# Coin Tossing: The Hydrogen Atom of Probability

Stefan Hollos and J. Richard Hollos

**Format and pricing:** Paperback (193 pages) $24.95, pdf $12.95

**ISBN:** 9781887187381 (paperback), 9781887187398 (ebook)

**Publication date:** May 2019

## Table of Contents

- Chapter 1
**Introduction** - Chapter 2
**Probability Distributions** - 2.1
**Bernoulli Distribution** - 2.2
**Binomial Distribution** - 2.3
**Beta Distribution** - 2.4
**Normal Distribution** - 2.5
**Geometric Distribution** - 2.6
**Negative Binomial Distribution** - 2.7
**Poisson Distribution** - 2.8
**Exponential Distribution** - Chapter 3
**Betting on Coin Tosses** - 3.1
**Known Bias** - 3.2
**Unknown Bias** - 3.2.1
**BSP Strategy** - 3.2.2
**Majority Rule Strategy** - Chapter 4
**Coin Tosses as Random Walks** - 4.1
**Walks Returning to the Origin** - 4.2
**Walks from the Origin to***m* - 4.3
**Gambler's Ruin** - Chapter 5
**Coin Toss Runs and Patterns** - 5.1
**Recurrence Times for Runs** - 5.1.1
**Recursion Equations for***f*_{n} - 5.1.2
**Generating Functions for***f*_{n} - 5.1.3
**Calculating Recurrence Time Statistics** - 5.1.4
**Examples** - 5.1.5
**Approximations and Other Methods** - 5.2
**Multiple Recurrence Times for Runs** - 5.3
**Run Occurrence Probability** - 5.3.1
**Examples** - 5.4
**Head or Tail Run Probabilities** - Chapter 6
**Runs and Patterns as Markov Chains** - 6.1
**Head Run Markov Chain** - 6.2
**Head or Tail Run Markov Chain** - 6.3
**Pattern Markov Chain** - Chapter 7
**Run Distributions** - 7.1
**Number of Runs in a Sequence** - 7.1.1
**Probability Generating Function** - 7.1.2
**Expected Number of Runs** - 7.1.3
**Runs in a Markov Chain** - 7.1.4
**Markov Chain Expected Number of Runs** - 7.2
**Number of Runs of a Given Length** - 7.3
**Longest Runs** - 7.3.1
**Expected Length of Longest Run** - 7.4
**Combinatorics of Longest Runs** - 7.4.1
**Fair Coin Head Runs** - 7.4.2
**Fair Coin Head or Tail Runs** - 7.4.3
**Biased Coin Head Runs** - Chapter 8
**Problems** - Appendix A
**Review of Discrete Probability**

The coin toss is to probability theory what the hydrogen atom is to quantum mechanics. It is the simplest random event that you can imagine. There are only two possible outcomes: heads or tails. This simplicity means that many questions about coin tossing can be asked and answered in great depth, serving as a gateway for exploring probability and statistics, and a foundation for understanding many kinds of probability distributions.

This book is an update to the author's original coin toss book (The Coin Toss: Probabilities and Patterns), expanding on run distributions and statistics, as well as a new chapter containing 26 problems and solutions. The page count has increased by over 40 percent.

The book contains material for both the beginning student and the advanced researcher. We suspect that the beginner will find some of the material quite difficult and not accessible on a first reading. This is a book that needs to be read more than once. There is more material here than anyone could absorb on a first reading. We hope that researchers on the other hand find the book to be a valuable reference and a stimulus for new research.

#### About the authors

Stefan Hollos and J. Richard Hollos are physicists and electrical engineers by training, and enjoy anything related to math, physics, engineering and computing. They are brothers and business partners at Exstrom Laboratories LLC in Longmont, Colorado.

Send comments to: Richard Hollos (richard[AT]exstrom DOT com)

Copyright 2019 by Exstrom Laboratories LLC