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Understanding Probability: Chance Rules in Everyday Life

Henk Tijms
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
David Mazel
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"What are the odds I would understand that book?" a friend asked me when he saw me with this book. "Well," I replied, "pretty good. This is a good book and it's easy to read."

This book presents probability in a lucid and non-technical way that makes it easy to understand. The author mixes cartoons with the text along with plenty of examples and plenty of problems within the text. At the end of each chapter there are lots of problems and answers to odd-numbered problems are in an appendix. Even if the reader does not want to work the problems, just reading them adds insight and is worth the time. The combination of explanations, examples, and problems, makes for a fun and thorough book that's engaging and enjoyable. I found the topics to be apropos for an introductory book and the explanations clear and intuitive.

To whet your appetite, Tijms begins with twelve probability questions such as the birthday problem, lottery problems, and the now famous Monty Hall problem. For many seasoned readers these are classics but newer readers will find them stimulating and motivating. Solutions and methods to these problems appear throughout the text so that readers can see the problems first, a sort of preview, then read of methods for solutions later on. This is a good literary technique to keep the reader's interest alight.

The book then discusses simulation and how that can be used to glean insights into problems. (Interested readers will find Paul Nahin's book Digital Dice to be a better text for simulation. Nonetheless, Tijms's book is a welcome companion to give insights that simulations sometimes miss through pure number crunching.)

One example to show the spice of the book is the drunkard's walk in chapter two. The problem is a classic but I found the discussion for higher dimensions of interest. In particular the author discusses the time it would take a photon (modeled as a drunkard's walk in three-space), on average, to travel from the core of the Sun to the surface (answer: 10 million years) as compared to the time from the Sun to earth (answer: 8 minutes).

Further in the book, Tijms provides a brief description of the bootstrap method, which I found new relative to older books on probability I have read and used. The discussion is not mathematically deep but there is plenty of detail so one can appreciate the method and if one finds it appropriate, to try some ideas. It's a good way to touch on the topic.

Chapter four, entitled "Rare events and lotteries," is something that everyone, not just mathematicians or practitioners, should read. Readers will find a lovely discussion of the Poisson distribution, used to model event arrivals, and graphs of simulations with this distribution. What's so interesting is not just the graphs but the explanations. Essentially, a rare event as a specific occurrence is, indeed, rare. What's not rare is for some rare event to occur. Thus the idea that one sees a rare event and thinks "What are the odds of that?" is not the right question. The author notes that it's not that rare to see one person win a lottery twice. It is rare, however, to say that a particular person will win the lottery twice, before he actually wins.

Further in the book, we find chance trees. This was something I have not seen discussed in similar texts and I found it refreshing and useful. Tijms motivates them with the Monty Hall problem and he shows the reader how to produce them and compute probabilities for various outcomes. (As an aside, good investors use these trees frequently. Charlie Munger, vice-chair of Berkshire-Hathaway, notes this in his book, Poor Charlie's Almanack, where Munger discusses tools a good investor should have mentally ready.)

Part II of the book looks at the foundations of probability and discusses axioms, sets, and Bayes rule, to name a few topics. The discussion is easy to read and is deep enough to introduce the reader to the topic and leave him with a good understanding. Overall, it is very well done. The last chapter is about Markov chains and here, too, the author does an admirable job of explaining them at an introductory level and with enough detail to whet the reader's appetite.

Let me finish by mentioning the following example from the text that I found of interest, and rather surprising. How should a juror in the now famous O.J. Simpson trial view the fact that Simpson abused his wife, Nicole Brown? Is that relevant or not?

One of Simpson's attorneys argued that it is not relevant because only 0.1% of men who physically abuse their wives end up murdering them. Sounds clear, doesn't it? Well, not quite.

Through Bayes rule and the use of readily available statistics from the lawyer who made this claim, Tijms finds that "there is an estimated probability of 81% that the husband is the murderer of his wife in light of the knowledge that he had previously physically abused her." Thus this fact is pretty relevant as is the method Tijms employs to arrive at this result.

This is just one of the many wonderful examples that pepper and illustrate the text in this fine book.

David S. Mazel received his doctorate from the Georgia Institute of Technology in electrical engineering and is a practicing engineer in Washington, DC. His research interests are in the dynamics of billiards, signal processing, and cellular automata.

Preface; Introduction; Part I. Probability in Action: 1. Probability questions; 2. The law of large numbers and simulation; 3. Probabilities in everyday life; 4. Rare events and lotteries; 5. Probability and statistics; 6. Chance trees and Bayes’ rule; Part II. Essentials of Probability: 7. Foundations of probability theory; 8. Conditional probability and Bayes; 9. Basic rules for discrete random variables; 10. Continuous random variables; 11. Jointly distributed random variables; 12. Multivariate normal distribution; 13. Conditional distributions; 14. Generating functions; 15. Markov chains; Appendix; Recommended readings; Answers to odd-numbered problems; Bibliography.