Introduction to Probability

At Iowa State University, where I teach, the two undergraduate courses in probability theory for mathematics majors are cross-listed as statistics courses and taught by members of that department. Thus, I have never taught either of these courses, and never will. This has never bothered me greatly in the past, since my prior exposure to upper-level probability is limited to a couple of undergraduate and graduate courses. However, as a result of this book, and also a comparable book by Tijms called Probability: A Lively Introduction (also published by Cambridge University Press, within a month of this one), I found myself thinking that it might be fun to teach a course on this material.

This book is aimed at junior/senior undergraduates with a background in calculus (single variable, infinite series, and, for parts of the book, multivariable). It contains all the topics one would want to cover in an introductory course: the axioms for probability, classical probability (i.e., a finite number of equally likely outcomes), conditional probability, random variables (continuous and discrete being treated simultaneously), various specific distributions (normal, binomial, etc.), joint distributions, limit theorems, and so on. There is more than enough material in this book for one semester, though likely not enough for two, so an instructor has some flexibility in its use. It is a nicely written book, with clear explanations and lots of examples and exercises.

I must confess that I was a bit surprised that Cambridge University Press would publish two fairly similar and obviously competing books within a month of each other; this almost seems like McDonalds authorizing two franchise restaurants on the same block. However, since the books do compete with each other, some comparison between them seems appropriate.

In terms of topic coverage, the scope of this book is somewhat more modest than is true of Tijms’s text. Tijms covers a number of topics that are not covered here. Some are fairly non-standard topics (e.g., Kelly betting, renewal-reward processes, law of the iterated logarithm, modes of convergence) that may not likely be covered in most introductory courses. Some instructors, however, may regret the loss of an extended discussion of Markov chains. Tijms spends about a hundred pages discussing, in two chapters, both discrete and continuous Markov chains; this book does not discuss the continuous theory at all and only spends a page or so on the discrete case.

On the other hand, the book under review spends more time than does Tijms on the topic of basic counting techniques (combinations and permutations). Tijms covers this material in only four pages of text with four worked-out examples; Anderson and his co-authors spend about three times as many pages on this topic, with about three times as many examples.

Both books struck me as readable and accessible to the intended audience. They seem quite comparable in this regard, but if I had to pick a winner, I would say that the book now under review, perhaps because it covers less topics, is one that students might find slightly easier to read than Tijms’s book. Also, for professors who care about such things, this book employs color in a way that the Tijms text does not: all definitions and important facts are set off in colored boxes, for example, and a number of the illustrations in the text are in color.

One good feature of both books is the number of exercises. Both books have quite a lot of them, this text not as many as Tijms’s. The exercises in this text all appear at the end of a chapter (in contrast to Tijms’s text, where they appear at the end of each section). Here, each chapter’s exercise set is divided into three parts: there are “warm-up” problems, arranged by section, followed by “further exercises” that are not tagged to a particular section, followed by a set marked “challenging problems”. Solutions to selected exercises appear at the end of the book (mostly these are just numerical answers, without explanations or justifications); in addition, a detailed 254-page password-protected solutions manual can be downloaded, by text adopters, from the book’s webpage. One unfortunate aspect of the manual is that it does not provide solutions to the “challenging problems”. One would think that these are the problems for which solutions would be most helpful.

Both books strongly emphasize exercises of a computational character: “find the probability of…”, “find the density function of the random variable ….”, etc. However, both books also provide at least some exercises that call for proofs or for construction of examples. Some of the exercises in this book introduce interesting ideas that many instructors might think are worthy of classroom discussion; the Monty Hall problem, for example, appears as an exercise in this book, and the Buffon Needle Problem is a “challenging problem”.

Both texts also have an impressive number of worked-out examples, another excellent pedagogical feature. In addition, both books make an effort to make these examples interesting and “real world relevant”. In the book now under review, for example, there is a very nice discussion of the Sally Clark case from England, where a woman was wrongfully convicted of murder on the basis of faulty mathematical evidence (from a non-mathematician pediatrician) after two of her children died from sudden infant death syndrome. This is a great vehicle for explaining why multiplying probabilities for non-independent events is a bad thing to do, and also for discussing the “prosecutor’s fallacy”. For an American judicial decision raising similar mathematical issues in the context of a robbery, see People v Collins.

In connection with the Sally Clark example, I can’t resist putting in a plug for a very interesting and useful book called Math on Trial by Schneps and Colmez, which discusses this, and a number of other, examples of faulty mathematical reasoning (usually involving probability) in courtrooms. Anybody who teaches probability, or who is looking for a good topic for a talk to a math club, might want to make the acquaintance of this book. It is listed in a short but decent bibliography at the end of this text; Tijms’s book has no bibliography at all, an unfortunate omission that I should have mentioned in my earlier review of that book.

One interesting feature that this book has in common with Tijms’s is avoidance of the usual “theorem/proof” format; in fact, both books even avoid use of the very word “theorem”. Tijms refers to theorems as “rules”; Anderson and his coauthors use the term “fact”.) It appears that both books, though wanting to convey that probability is a field of mathematics, also want to stress the use of probability. The various theorems (or rules, or facts) that they cite are of interest to them primarily because of the way they can be applied to actual real-world problems.

This is not to say, however, that mathematical rigor is ignored in this text. Proofs of basic results are often given and proofs of more sophisticated results are generally at least sketched. Moreover, every chapter of this text but one ends with a section titled “Finer points”, in which some mathematically subtle or sophisticated topics are discussed. For example, in chapter 1, the authors look at the axioms for probability in more detail, proving the continuity of a probability measure, noting that there are difficulties with defining a probability measure on all the subsets of an arbitrary set, and defining a \(\sigma\)-algebra of subsets of a sample space.

The authors use a “clubs” symbol (§) in the main body of the text to signify that the discussion is elaborated on in this final section, and point out this convention in the preface of the book; unfortunately, students who don’t bother reading the preface (probably most of them, if my experience is typical) will not know what this symbol means. The result is that a reader of the book may not realize that what he or she is first encountering is not the complete story.

To summarize and conclude: I ended my review of Tijms’s book by saying that it should be on the short list of anybody looking for a text for an undergraduate calculus-based course in mathematical probability. At the risk of repeating myself, I must say the same about this book. Either book would make a fine textbook; I think that I would lean towards this one, if only because it seems to me to be more tightly focused on the basics and doesn’t include a lot of topics that I would view as peripheral to the course, but the Tijms book certainly has some features that, based on personal preference, might make it the preferred candidate.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.