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# Bet Smart: The Kelly System for Gambling and Investing

Format and pricing: Paperback (133 pages) \$28.95, Kindle/pdf \$9.95
ISBN: 9781887187015 (paperback), 9781887187022 (ebook)
Publication date: October 2008

This book is also available in paperback at most bookstores and distributors, including these:

 Booktopia (Australia) Amazon.ca (Canada) Amazon.fr (France) libreriauniversitaria.it (Italy) Amazon.co.jp (Japan) Amazon.co.uk (UK) Barnes and Noble (US)

In 1956, a physicist named John Kelly working at Bell Labs published a paper titled A New Interpretation of Information Rate. In the paper he draws an analogy between the outcomes of a gambling game and the transmission of symbols over a communications channel. For a positive expectation game, Kelly showed that a betting system based on a fixed fraction of the bankroll can make the bankroll grow at an exponential rate in the long run. The exponential growth rate in this case is directly analogous to the rate of information transmission through a communications channel.

This book examines the Kelly system in detail. Applications of the Kelly system in both gambling and investing are considered. Python code for calculating the Kelly fractions for both a single stock investment and an investment in two stocks simultaneously is included.

Included is an introductory review chapter on the probability theory needed to analyze gambling systems in general. There is also a chapter on some of the more commonly used gambling systems such as the Martingale system.

This book will be useful for anyone interested in a good mathematical introduction to gambling systems in general, and the Kelly system in particular.

#### 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.

1  Mathematical Introduction
1.1  Random Variables
1.2  Mean of a Random Variable
1.3  Properties of the Mean
1.4  Variance and Standard Deviation
1.5  Chebyshev's Theorem
1.6  Moment Generating Functions
1.7  Binary Random Variables
1.8  Binomial Random Variables

2  Gambling Systems
2.1  General Framework
2.2  Single State System
2.3  Two State System
2.4  m State System
2.5  Martingale System
2.6  Cancellation System

3  The Kelly System
3.1  Fixed Fraction Betting
3.2  Choosing a Betting Fraction
3.2.1  The Bernoulli Approach
3.2.2  The Kelly Approach
3.3  Expectation and Variance
3.4  Bankroll Probabilities
3.5  Multiple Simultaneous Games
3.5.1  Two Games
3.5.2  Three Games
3.5.3  Identical Games
3.6  Kelly Plays Powerball

4  Investing with Kelly
4.1  Single Stock Investment
4.2  Single Stock and Risk Free Bond
4.3  Two Stock Investment

Bibliography

Index

Send comments to: Richard Hollos (richard[AT]exstrom DOT com)
Copyright 2021 by Exstrom Laboratories LLC