of primero (loosely similar to poker). A century before Antoine de Méré, he posed the problem of the number of throws of two dice necessary to obtain at least one roll of two aces with an even chance (he answered 18 rather than the correct value of 24.6). Of this brilliant but erratic Milanese, the English historian Henry Morley wrote, “He was a genius, a fool, and a charlatan who embraced and amplified all the superstition of his age, and all its learning.” Cardano was imbued with a sense of mysticism; his undoing came through a pathological belief in astrology. In a much-publicized event, he cast the horoscope of the frail 15-year-old Edward VI of England, including specific predictions for the 55th year, third month, and 17th day of the monarch’s life. Edward inconsiderately expired the following year at the age of sixteen. Undismayed, Cardano then had the temerity to cast the horoscope of Jesus Christ, an act not viewed with levity by 16th-century theologians. Finally, when the self-predicted day of his own death arrived, with his health showing no signs of declining, he redeemed his reputation by committing suicide.

Following Cardano, several desultory assaults were launched on the incertitudes of gambling. Kepler issued a few words on the subject, and shortly after the turn of the 17th century, Galileo wrote a short treatise titled, Considerazione sopra il Giuoco dei Dadi.¹ A group of noblemen of the Florentine court had consulted Galileo in a bid to understand why the total 10 appears more often than 9 in throws of 3 dice. The famous physicist showed that 27 cases out of 216 possible total the number 10, while the number 9 occurs 25 times out of 216.

Then, in 1654, came the most significant event in the theory of gambling as the discipline of mathematical probability emerged from its chrysalis. A noted gambler and roué, Antoine Gombaud, Chevalier de Méré, posed to his friend, the Parisian mathematician Blaise Pascal, the following problem: “Why do the odds differ in throwing a 6 in four rolls of one die as opposed to throwing two 6s in 24 rolls of two dice?” In subsequent correspondence with Pierre de Fermat (then a jurist in Toulouse) to answer this question, Pascal constructed the foundations on which the theory of probability rests today. In the discussion of various gambling problems, Pascal’s conclusions and calculations were occasionally incorrect, while Fermat achieved greater accuracy by considering both dependent and independent probabilities.

Deriving a solution to the “problem of points” (two players are lacking x and y points, respectively, to win a game; if the game is interrupted, how should the stakes be divided between them?), Pascal developed an approach similar to the calculus of finite differences. Pascal was an inexhaustible genius from childhood; much of his mathematical work was begun at age 16. At 19 he invented and constructed the first calculating machine in history.² He is also occasionally credited with the invention of the roulette wheel. Whoever of the two great mathematicians contributed more, Fermat and Pascal were first, based on considerations of games of chance, to place the theory of probability in a mathematical framework.

Curiously, the remaining half of the 17th century witnessed little interest in or extension of the work of Pascal and Fermat. In 1657, Christiaan Huygens published a treatise titled, De Ratiociniis in Ludo Aleae (Reasonings in Games of Chance), wherein he deals with the probability of certain dice combinations and originates the concept of “mathematical expectation.” Leibnitz also produced work on probabilities, neither notable nor rigorous: he stated that the sums of 11 and 12, cast with two dice, have equal probabilities (Dissertatio de Arte Combinatoria, 1666). John Wallis contributed a brief work on combinations and permutations, as did the Jesuit John Caramuel. A shallow debut of the discipline of statistics was launched by John Graunt in his book on population growth, Natural and Political Observations Made Upon the Bills of Mortality. John de Witt analyzed the problem of annuities, and Edmund Halley published the first complete mortality tables.³ By mathematical standards, however, none of these works can qualify as first-class achievements.

More important for the comprehension of probabilistic concepts was the pervasive skepticism that arose during the Renaissance and Reformation. The doctrine of certainty in science, philosophy, and theology was severely attacked. In England, William Chillingworth promoted the view that man is unable to find absolutely certain religious knowledge. Rather, he asserted, a limited certitude based on common sense should be accepted by all reasonable men. Chillingworth’s theme was later applied to scientific theory and practice by Glanville, Boyle, and Newton, and given a philosophical exposition by Locke.

Turning into the 18th century, the “Age of Reason” set in, and the appeal of probability theory once again attracted competent mathematicians. In the Ars Conjectandi (Art of Conjecturing), Jacob Bernoulli developed the theory of permutations and combinations. One-fourth of the work (published posthumously in 1713) consists of solutions of problems relating to games of chance. Bernoulli wrote other treatises on dice combinations and the problem of duration of play. He analyzed various card games (e.g., Trijaques) popular in his time and contributed to probability theory the famous theorem that by sufficiently increasing the number of observations, any preassigned degree of accuracy is attainable. Bernoulli’s theorem was the first to express frequency statements within the formal framework of the probability calculus. Bernoulli envisioned the subject of probability from the most general point of view to that date. He predicted applications for the theory of probability outside the narrow range of problems relating to games of chance; the classical definition of probability is essentially derived from Bernoulli’s work.

