The coin toss is really just a metaphor for a random event that has only two possible outcomes. The actual tossing of a real coin is just one way to realize such an event. There are many examples of questions that are equivalent to a coin toss:

- Will the stock market close up or down tomorrow?
- Will a die roll come up with an even or odd number?
- Will we make contact with extraterrestrials within the next ten years?
- Will a car drive by in the next minute?
- Will tomorrow be sunny or cloudy?
- Will my medical test result be negative or positive?
- Will I enjoy this movie?
- Will the next joke be funny?
- Will the Earth's average temperature go up next year?

Because a coin toss is equivalent to such a wide variety of questions, the results in this book are widely applicable.

With this book, you can answer questions like:

- Is it unusual to get only 6 heads in 20 tosses of a fair coin?
*(pg 12)* - In 100 tosses of a fair coin, what is the probability there will be no
more than 40 heads?
*(pg 22)* - What is the probability that a fair coin has to be tossed 5 times
before the first head appears?
*(pg 23)* - What is the probability that it takes 10 tosses to get 3 heads, with probability of heads=0.4?
*(pg 24)* - How is the coin toss related to the amount of time it takes for an
unstable nucleus to decay?
*(pg 30)* - If you are betting on a coin toss whose probability of heads varies
over time, what is the best strategy to use?
*(pg 33)* - How are the Catalan numbers related to coin tosses?
*(pg 45)* - How is the probability of losing all your money in a gambling game
related to coin tosses?
*(pg 46)* - Playing an unfavorable game against an opponent with unlimited
resources, what's the average time before you're broke?
*(pg 60)* - If you're tossing a coin once per second, and after about 40 minutes
you suddenly get 10 heads in a row, should you be surprised?
*(pg 65)* - What is the probability that a run of length 5 heads occurs somewhere
in the first 25 tosses of a fair coin?
*(pg 71)* - How are the Fibonacci numbers related to coin tosses?
*(pg 73)* - What is the probability that it takes 100 tosses to get more than one
run of 5 heads with probability of heads = 0.49?
*(pg 83)* - What is the probability that on the 50th toss we get 5 heads in a row with a fair coin?
*(pg 88)* - What is the average number of tosses needed to get either 10 heads in a
row or 10 tails in a row using a coin with probability of heads = 0.55?
*(pg 92)* - What is the probability that a run of 10 heads or 6 tails occurs for
the first time on the 50th toss using a coin with probability of heads =
0.7?
*(pg 94)* - What is the probability that the pattern HTHTH occurs on toss 50 using
a fair coin?
*(pg 104)* - For a coin with heads probability = 0.55, what is the probability that in a sequence of 50
tosses, a head never comes up more than 6 times in a row?
*(pg 105)* - For a fair coin, what is the probability that the longest run of heads
or tails in a sequence of 30 tosses is less than or equal to 5?
*(pg 107)*

Because the coin toss is the simplest random event you can imagine, many questions about coin tossing can be asked and answered in great depth. The simplicity of the coin toss also opens the road to more advanced probability theories dealing with events with an infinite number of possible outcomes.

This book is very mathematical. Some knowledge of calculus, discrete
math, and generating functions is helpful to get the most out of it. A
review of discrete math is provided in the index, and a
free copy
of Herbert Wilf's book *generatingfunctionology* can get you up to speed on
generating functions.

You can buy this ebook now at Amazon as a Kindle book.

You can also get this ebook instantly as a pdf from Gumroad.

**About the authors:**
Stefan Hollos
and J. Richard
Hollos are physicists by training, and enjoy finding patterns and
information in data, and anything related to the calculation of
probabilities. They are the authors
of Probability
Problems and Solutions,
Combinatorics
Problems and Solutions,
Bet Smart: The
Kelly System for Gambling and Investing, as well as
Simple Trading Strategies That Work
and Pairs
Trading: A Bayesian Example, and are brothers and business
partners at Exstrom Laboratories LLC in Longmont, Colorado. The
websites for their work
are Exstrom.com
and QuantWolf.com.

## 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
**The Longest Run Probability Distribution** - 7.1
**Fair Coin Head Runs** - 7.2
**Fair Coin Head or Tail Runs** - 7.3
**Biased Coin Head Runs** - Appendix A
**Review of Discrete Probability**

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

Copyright 2012-2014 by Exstrom Laboratories LLC