Chance, Logic and Intuition: An Introduction to the Counter-Intuitive Logic of Chance

World Scientific, 2021, 256 pp., US$ 68.00

“A rational judgement about a chance event must not only be based on one’s hope or fear. It also must take into account the probability of the event”

hat probability and statistics increasingly pervade our daily lives is illustrated, for instance, by the mathematics related to the ongoing COVID crisis. In his lively book Chance, Logic and Intuition, Steven Tijms has embedded that mathematics in the context of an everyday narrative, one with a minimum of formulas. Here, I will discuss some observations that struck me most, or that, in my opinion, form the basis for the underlying message of this book, a book whose contents are richer than what can be discussed in a brief review.

The first part of the book, entitled “The Birth of Probability,” offers an exciting bird’s-eye view of the history of probability—or the theory of chance—from antiquity to the seventeenth century, when it gained a firm mathematical footing thanks to Jacob Bernoulli’s so-called golden theorem, that is, the mathematical law of large numbers. In antiquity, chance was believed to be the fickle and blind goddess $T\upsilon \chi \eta$, or Fortuna, who determines our fates. Divination was used as an attempt to extract information about the future. However, more rational approaches to the taming of chance emerged among the skeptic philosophers of the New Academy in the third century BCE. For instance, Carneades already believed in a form of the law of large numbers, as testified by his assertions on predictive dreams. Indeed, according to him, predictive dreams are not predictive at all. Since we sleep and have dreams every night, once in a while we are bound to have a dream that comes true solely by chance. He made his point via a comparison with the outcomes of a game of dice.

Even if such games were popular in ancient Greece, it is unknown whether contemporaneous mathematicians had a theory for how to calculate odds. This, however, does not prove, as Tijms points out, that Greek mathematicians never considered such issues. Since we also lack documentation on the rules of the various games of dice they might have played, Tijms suggests that it would have been a lucky coincidence for a text with calculations of odds to have survived. Or would it? Wouldn’t it have been more profitable not to reveal to others how to calculate odds? This might also be the reason for the relative silence on this topic in the Middle Ages, even though games of dice continued to be immensely popular.

The earliest known medieval calculation of odds appears in a love poem from the thirteenth/fourteenth century. Its correctness is all the more striking, since some four centuries later, the great mathematician Leibniz (!) made a mistake in calculating odds for two dice (p. 23). He failed to realize that two dice are different but identifiable objects, an observation that does not apply, say, to photons and nuclei.1 No wonder, then, that our students still have difficulties with combinatorics.

Although the word “probability” had already been used by Cicero in the first century BCE to refer to the plausibility of a judgment, the concept of mathematical probability was first introduced only in 1683, by Antoine Arnauld. A Jansenist theologian who had taken refuge in the monastery of Port-Royal des Champs and who became a friend of Blaise Pascal, Arnauld wrote the influential philosophical handbook often referred to as the Port-Royal Logic.2 In his book, Tijms discusses the Port-Royal Logic’s last chapter “dedicated to the question how one can arrive at a rational judgement about something that is unpredictable” (p. 84). Arnauld argued that “a rational judgement about a chance event must not only be based on one’s hope or fear. It also must take into account the probability of the event” (p. 84). He did not, however, give an explicit definition of probability.

For him, the probability of an event was, in fact, “some kind of inherent plausibility” that is quantified (p. 88). According to Tijms, the mathematical concept of probability (aleatory probability) is defined by analogy to the earlier concept of plausibility (epistemic probability). Tijms attributes Arnauld’s “mathematical wisdom” to Pascal (pp. 90–91), but it was Jacob Bernoulli in his Ars Conjectandi (1713) who first explicitly defined probability as a number between 0 and 1 that expresses the degree of certainty of an event and who established the law of large numbers for a particular case.

Tijms moves in his book’s second part, entitled “The Logic of Chance,” to discuss five cases illustrating wrong intuitions about chance events, and he very clearly explains the fallacies. The first concerns a rare event that occurred at the Grand Casino of Monte Carlo in 1913: the ball of a roulette wheel fell on black 26 times in a row. Great sums of money were lost as gamblers bet more and more on red, intuitively assuming that the law of large numbers applies to small numbers. An account of amazing coincidences that readily make the headlines—like winning the jackpot twice—follows. As Tijms explains, even if the probability that a given winner of the jackpot wins it a second time is extremely small, the probability that someone among a very large number of players will win it twice can be relatively large.

“Test, Test, Test,” the third case, contains amusing examples in which the notion of conditional probability is not correctly understood. For instance, suppose there is a test for detecting past infections of COVID-19, or any other disease, with an accuracy of 94%. Suppose that ten out of a population of 1000 people have been infected. Mass screening will then detect six times as many false positives as true positives, thus producing a large number of wrongly confident people who trust that they are already immune.

This is followed by a discussion of several renowned court cases, the first of which—the 1994 O. J. Simpson case—involved the murder of the celebrity’s ex-wife Nicole and her friend. Simpson had a history of spousal abuse. According to Tijms, the defense hoped to win its case by calculating that fewer than one out of 2500 cases of spousal abuse in the United States had resulted in murder. The prosecution, on the other hand, failed to realize that the relevant calculation would have been something different, namely, the probability that the murderer of an abused woman is her abusive partner. Under some mild assumptions, Tijms estimates that this probability is 90%, which might have won the case for the prosecution. Even so, as Tijms rightly points out, conviction should be based on tangible evidence and good judgment. This represents a new element in the discussion, which is further elaborated in the epilogue.

The fifth and final case is the well-known Monty Hall dilemma. About such questions, Persi Diaconis remarked, “But I do know that my first reaction has been wrong time after time on similar problems. Our brains are just not wired to do probability problems very well” (p. 197). The second part concludes with ten challenging problems that aim to test the reader’s understanding of cases similar to those presented. Solutions are helpfully provided.

In the epilogue, Tijms addresses the question why our intuition fails us again and again in probabilistic problems. Drawing from history, he illustrates how epistemic probability and aleatory probability came to be interpreted as “branches of the same tree” (p. 216) or as “a double-edged sword” (the subtitle of the epilogue). The first to do so was Jacob Bernoulli. This initial confusion has led to the extension of the use of probability to statistical data, in other words, to quantification of epistemic probability, that is, the all too often occurring substitution of numbers for “good judgment.”

No wonder, then, that the author concludes with a plea for exercising good judgment rather than pursuing the ideal of developing a calculus of epistemic probability. As he states in his preface, though, “the main reason I wrote this book is [so] that the reader will enjoy it” (p. x). I, for one, enjoyed it immensely, discovering in the book refreshing views, amusing facts, and much food for thought.