How to analyse chance, ignorance and risk

The Art of Uncertainty: How to Navigate Chance, Ignorance, Risk and Luck. By David Spiegelhalter. Pelican; 512 pages; £22. To be published in America by W.W. Norton in March; $32.99

A hazard of teaching mathematics rather than, say, history is that the homework is a lot harder to come up with. After all, “was Henry VIII a good king?” is a reasonable question to ask either a classroom of nine-year-olds or a lecture theatre of postgraduates. But “solve this quadratic equation” would leave the classroom nonplussed and the lecture theatre unimpressed. Maths is learned by doing, and devising a problem that is easy enough to be accessible, yet hard enough to be satisfying, is a devilish problem itself.

Partly for this reason, books that successfully communicate how mathematicians think, but are aimed at those not already in the tribe, are both valuable and rare. Over the decades many eager students have devoured Thomas Körner’s “The Pleasures of Counting” (1996) and Sir Timothy Gowers’s “Mathematics: A Very Short Introduction” (2002). Now Sir David Spiegelhalter, emeritus professor of statistics at the University of Cambridge, has added to the genre with “The Art of Uncertainty”. His new book will appeal to many more than just aspiring mathematicians, for its topic is universal: how to analyse chance, ignorance and risk.

With the covid-19 pandemic so recently in the past, few will need reminding of how vital such analysis can be. If a new virus is running rampant and the majority of deaths are among those who have received an even newer vaccine, is that evidence that the vaccination scheme is harmful? (No. Those at highest risk were vaccinated first; with an imperfect vaccine this group was always likely to have the most deaths.)

Yet the book finds space for lighter fare, too. How did Giacomo Casanova, a notorious adventurer and lover in the 18th century, manage to design a lottery that was guaranteed to make the French government money? How much of top football teams’ performance comes down to luck rather than skill? How certain are scientists about the value of physical constants like the speed of light?

Professor Spiegelhalter’s exploration of such questions is delightful for three reasons. First, he uses them to illuminate broader ideas about how probability and statistics work. So a discussion of vaccine safety proceeds to Bayes’s theorem, a procedure for refining one’s judgment of probabilities as new evidence comes to light. Bayes’s theorem is crucial for everything from the scientific method to evaluating legal evidence and progress in artificial intelligence. The joy of Professor Spiegelhalter’s approach is that he reaches this deep truth via nothing more than some intuitive assumptions and (very) simple algebra. It is this sort of elegance that makes mathematicians enjoy maths.

The second reason to read this book is the author’s fascinating philosophical questioning. What, after all, does “probability” actually mean? Is it a basic property of the world, or a mere expression of our ignorance? When we talk about the probability of rolling a six on a dice, is that the same concept as the probability of there being life on Mars? The author’s personal view—that uncertainty is a subjective relationship between observer and observed rather than a fundamental quantity—is a welcome corrective for those who think of maths as a black-and-white subject.

Most important, though, is Professor Spiegelhalter’s skill at communicating these ideas. Much of probability and statistics can be highly counterintuitive, and the maths underlying it is often fearsome. But this is not a difficult book to read or understand. For that reason alone, it is a coup—and, no doubt, a beacon for the author’s successors.

For more on the latest books, films, TV shows, albums and controversies, sign up to Plot Twist, our weekly subscriber-only newsletter

© 2024, The Economist Newspaper Limited. All rights reserved. From The Economist, published under licence. The original content can be found on www.economist.com