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Probability: A Lively Introduction

Henk Tijms
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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This is an attractive textbook for an introductory probability course at the upper undergraduate level. It covers not only the standard material for such a course (discrete probability, the axioms of probability, conditional probability, discrete and continuous random variables, jointly distributed random variables, limit theorems, Markov chains, etc.) but also some topics that might be considered more unusual, such as Kelly betting, renewal-reward stochastic processes, and the law of iterated logarithms. Topics from statistics (confidence intervals, Student-t distribution, Baysian inference, etc.) also appear. The book is quite well-written, nicely motivated, demonstrates considerable enthusiasm for the material, and gives lots of examples of the usefulness of probability.

This book (hereafter referred to as Lively Introduction) may be described as the big brother of the third edition of Understanding Probability, an earlier book by the same author. Understanding Probability has an interesting format: Part I was mostly example-driven, at a reasonably elementary level, and Part II was more of a traditional probability text, covering both discrete and continuous probability, with calculus used as appropriate.

Lively Introduction is an expanded version of Part II of the earlier book. The chapters of Lively Introduction correspond almost precisely with the chapters of this part of the earlier book with almost precisely the same titles. There is one extra chapter titled “Additional Topics in Probability”, but even here the material is not all new; some of this chapter comes from sections of the older book that have been moved around. Moreover, much of the content of Part II Understanding Probability appears verbatim, or nearly verbatim, in Lively Introduction.

However, Lively Introduction consists of more than just the chapters of Part II of the earlier book. More than 100 pages of new content have been added. The increased heft is attributable to both an expanded set of exercises and some additions to the textual material as well.

To deal with the latter point first: the author has added quite a few new worked out examples to Lively Introduction, and has also taken the opportunity to modify some of the textual discussions in the older book. For example, the first Borel-Cantelli lemma is now proved in the text, and the second left as an exercise; in Understanding Probability, only one version was given, and that was an exercise. Also, Lively Introduction deals a bit more precisely with the Kolmogorov axioms, stating in the main body of the text that there are, for large sets, set-theoretic difficulties with defining probability for all subsets and then, in an appendix, discussing \(\sigma\)-algebras. In addition to all these (and more) changes, the author has added a substantial number of new worked-out examples to the text.

Parts of this book (e.g., Kelly betting) are taken from Part I, rather than Part II, of the earlier book; some other material from Part I (such as probabilities and expected value in roulette and craps) now appear as exercises. Unfortunately, there is some interesting material from Part I (notably the Monty Hall problem, which students always seem to find interesting) that did not seem to find its way into this text.

And speaking of things that did not find their way into the text, let me record one mild quibble here: I think the author’s discussion of basic combinatorics, which basically comprises a five-page (six, counting some problems) appendix, could have been beefed up. The fact that this material is in an appendix rather than the main body of the text doesn’t bother me, but I think five pages (and only four simple examples) is likely not a sufficient amount of time to spend on this topic, particularly since it is my experience (from teaching combinatorics, rather than probability) that many students may not be as familiar with this material as the author assumes they are. One particular thing the author could have done is to give some examples of common incorrect combinatorial reasoning. This is done, for example, in Ash’s Basic Probability Theory, a book I used as an undergraduate (now a Dover paperback); I remember thinking, even back then, that this was an informative discussion.

As noted above, the number of exercises in this book has increased from Understanding Probability. There are now a lot of them, and they seem quite well-chosen: some are reasonably easy, but others will challenge the best students. They also struck me as interesting and informative, rather than busy-work problems. Solutions to the odd-numbered problems appear at the end of the book; these are not just numerical answers but worked-out solutions showing how the answer was obtained. This is a good compromise between the competing desires to have the book be one that can be read for self-study while at the same time making a good number of problems available for an instructor who wants to assign them as graded homework.

In addition to these back-of-the-book solutions, the publisher offers a password-protected solutions manual to all the problems in the book. This 265-page manual is also an impressive work, and should be of considerable assistance to faculty teaching from this book.

The back cover of this text advertises it as intended for “undergraduate or first-year-graduate-level courses”. While the book’s suitability as an undergraduate text is beyond dispute, there may be some issues with using it as a graduate text. Most graduate courses, I would think, would want to discuss the measure-theoretic aspects of probability at a deeper level than is presented here. A graduate course instructor might want to prove the central limit theorem in detail (in this book, there is a sketch of a proof). Moreover, a graduate course might include topics such as ergodic theory, Brownian motion and martingales, none of which are discussed here.

One amusingly quirky aspect of the book is that the author seems to dislike the word “theorem”. Most upper-level mathematics texts consist of lots of theorems and proofs, denominated as such and set off from the rest of the text. This book isn’t written that way. Mostly the author just tries to incorporate everything into the narrative, and when certain specific results are set off, they are generally referred to as “rules”: the strong law of large numbers, for example, is labelled “Rule 9.5 (the strong law of large numbers)”, even though it is followed by a sketch of the proof.

One question that naturally arises is whether this book is a better candidate as a text for an undergraduate course than is Understanding Probability. I think so, for several reasons. For one thing, the additional examples and homework problems are valuable. In addition, the unconventional two-part format of the earlier book may not work as well in a class.

A principal competitor of this text is A First Course in Probability by Sheldon Ross (now in its 9th edition, with a 10th apparently on the way). Both are very good textbooks and both are well-suited for use as a text in an upper-level course. In terms of overall student readability, though, I would give the edge to Tijm’s text. Moreover, in these days of high textbook prices (as well as high tuition), it is not inappropriate to consider the cost of a text in making adoption decisions. As I write this, the publisher of Ross’ book, Pearson, advertises that text as having a suggested retail price in excess of 200 dollars; Cambridge University Press offers a paperback version of Tijm’s text for about 45.

Bottom line: this book should be on the short list of any instructor who is shopping around for a text for an upper-level probability course. University librarians should also take a look at this text.

Mark Hunacek ( teaches mathematics at Iowa State University.

1. Foundations of probability theory
2. Conditional probability
3. Discrete random variables
4. Continuous random variables
5. Jointly distributed random variables
6. Multivariate normal distribution
7. Conditioning by random variables
8. Generating functions
9. Additional topics in probability
10. Discrete-time Markov chains
11. Continuous-time Markov chains.