In 1708, Pierre Remond de Montmort published his work on chance titled, Essai d’Analyse sur les Jeux d’Hazards. This treatise was largely concerned with combinatorial analysis and dice and card probabilities. In connection with the game of Treize, or Rencontres, de Montmort was first to solve the matching problem (the probability that the value of a card coincides with the number expressing the order in which it is drawn). He also calculated the mathematical expectation of several intricate dice games: Quinquenove, Hazard, Esperance, Trois Dez, Passedix, and Rafle, inter alia. Of particular interest is his analysis of the card games le Her and le Jeu des tas. Following de Montmort, Abraham de Moivre issued a work titled, Doctrine of Chances, that extended the knowledge of dice combinations, permutation theory, and card game analysis (specifically, the distribution of honors in the game of whist) and, most important, that proposed the first limit theorem. In this and subsequent treatises, he developed further the concepts of matching problems, duration of play, probability of ruin, mathematical expectation, and the theory of recurring series. On the basis of his approximation for the sum of terms of a binomial expansion (1733), he is commonly credited with the invention of the normal distribution. De Moivre lived much of his life among the London coffeehouse gamblers.⁴ It was likely this environment that led him to write what is virtually a gambler’s manual.

Miscellaneous contributors to gambling and probability theory in the first half of the 18th century include Nicolas Bernoulli and Thomas Simpson, the latter responsible for introducing the idea of continuity into probability (1756). These mathematicians and their contemporaries analyzed card games and solved intricate dice problems. Little additional understanding of probability was furnished to the orbis scientiarum.

Daniel Bernoulli advanced the concepts of risk and mathematical expectation by the use of differential calculus. He also deserves credit, with de Moivre, for recognizing the significance of the limit theorem to probability theory. Leonhard Euler, far more renowned for contributions to other branches of mathematics, worked for some time on questions of probabilities and developed the theory of partitions, a subject first broached in a letter from Leibnitz to Johann Bernoulli (1669). He published a memoir, Calcul de la Probabilité dans le Jeu de Rencontre, in 1751 and calculated various lottery sequences and ticket-drawing combinations. Jean le Rond d’Alembert was best known for his opinions contrary to the scientific theories of his time. His errors and paradoxes abound in 18th-century mathematical literature. According to d’Alembert, the result of tossing three coins differs from three tosses of one coin. He also believed that tails are more probable after a long run of heads, and promulgated the doctrine that a very small probability is practically equivalent to zero. This idea leads to the progression betting system that bears his name. His analyses of card and dice games were equally penetrating.

The Rev. Thomas Bayes (his revolutionary paper, “An Essay Towards Solving a Problem in the Doctrine of Chances,” was published in 1763, two years after his death) contributed the theorem stating exactly how the probability of a certain “cause” changes as different events actually occur. Although the proof of his formula rests on an unsatisfactory postulate, he was the first to use mathematical probability inductively—that is, arguing from a sample of the population or from the particular to the general. Bayes’ theorem has (with some hazard) been made the foundation for the theory of testing statistical hypotheses. Whereas Laplace later defined probability by means of the enumeration of discrete units, Bayes defined a continuous probability distribution by a formula for the probability between any pair of assigned limits. He did not, however, consider the metric of his continuum.

Joseph Louis Lagrange contributed to probability theory and solved many of the problems previously posed by de Moivre. Beguelin, George Louis Buffon,⁵ and the later John Bernoulli published sundry articles on games of chance and the calculus of probability during the second half of the 18th century. Jean Antoine de Caritat, Marquis de Condorcet, supplied the doctrine of credibility: The mathematical measure of probability should be considered as an accurate measure of our degree of belief.

Quantitatively, the theory of probability is more indebted to Pierre Simon, Marquis de Laplace, than to any other mathematician. His great work, Théorie Analytique des Probabilités, was published in 1812 and was accompanied by a popular exposition, Essai Philosophique sur les Probabilités. These tomes represent an outstanding contribution to the subject, containing a multitude of new ideas, results, and analytic methods. A theory of equations in finite differences with one and two independent variables is proposed, the concomitant analysis being applied to problems of duration of play and random sampling. Laplace elaborated on the idea of inverse probabilities first considered by Bayes. In subsequent memoirs, he investigated applications of generating functions, solutions and problems in birth statistics and the French lottery (a set of 90 numbers, five of which are drawn at a time), and generalized Montmort’s matching problem. Although he employed overly intricate analyses and not always lucid reasoning, he provided the greatest advance in probabilistic methodology in history. Laplace can claim credit for the first scientifically reasoned deterministic interpretation of the universe.

In the mid-18th century, the empiricist philosopher and economist David...

Table of contents

• Cover Image
• Table of Contents
• Title
• Copyright
• Dedication
• Preface
• Chapter One. Kubeiagenesis
• Chapter Two. Mathematical Preliminaries
• Chapter Three. Fundamental Principles of a Theory of Gambling
• Chapter Four. Parrondo’s Principle
• Chapter Five. Coins, Wheels, and Oddments
• Chapter Six. Coups and Games with Dice
• Chapter Seven. The Play of the Cards
• Chapter Eight. Blackjack
• Chapter Nine. Statistical Logic and Statistical Games
• Chapter Ten. Games of Pure Skill and Competitive Computers
• Chapter Eleven. Fallacies and Sophistries
• Epilogue
• Appendix
• Index