Introduction to Probability with Texas Hold 'Em Examples

60 Each chapter of this book covers a part of basic probability theory, most of the time using as illustrative examples poker hands that are famous among poker players, which makes the book entertaining to read. Chapter 1 deals with Kolmogorov’s axiomatic and shortly digresses into the actual meaning of probability in a way that was new to me. Like all the chapters of the book, it ends up with a couple of poker-based exercises. The first illustrative hand of the book is, in my opinion, particularly well chosen, since it features a decision worth several millions of dollars that can be mathematically solved in a simple way using basic probability theory, which is discussed throughout the book. I think it could be a good starting point for discussing utility function in a different work. Chapter 2 focuses on counting examples. I have to admit that all the combinatorics examples I got as a student were kind of dry and unappealing, whereas the poker frame renders the whole chapter entertaining to read. Some of the questions solved in this chapter turn out to be surprisingly challenging (What is the probability that Ace-King offsuits splits with Ace-Queen offsuit all in preflop? What is the probability of flopping the unbreakable nuts (left as an exercise)? ), even if the solutions just involve basic mathematical arguments. Chapter 3 is devoted to conditional probability and independence. Poker examples of independence become, in this case, more linked to different dealt hands, which like the different outcomes of a die flip, are supposed to be independent. The author provides numerous examples of dependence, as well (e.g., What is the probability that the player next to me has AA given that I was dealt A7o? ). Chapter 4 deals with the fundamental notions of expected value and variance, which translates into expected profit a player gets from a hand when facing a zero or one decision (call for all his chips or fold) and knowing the opponent’s cards. This is the longest chapter of the book, and rightfully so. The examples are generally well chosen (e.g., the author investigates whether Wasicka would have had to call in the first hand of the book, the answer being yes if one focuses purely on the expected numbers of chips, then more purely poker considerations play a huge role in this decision, prize scale of the tournament in particular). The author discusses both the influences of luck and skill in poker. Although this section is interesting from a poker player’s point of view (I never heard about this kind of computation before), this particular section is of less interest to the undergraduate student. Chapter 5 introduces discrete random variables and defines the standard ones (Bernoulli, binomial, geometric, Poisson). It also shows the convergence in law of the binomial variable with parameter (n, /n) as n → ∞ to a Poisson of parameter . Poker examples become more scarce, but still interesting and pertinent. Chapter 6 focuses on continuous random variables. The author introduces density and the straightforward extensions of variance and expectation from the discrete case. He then defines the uniform density, using it to discuss basic examples issued from game theory due to Von Neumann and Morgenstern (and developed much more in Chen and Ankenmann’s The Mathematics of Poker). Exponential, normal, and Pareto’s densities are explored with more purely poker-based examples. Let us note that the exercises are much less poker-based than in the previous chapters, mainly because it is not as easy to link continuous random variables to relevant poker questions. Chapter 7 deals with collections of random variables, beginning with expectation and variance of the sum of random variables. The author follows with the notion of conditional expectation, which he explains well. (I remember struggling with the intuition of this notion when I was a student.) Then, the weak and strong laws of large numbers are discussed, the weak one being the only one to be proved. It follows a fascinating discussion on Sklansky’s well-known Fundamental Theorem of Poker, which I particularly enjoyed and whose conclusion is that the relevance of probability theory in the poker framework should not be overestimated (poker being a game). Once again, I am not totally convinced by this digression in the framework of a basic probability book. The book then focuses on the Central Limit Theorem, its natural statistical counterpart, the notion of confidence intervals, and the estimation of the win rate of a player. The chapter ends with the introduction of random walks and their most famous properties (the reflection principle, the ballot theorem, the arcsine law, and Kelly’s criterion) without poker examples (but these seem to be challenging to get in this context). Introduction to Probability with Texas Hold ’Em Examples