# Review of Ten Great Ideas About Chance

Princeton University Press, Princeton and Oxford, 2017

272 pp., Illus. 44 b/w. Trade, $27.95

ISBN: 978-0-691-17416-7

On the face of it this is a book about probability, but it is a lot more than that. The book is written with a great deal of authority by two academics who know the subject well.

As the title implies, there are 10 chapters, each concerning chance or probability using its broadest definition. In general, the readership is assumed to be familiar with the technicalities as well as willing to read quite deep philosophical concepts. The 10 concepts are distinct without being completely separate. The first called measurement is about using counting to calculate probabilities and assumes some mathematical knowledge. The second chapter leads us into the murkier waters of judgement and how we can fool ourselves. The natural step from this is psychology, a more formal look at ways our judgement is misled by the environment or the words used. At this stage the reader begins to doubt that probability means what they think, or that it can be usefully pinned down at all. The next two chapters on frequency and mathematics are more reassuring. The frequency chapter goes back to using repetition and frequency of success to measure probability although this can never be 100% accurate, and then along comes mathematics to put probability on an axiomatic footing and give reassurance. Chapter six introduces Bayesian statistics that is old (eighteenth century) but is now a resurgent and practical way of, for example, assessing the outcomes of clinical trials. It is all about inference from data. The Bayesian view is cemented in the next chapter called unification. Here we meet hypothesis testing and exchangeability leading to ergodic theory. The next chapters could be considered slightly off topic. There are the computability ideas of Church and Turing and a look at randomness in algorithms. The discussion involving martingales is quite difficult. Next, a more physics approach via Boltzmann and thermodynamics; deterministic chaos vs. randomness, and finally quantum mechanics. A final more unifying chapter on induction follows that draws upon the belief side of chance as well as more systematic ideas of mathematical proof. The authors have included an appendix on the basics of probability that is useful, as are short summaries at the end of each chapter.

This is a book well worth a wide readership. It is extremely scholarly and could be used as the basis for many a thesis at final year undergraduate or masters level. It is enjoyable, but you have to enjoy fierce thinking.