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There are many non-technical (i.e. non-textbook) books on ”mathematics in general” or ”statistics in general”, but let us focus more specifically on probability. I have read and reviewed (the reviews, and other material marked (*), can be found on my web site www.stat.berkeley.edu/~aldous) over 70 such non-technical books relating to probability. Most of these books have almost no mathematics, so what is in them? Well, here are some titles, chosen to illustrate the breadth of topics.
• Dicing With Death. Chance, risk and health
• A Random Walk Down Wall Street
• Risk. A practical guide for deciding what's really safe
• The Black Swan: The impact of the highly improbable
• Chance: the life of games and the game of life
• Dance with Chance: Making luck work for you
• The Drunkard's Walk: How Randomness Rules
• Luck: the brilliant randomness of everyday life
• The Cult of Statistical Significance
• Rock, Paper, Scissors: Game Theory in Everyday Life
• God, Chance and Purpose
Clearly the notion of “chance” in these books is much broader than what mathematically-oriented readers regard as “probability”. After thinking about this contrast for several years, partly through developing a “Probability in the Real World” course (*), I have come to some conclusions – just personal opinions, of course – that I would like to share with readers in this article.
Let me point out two features of a first undergraduate course in probability. What we actually teach is mathematics – random variables and independence and so on. And underlying the mathematics is some historically-derived view of what probability is about. Coin tossing and random sampling, LLNs and CLTs, regression and hypothesis testing, and so on. A student who takes more courses in probability, does so within pure mathematics or mathematical statistics or one of a familiar list of academic “applied probability” subjects – finance, information theory, population genetics, queueing, statistical physics, algorithms, etc. Over the last 75 or so years since probability and mathematical statistics became established disciplines, the list of applications has grown but the underlying ”what probability is about” viewpoint remains the same. To caricature a little, we have built a vast intellectual edifice upon the observation that there's a mathematical model for the results of throwing dice.
Of course there's nothing wrong with this edifice, but let me make an analogy with History. It would be much easier to teach human history in terms of kings and battles – very concrete matters – rather than (say) the evolution of ideas about how societies should be run or the effect of technological advances. Analogously, I believe that much of what we traditionally teach about chance, is taught because it is easy to teach by virtue of being definite mathematics. If you doubt this, try teaching from The Black Swan or another nonmathematical book!
Here's my view. We live in several overlapping worlds. There's the natural world that exists independently of humans; the human social and economic worlds; the world of human artifacts; the world of ideas and perceptions and motivations. Outside the classroom, we know that chance enters all these worlds in many ways; from the way we first meet a future spouse to the spatial fluctuations in mass density of the early universe that led to galaxy formation; from the chance of Sarah Palin becoming the 2012 Republican Presidential nominee (currently 20% on the prediction market) to the event that Kokura was covered by clouds on August 9, 1945. Inside the classroom we forget all this, and revert to the mathematical tradition that implicitly views probability as “things that are similar to dice”.
Amongst the books I have reviewed, the majority (quite sensibly) explicitly address some particular part of the world of chance, but a dozen or so with titles like How Randomness Rules our Lives (I categorize these as “popular science”) seem to attempt a big picture. But – to my taste – the result is a sampling of stories rather than an articulation of a big picture. As an alternative I am working on a project to make a list of 100 representative instances of chance, and thereby give one illustration of “a big picture” by examples. Let me encourage readers to look at the current draft list (*) and send me additional suggestions.
Returning to the topic of books, my opinion is that there are too many undergraduate textbooks on probability – “too many” in the sense that they are insufficiently distinctive. With non-technical books also, I sometimes get the impression that authors have not made the effort to read what already exists before writing their own book! But those books which uncover a novel topic are often remarkably good. A nice example is Fortune's Formula, which combines a clear non-mathematical exposition of the Kelly criterion with entertaining anecdotal history of personalities from Shannon and Thorp to Boesky and the principals of Long Term Capital Management.
Two topics unexpectedly lack books. Dawkins' Climbing Mount Improbable is a great sequel to The Origin of Species but, like the latter, doesn't actually engage probabilities. I don't know any non-technical book relating chance to evolution, genetics or wider areas of biology in any substantive way. Haigh's Taking Chances is a wonderful collection of elementary probability calculations involving real-world sports and games, and many books give detailed analyses of strategies for particular sports or games, but no-one seems to have tried to engage the bigger picture of the role of chance in sport.
Only a few of the reviewed books are popular exposition by mathematicians. These tend to combine the interesting parts of undergraduate courses with popular topics such as the Monty Hall problem, a style which I find competent but rather unimaginative.
Finally, on the topic of popular exposition of what we academics know, I view Wikipedia as an underexploited venue. Currently there are around 600 pages relating to Probability, listed on an obscure page Catalog of articles in probability theory. Many are useful, giving a definition (Contiguity (probability theory) or a theorem statement (Lovasz local lemma) or a set of formulas (Beta distribution) or an explanation (Benford's law). But as one moves away from such specific topics I find the coverage less complete and less satisfactory. In particular, even for the usual academic “applied probability” subjects mentioned before, I don't find the entries very authoritative. If it is a fact that the thousands of academic papers involving probability models in Queueing theory or Population genetics have contributed substantially to human knowledge, then this fact is not apparent from the current Wikipedia articles! Of course, to write a short article on a huge topic is very difficult. It should be less difficult to choose some level intermediate between Population genetics and Ewens's sampling formula, or intermediate between Queueing theory and M/M/1 model, and write authoritative Wikipedia articles at that level. Arguably, one such article represents a much more valuable contribution to scholarship than the average research paper! So let me encourage readers to write such articles.
David Aldous, Berkeley
Editor's note: This is the first installment of a regular opinion column by David Aldous (U.C. Berkeley).