Reasoning About Luck

Writing a physics or mathematics book for an audience with little background in these areas is a difficult undertaking, fraught with dual perils. On the one hand, if the book is too technical, it may be incomprehensible to the intended audience; on the other hand, if the book is too watered down, it may provide only an illusion of understanding. Many of us have probably had, at some point, the experience of hearing somebody use technical words in a way that indicated the speaker had no idea what he or she was talking about; it is not a pretty sight.

The book under review, I am happy to report, generally manages to avoid both of these two pitfalls. The goal of the book is to discuss the rudiments of probability and statistics and to illustrate how these ideas are used in physics, specifically in relation to entropy, heat and (somewhat more briefly) quantum mechanics. Along the way the author also spends some time explaining Newton’s laws of motion. The book, according to the author, is intended to be accessible by a reader “who knows little mathematics and little or no physics”. In particular, calculus is not used, but the author does develop mathematics to the point that, by page 65, we encounter limits and infinite series.

This illustrates a point that needs to be stressed: the text does assume a reasonably high level of intelligence on the part of the reader, and a willingness to work hard. (The author states that the students, in the course that spawned this book, “have typically been free of any background in science, but also free of math-anxiety.” He also acknowledges that the book “is not light reading. Only by working through it with a pencil and paper and understanding the solved problems at the ends of the chapters are you likely to get very far.”) With this caveat (about which, more later), the author has, I think, largely achieved his goals.

The book has its origins in a “general education” physics course at Cornell University that the author taught a number of times, the goal of which was to expand on the students’ understanding of high school mathematics to the point where he or she could reason quantitatively about the uses of probability in physics (and fulfill a graduation requirement). The choice of topics might be viewed as a little eclectic for a general education physics course, but as the author points out, “I saw no particular virtue in completeness and no particular vice in the unconventional.”

The first four chapters of the text do not involve physics at all, and thus should be of interest to people who teach mathematics as well as to physics faculty. In the space of eighty pages — roughly a third of the book — the author discusses a number of issues relating to probability and statistics (the definition and interpretation of probability, basic statistics, elementary combinatorics, the law of large numbers, the exponential and natural logarithm functions, the binomial, normal and Poisson distributions) along with some examples (radioactive decay, statistics in baseball, testing a vaccine, polling, and even knowing how much lobster to buy for a lobster dinner for a large crowd.)

It is perhaps a measure of the author’s sense of humor that the word “lobsters” appears in the index; unfortunately, though, the page reference is wrong: it should be 51 instead of 5. This may be the most serious error I noticed in the book.

Physics enters the picture in chapter five, where the author points out that in order to understand heat, one must understand motion. So, he provides a chapter on Newton’s laws of motion. This is then followed by chapters on molecular motion, statistical mechanics and thermodynamics, and the direction of time (i.e., why does ice not grow in your soft drink?). Two final chapters, on chaos and quantum mechanics, again looked at from the perspective of probability and statistics, round out the book.

Each chapter ends with a selection of worked-out problems. (One of them is the famous Monty Hall problem, albeit not with that name.) Unfortunately, there are no unsolved exercises, so an instructor teaching out of this text will have to look elsewhere for homework problems to assign. The book also lacks a bibliography, but a few of the chapters do have, at the end, a few paragraphs titled “Further Reading” that list some additional sources.

The author’s style of writing is very pleasant — erudite, witty and conversational. Well-motivated students (or even faculty members, for that matter) should find reading this book to be an enjoyable experience as well as an educational one.

Now, about that caveat I mentioned in the third paragraph, above: this text is a Dover reprint (with a brief addendum added, making three comments and corrections on points in the original) of a book that was first published in 1996. The twenty years that have passed since it was first written have, unfortunately, seen a dramatic decline in the mathematical preparedness of students in college. What we used to be able to assume, mathematically, of a high school graduate, we no longer can; this is particularly true, of course, of students who do not intend to major in a STEM discipline. Moreover, this lack of preparation is accompanied by an increase in fear and loathing of mathematics; the author’s students may have been largely free of math anxiety, but my lower level non-major students have enough to share. These students also seem to have difficulty reading any serious book about mathematics that does not come equipped with cartoons, pictures, multi-colored illustrations, boxed definitions, and the other trappings of a much more elementary textbook.

For these reasons, I suspect that this book may, notwithstanding its many virtues, be now simply beyond the ken of most of these students. And I definitely think that the author is being overly optimistic when he suggests in the preface that high school students could profitably read the first four chapters. On the other hand, at a different level — students in an honors seminar, for example — the book should serve nicely. It could also be useful as supplemental reading for a course in probability and statistics, or an upper-level physics course. And finally, as noted earlier, even faculty members might enjoy reading it; I did.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University. Thanks to some very ill-advised academic decisions in his youth, he knows much less physics than a person in his position should.