Book Review. Probability: A Very Short Introduction, by John Haigh

Probability:  A Very Short Introduction by John Haigh (2012) is a nontechnical and quite brief (main text:  117 pages) overview of probability theory and applications.  Only elementary arithmetic, and occasionally high school algebera are used, but equations are absent except in a 2-page appendix.  Most of the math is explained verbally.  There are occasional excusions into stochastic processes, decision theory, game theory, information theory, and statistics.  (Statistical physics is notably absent among the applications discussed.)  Readers should note that the book is written from a British  perspective, and references to British sports (cricket) and televsion shows, and King Richard III, may be lost on US readers.

Chapter 1, "Fundamentals", begins by explaining and comparing three interpretations of the concept of probability:  classical ("objective"), frequentist, and subjective.  The classical approach is based on first principles, such as symmetry.  For instance, one could decree that each face of a (fair) die is equally probable.  The frequentist approach states that probabilities emerge as empirical, long run frequencies of trial replications.  The subjective approach refers to a degree of belief, or a personal opinion.  The mathematics of probability are largely independent of which interpretation is being used.  Chapter 2, "Workings of Probability", introduces a few basic mathematical ideas:  the addition law for disjoint events, the multiplication law (in the context of conditioning), independence, and ways to handle overlapping and multiple events. 

Chapter 3, "Historical sketch", uses a historical account to introduce (in lesser detail) the laws of large numbers, the central limit theorem, Bayes' theorem, probability distributions (including the Gaussian and Poisson), the Markov property, the law of the iterated logarithm, measure theory, and martingales.  I am persuaded by the author's argument (p. 35) that the normal distribution's name is unfortunate, and that "Gaussian" should be preferred.  Chapter 4, "Chance Experiments", introduces additional distributions (uniform, binomial, geometric, and exponential) and mentions several extreme value distributions.  The concepts of mean and variance are also introduced.

Chapter 5, "Making Sense of Probabilities", is perhaps the most interesting one in the book.  This chapter and those that follow describe a series of case studies where probabilistic thinking can be applied.  This chapter starts with explaining odds in gambling.  Next, biostatistics appears with a comparison of the risk difference with the risk ratio, a topic discussed at greater length by Gigerenzer (2002), a book with which I will make several comparisons.  After a short discussion on "combining tiny probabilities" (referring to the Borel-Cantelli lemmas), there is a very good 3-page section titled "Some Misunderstandings".  Several of the misunderstandings are connected to quoting probabilities without specifying the reference class, an issue emphasized in Gigerenzer (2002).  The chapter ends with sections on "Describing Ignorance" (which introduces the Beta family of distributions) and "Utility".

Chapter 6, "Games People Play", discusses various applications in gambling as well as TV game shows.  Proficient card counters are banned by casinos; the author finds that "No better tribute to the power of understanding probability has ever been paid" (p. 77). Chapter 7, "Applications in Science, Medicine, and Operations Research" tackles Brownian motion, pseudo-random number generators, Monte Carlo simulations, error correcting codes, and several case studies in health:  amniocentesis decision making, estimating probabilies of carrying a gene based on knowledge of phenotypes in one's family tree, modeling epidemics, and improving the efficiency of mass blood tests by pooling samples.  The chapter concludes with sections on airline overbooking and queuing theory.

Chapter 8, "Other Applications", discusses law (particularly the Prosecutor's Fallacy), randomized response in surveys, and diagnostic tests for the use of performance enhancing drugs in sports.  In this latter section, the subtleties of a diagnostic test's sensitivity and specificity are finally discussed (pp. 100-102).  (As most readers will eventually have personal experience with medical diagnostics, I much prefer the more in-depth treatment of this topic found in Gigerenzer, 2002.)  The chapter concludes with sections on applications in sports and finance, including the Black-Scholes equation. 

The final chapter, "Curiosities and Dilemmas", returns to games, including a discussion of Parrondo's Paradox.  It concludes with some dilemmas surrounding genetic testing.  The book ends with the aforementioned appendix as well as a brief list of References and Further Reading.

The first sentence in the book is "Probability is the formalization of the study of the notion of uncertainty".  I take issue with this claim.  At the very least, the phrase "a formalization" should replace "the formalization".  There are other kinds of uncertainty, such as those found in approximation theory (where the error bounds on a given approximation can be thought of as a kind of deterministic uncertainty) or fuzzy logic.  Moreover, Heisenberg's Uncertainty Theorem is nowhere to be found in the book; it can be thought of as a consequence of wave-particle duality in quantum physics.

There are many nontechnical books about probability theory and applications; many are reviewed by the eminent probabilist David Aldous, including an earlier and longer work by John Haigh.  I've read none of the books he has reviewed except for Taleb (2005), which covers some of the psychological aspects of applying probability in real life.  However I have read Gigerenzer (2002), not mentioned by Aldous, but which remains my own favorite work on probabilistic thinking. This is why I make several comparisons above between it and Haigh (2012).  To finish the comparison, Gigerenzer (2002) seems very much focused on risk and its psychology, whereas Haigh (2012) is concerned with probability at large, and gives a more systematic and wider overview of its theory and applications.  However, I do think Gigerenzer is stronger when it comes to explaining the pitfalls of probabilistic thinking.  One strength of Haigh (2012), compared to the works reviewed by Aldous, is that Haigh (2012) is a very quick read, appropriate for the VSI (Very Short Introductions) series.  From that perspective alone, it may have a niche audience. 

References

 

Gerd Gigerenzer, 2002:   Calculated Risks:  How to Know When the Numbers Deceive You.  Simon & Schuster.

John Haigh, 2012:  Probability:  A Very Short Introduction.  Oxford University Press.  (Very Short Introductions, Vol. 310.)

Nassim Nicholas Taleb, 2005: Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. Second edition. Random House.