Mathematics of The Big Four Casino Table Games

Blackjack, Baccarat, Craps, & Roulette

Mark Bollman

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000423891
• Edition: 1
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

Mathematics is the basis of casino games, which are the bedrock of a $100 billion/year industry. Mathematics of the Big Four Casino Table Games: Blackjack, Baccarat, Craps, & Roulette takes an in-depth look at the four biggest table games in casinos: blackjack, baccarat, craps, and roulette. It guides readers through the mathematical principles that underpin these games and their different variations, providing insights that will be of huge interest to gamblers, casino managers, researchers, and students of mathematics.

Features

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• A valuable teaching resource, replete with exercises, for any course on gambling mathematics

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• Suitable for a wide audience of professionals, researchers, and students

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• Many practical applications for the gambling industry

Mark Bollman is Professor of Mathematics and chair of the Department of Mathematics & Computer Science at Albion College in Albion, Michigan, and has taught 116 different courses in his career. Among these courses is "Mathematics of the Gaming Industry, " where mathematics majors carefully study the math behind games of chance and travel to Las Vegas, Nevada, in order to compare theory and practice. He has also taken those ideas into Albion's Honors Program in "Great Issues in Humanities: Perspectives on Gambling, " which considers gambling from literary, philosophical, and historical points of view as well as mathematically. Mark has also authored Basic Gambling Mathematics: The Numbers Behind the Neon, Mathematics of Keno and Lotteries, and Mathematics of Casino Carnival Games.

Chapter 1

Essential Probability

DOI: 10.1201/9781003156680-1

The mathematics behind roulette, craps, baccarat, and blackjack is drawn from probability theory at various levels of complexity. This is appropriate, because the origins of this branch of mathematics lie with the analysis of simple games of chance. This began with a series of letters between Blaise Pascal and Pierre de Fermat in 1654 that raised and answered several questions at the foundation of probability theory while addressing questions that arose in resolving gambling games [35]. Later mathematicians have developed and expanded this topic into a rigorous field of mathematics with many important applications unrelated to gambling. In this chapter, we shall outline the fundamental ideas of probability that are essential for analyzing casino games.

1.1 Elementary Ideas

We begin our study of probability with the careful definition of some important terms.

Definition 1.1. An experiment is a process whose outcome is determined by chance.

This may not seem like a useful definition. We illustrate the concept with several examples.

Example 1.1. Roll a standard six-sided die and record the number that results. ■

Example 1.2. Roll two standard six-sided dice (abbreviated as 2d6) and record the sum. ■

Example 1.3. Roll 2d6 and record the larger of the two numbers rolled (or the number rolled, if both dice show the same number). ■

Example 1.4. Deal a five-card video poker hand and record the number of aces it contains. ■

One trait of an experiment is that it results in a single definite outcome. While we will eventually concern ourselves with individual outcomes, we begin by looking at all of the possible results of an experiment.

Definition 1.2. The sample spaceS of an experiment is the set of all possible outcomes of the experiment.

Example 1.5. In Example 1.1, the sample space is $S=\{1,2,3,4,5,6\}$. The same sample space applies to the experiment described in Example 1.3. ■

Example 1.6. In Example 1.2, the sample space is $S=\{2,3,4,\dots ,12\}.$

It is important to note that the 11 elements of S are not equally likely, as this will play an important part in our explorations of probability. Rolling a 7 is more likely than rolling any other sum on two dice; 2 and 12 are the least likely sums. ■

When we're only interested in some of the possible outcomes of an experiment, we are looking at subsets of S. These are called events.

Definition 1.3. An eventA is any subset of the sample space S. An event is called simple if it contains only one element.

Example 1.7. In Example 1.2, we rolled 2d6 and considered the sum as the outcome. The event “the roll is an even number” is the subset $\{2,4,6,8,10,12\}.$

The event “The roll is a 7” is a simple event. Although there are 6 different ways to roll a sum of 7 on 2d6, we are ...