The Doctrine of Chances (eBook)

Preface 6
Contents 10
List of Notation 14
Part I Theory 16
Review of Probability 17
Combinatorics and Probability 17
Independence and Conditional Probability 29
Random Variables and Random Vectors 35
Expectation and Variance 41
Law of Large Numbers and Central Limit Theorem 56
Problems 66
Notes 78
Conditional Expectation 88
Conditioning on an Event 88
Conditioning on a Random Vector 92
Problems 101
Notes 105
Martingales 108
Definitions and Examples 108
Optional Stopping Theorem 112
Martingale Convergence Theorem 119
Problems 124
Notes 128
Markov Chains 131
Definitions and Examples 131
Transience and Recurrence 139
Asymptotic Behavior 148
Renewal Theorem 157
Problems 162
Notes 169
Game Theory 171
Matrix Games 171
Minimax Theorem 186
Utility Theory 197
Problems 205
Notes 209
House Advantage 211
A Single Wager 211
Composite Wagers 231
Volatility 237
Problems 243
Notes 248
Gambler's Ruin 253
Even-Money Payoffs 253
Integer Payoffs 264
Arbitrary Payoffs 271
Problems 279
Notes 283
Betting Systems 287
Examples 287
Conservation of Fairness 310
Problems 317
Notes 323
Bold Play 329
Red-and-Black 329
Red-and-Black with a House Limit 342
Primitive Casinos 350
Problems 362
Notes 367
Optimal Proportional Play 369
A Single Wager 369
Simultaneous Wagers 375
Optimality Properties 383
Problems 396
Notes 400
Card Theory 403
Shuffling 403
Dealing 412
Card Counting 420
Problems 433
Notes 436
Part II Applications 438
Slot Machines 439
Expected Payout 439
Volatility and Ruin 451
Problems 460
Notes 465
Roulette 471
Unbiased Wheels 471
Biased Wheels 479
Problems 484
Notes 487
Keno 493
The m-Spot Ticket 493
Way Tickets 499
Problems 503
Notes 506
Craps 511
Line Bets and Free Odds 511
The Shooter's Hand 515
Problems 523
Notes 527
House-Banked Poker 535
Let It Ride 535
Three Card Poker 544
Problems 549
Notes 552
Video Poker 555
Jacks or Better 555
Deuces Wild 563
Problems 576
Notes 580
Faro 583
The Denomination Bet 583
From Soda to Hock 588
Problems 596
Notes 599
Baccarat 606
Player vs. Banker 606
Card Counting 614
Problems 617
Notes 622
Trente et Quarante 631
Red, Black, Color, Inverse 631
Card Counting 636
Problems 642
Notes 645
Twenty-One 651
Rules 651
Basic Strategy 658
Card Counting 671
Problems 678
Notes 684
Poker 696
Rules and Pot Odds 696
Poker Models and Game Theory 701
Texas Hold'em 711
Problems 732
Notes 738
Results Cited 752
Algebra and Number Theory 752
Analysis and Probability 754
Bibliography 758
Index 789

"Preface (p. v-vi)

I have found many thousands more readers than I ever looked for. I have no right to say to these, You shall not ?nd fault with my art, or fall asleep over my pages; but I ask you to believe that this person writing strives to tell the truth. If there is not that, there is nothing.
William Makepeace Thackeray, The History of Pendennis

This is a monograph/textbook on the probabilistic aspects of gambling, intended for those already familiar with probability at the post-calculus, premeasure- theory level. Gambling motivated much of the early development of probability theory (David 1962).1 Indeed, some of the earliest works on probability include Girolamo Cardano’s [1501–1576] Liber de Ludo Aleae (The Book on Games of Chance, written c. 1565, published 1663), Christiaan Huygens’s [1629– 1695] “De ratiociniis in ludo aleae” (“On reckoning in games of chance,” 1657), Jacob Bernoulli’s [1654–1705] Ars Conjectandi (The Art of Conjecturing, written c. 1690, published 1713), Pierre R´emond de Montmort’s [1678– 1719] Essay d’analyse sur les jeux de hasard (Analytical Essay on Games of Chance, 1708, 1713), and Abraham De Moivre’s [1667–1754] The Doctrine of Chances (1718, 1738, 1756).

Gambling also had a major in?uence on 20thcentury probability theory, as it provided the motivation for the concept of a martingale. Thus, gambling has contributed to probability theory. Conversely, probability theory has contributed much to gambling, from the gambler’s ruin formula of Blaise Pascal [1623–1662] to the optimality of bold play due to Lester E. Dubins [1920–2010] and Leonard J. Savage [1917–1971]; from the solution of le her due to Charles Waldegrave to the solution of chemin de fer due to John G. Kemeny [1926–1992] and J. Laurie Snell [1925–]; from the duration-of-play formula of Joseph-Louis Lagrange [1736–1813] to the optimal proportional betting strategy of John L. Kelly, Jr. [1923–1965]; and from the ?rst evaluation of the banker’s advantage at trente et quarante due to Sim´eon-Denis Poisson [1781–1840] to the ?rst published card-counting system at twenty-one due to Edward O. Thorp [1932–]. Topics such as these are the principal focus of this book.

Is gambling a subject worthy of academic study? Let us quote an authority from the 18th century on this question. In the preface to The Doctrine of Chances, De Moivre (1718, p. iii) wrote,

Another use to be made of this Doctrine of Chances is, that it may serve in Conjunction with the other parts of the Mathematicks, as a ?t introduction to the Art of Reasoning; it being known by experience that nothing can contribute more to the attaining of that Art, than the consideration of a long Train of Consequences, rightly deduced from undoubted Principles, of which this Book a?ords many Examples.

We also quote a 20th-century authority on the same question. In Le jeu, la chance et le hasard, Louis Bachelier [1870–1946] (1914, p. 6) wrote,2 It is almost always gambling that enables one to form a fairly clear idea of a manifestation of chance; it is gambling that gave birth to the calculus of probability; it is to gambling that this calculus owes its ?rst faltering utterances and its most recent developments; it is gambling that allows us to conceive of this calculus in the most general way; it is, therefore, gambling that one must strive to understand, but one should understand it in a philosophic sense, free from all vulgar ideas."