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

As noted in Preface, human mind often judges by similarity to already known cases, that in psychology is called the system 1 fast thinking. However, the fast thinking, intuition and common-sense work poorly in estimating chances and probabilities. This book shows on multiple examples how to employ a meaningful reasoning to get correct solutions without using mathematical formulas and symbols. The book consists of 2 parts and 10 chapters, each divided into many sections.

Part I “The Birth of Probability” contains four chapters on the history of the probability concept. Chapter 1 “The Silence of Antiquity” explains why this idea was not developed in ancient Greco-Roman times, in spite of philosophical discussions on the fate. Predictive dreams and divinations were also supported by dice casting, which was known in the most ancient civilizations from 6000 BCE.

Chapter 2 “Dice and Odds” covers the Medieval Europe dice gambling. French cleric of 13th century Richard de Fournival reveals his systematic calculations of the odds of possible sums of three dice in the game Zara, an early version of the Craps. A single die has the same chances for any of its six sides, but in rolling two or three dice some sums are more often, and he suggested tables of odds for all of 6 × 6 × 6 = 216 ordered possible combinations of dice outcome. The famous physicist, astronomer, and mathematician Galileo Galilei (1564–1642), whom Einstein called the father of modern science, worked on evaluations of the ordered combinations for Archduke of Tuscany Cosimo II de’ Medici and his gambling friends. The eminent Italian polymath of the 16th century Girolamo Cardano wrote books on mathematics and gambling where he argued that a game can be fair and equal to the players by chances, and calculated the odds, though with some errors. The mathematical genius Pierre-Simon Laplace (1749–1827) in his Philosophical Essay on Probabilities later defined probability by dividing the favorable outcomes by the total number of possible outcomes.

Chapter 3 “The Mathematics of Chance” describes the life of Blaise Pascal (1623–1662) who wrote his first mathematical works from the age of sixteen. In 1653 Chevalier de Méré, a courtier at the court of the Sun King Louis XIV, presented Pascal with gambling problems, one of which formulated as following: two players make a bet, and the player who is the first to win three times wins the bet and gets the pot; but the bet is prematurely broken off at a score of 1–0, so how to give each player a fair share of the pot? Pascal suggested a solution based on the average expectation, and discussed it with another brilliant mathematician Pierre Fermat who came with different approach, but both methods led to the same result. A great Dutch–French scientist Christiaan Huygens (1629–1695) wrote a treatise On the Calculus of Games of Chance, formulating the concept of expected value, the unfinished games, two- and three-players games, and the gambler’s ruin problems.

Chapter 4 “The Birth of Probability” continues with development of the concept of probability in the book of 1662, Logic or the Art of Thinking, by theologian Antoine Arnauld, coauthored by Pierre Nicole. The book became known as the Port-Royal Logic, was translated into many languages, and studied at universities throughout Europe. Its last chapter describes the rational judgment about something unpredictable, and proposes to take into account the probability of the event. Actually, the word probable had been coined by the Roman statesman Cicero, who translated to Latin the old Greek word plausible used for reasonable judgements. The term probability as a value from 0 to 1 was introduced by the famous Swiss mathematician Jacob Bernoulli (1655–1705), who had written a textbook on probability theory, was in correspondence with Leibniz, introduced the term integral, the concepts of a priori and a posteriori probability, proved the law of large numbers, founded the binomial distribution and Bernoulli processes. Abraham de Moivre derived the central limit theorem of probability theory.

Part II “The Logic of Chance” in its six chapters demonstrates on various examples how to deal with the counter-intuitive logic of chance. It starts from Chapter 1 “The Capricious Ways of Chance” which describes origins of casino business throughout Europe in 17th–18th centuries till nowadays, when only in 2015 the American casinos players spent 357 billion dollars. The casino games are described, together with lotteries, and collective psychosis around them. The gambler’s fallacy is defined as a delusional reasoning that if a chance event occurs less often than average during some time, then its probability should increase in the nearest future, although the events are independent. The prominent psychologists Tversky and Kahneman called this persistent probability misconception the belief in the law of small numbers.

Chapter 2 “Amazing Coincidences” discusses events defying the odds, for example, collected at the website The Cambridge Coincidences Collection. Improbable things happen, like the same person winning millions in consecutive jackpots, when the chances are 1 in 17 trillion as told by the New York Times. Such rare cases are analyzed by mathematicians Mosteller and Diaconis who introduced the law of truly large numbers which states that extremely unlikely coincidences are likely to happen if a sample is large enough, for instance, in millions of a population. The same day birthdays, hits by meteorites, appearance of black swans, predictive dreams and disasters, ancient oracle Pythia and modern Paul the Octopus or other “psychic” animals, miracles of different kind and problems with extraterrestrials are clarified as well.

Chapter 3 “Test, Test, Test” considers a positive test for a disease, and related to this situation conditional probabilities, their inverse, and fallacy of the transposed conditional, explained by the Bayes’ Rule. Thomas Bayes (1702–1761) Essay Toward Solving a Problem in the Doctrine of Chances nowadays became the famous Bayes’ theorem applied for various problems. Multiple examples are described, including the false-positive paradox, with application to different aspects of Covid-19 pandemics and testing.

Chapter 4 “Trials by Numbers” discusses judiciary investigations and testimonies. It starts with O.J. Simpson double murder case, and considers statistical argumentation by the prosecution and defense on conditional probabilities. A case of misused probability theory was the famous Dreyfus Affair, when in 1894 a Jewish Captain of French Army was wrongly blamed and convicted for treason. A mathematical “proof” was based on handwriting identification, which was done incorrectly, and based on the so-called prosecutor’s fallacy of the following wrong reasonings: if it is highly improbable that an innocent person is by chance involved in a crime, then this person must be guilty. In 1902, the prominent French scientist Henri Poincaré (1854–1912) concluded that that “proof” was totally absurd and illegitimate, and finally in 1906 Dreyfus was fully rehabilitated. Other situations with a killer nurse and witch trial are analyzed by probabilities of the incidents, with explanations offallacies.

Chapter 5 “The Monty Hall Dilemma” describes a hit show hosted during 30 years by the legendary Monty Hall (1921–2017). The show presented three identical closed doors on stage. Behind one of them is the grand prize—a new car. Behind the two other doors there is nothing or a bizarre consolation prize such as a goat. The contestant chooses one of the doors and stands in front of it. The host opens one of the other doors, always a losing door, and the contestant is offered a final choice: either to stick with the same door, or to switch to the other remaining door. The contestant makes the final choice—and the host opens the other remaining door. So, to switch or not to switch—that’s the question! Marilyn vos Savant, mentioned in the Guinness Book of Records due to her exceptionally high I.Q., was the first person correctly answered this question and proved that switching doors increases the chance of winning twice.

Chapter 6 “Ten Challenging Problems” proposes exercises on the considered topics and their solutions. Epilogue “The Two-Edged Sword” discusses psychological mechanism of representativeness heuristics, original concept of epistemic probability and mathematical concept of aleatory probability, and concludes on problems of the counter-intuitive world of chances. References and Future Reading suggest a long list of sources for each chapter. List of Illustrations and Index finalize the book.

The book presents a fascinating reading on history and thoughts of the founders of probability theory. It can be interesting to students and instructors, researchers and laymen because of its thought-provoking content helping to understand games and various other real-life problems. Additional sources can be found in the references (Lipovetsky 2013, 2021, 2022).

Stan Lipovetsky
Minneapolis, MN
[email protected]