Mathematics of Casino Carnival Games

Mark Bollman

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9780367348656
• eBook ISBN: 9781000192032
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

There are thousands of books relating to poker, blackjack, roulette and baccarat, including strategy guides, statistical analysis, psychological studies, and much more. However, there are no books on Pell, Rouleno, Street Dice, and many other games that have had a short life in casinos!

While this is understandable — most casino gamblers have not heard of these games, and no one is currently playing them — their absence from published works means that some interesting mathematics and gaming history are at risk of being lost forever. Table games other than baccarat, blackjack, craps, and roulette are called carnival games, as a nod to their origin in actual traveling or seasonal carnivals.

Mathematics of Casino Carnival Games is a focused look at these games and the mathematics at their foundation.

Features

• Exercises, with solutions, are included for readers who wish to practice the ideas presented

• Suitable for a general audience with an interest in the mathematics of gambling and games

• Goes beyond providing practical 'tips' for gamblers, and explores the mathematical principles that underpin gambling games

Chapter 1

Mathematical Background

Before diving into carnival games, it will be useful to review the mathematics that is essential to analysis of games of chance. In this chapter, we shall do this through the lens of familiar casino games such as craps, roulette, and blackjack.

1.1Elementary Probability

An event, in a probabilistic sense, is simply the outcome of some sort of random experiment. If A is an event, the probability of A is a function P(A) that assigns a number to A, in the interval 0 ≤ P(A) ≤ 1, as a measure of how likely A is to occur. Informally, we might define P(A) as

P(A)=Number of ways that A can happenNumber of ways that something can happen.

The set of all things that can happen in a given experiment, counted in the denominator, is called the sample space and often denoted by S. If we define a function #(A) that counts the number of elements in the event A, we can write

P(A)=#(A)#(S).

Counting the numerator and denominator of the expression for P(A) is typically done either by appealing to pure mathematical reasoning, the theoretical probability, or by looking at real data, the experimental or empirical probability.

Example 1.1. In tossing a fair coin, the theoretical probability of Heads is ½: there is one way to throw Heads, and two ways for the coin to land.

If instead we were to toss a fair coin 500 times, achieving 256 Heads and 244 Tails, the experimental probability of Heads is 256500=.512.▄

The connection between theoretical and experimental probability is described in a mathematical result called the Law of Large Numbers, or LLN for short.

Theorem 1.1. (Law of Large Numbers) Suppose an event has theoretical probability p. If x is the number of times that the event occurs in a sequence of n trials, then as the number of trials n increases, the experimental probability x/n approaches p, or

limn→∞xn=p.

Informally, the LLN states that, in the long run, things happen in an experiment the way that theory says that they do. What is meant by “in the long run” is not a fixed number of trials, but will vary depending on the experiment. For some experiments, n = 500 may be a large number, but for others—particularly if the theoretical probability is small—it may take far more trials befo...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• 1. Mathematical Background
• 2. Mathematics and Casino Game Design
• 3. Wheel and Ball Games
• 4. Card Games
• 5. Dice Games
• Appendix A: Answers to Selected Exercises
• References
• Index