Probability, Choice, and Reason

The book presents a fascinating collection of multiple problems on Bayesian techniques, probability and statistics, paradoxes and puzzles, applications to various real-world subjects, including economics, finance, law, medicine, and much more. Human common sense and intuition are often flawed and yield false answers. This book teaches us using various examples of how to apply a scientific thinking and deep reasoning to infer correct conclusions. The author is a renowned specialist in economics, professor at Nottingham Business School of Trent University, who served as an expert for national and international courts of law, committees of the House of Commons and House of Lords, and senior adviser to the UK government.

The preface lists various challenging questions with no evident solution, but states that they will be resolved in the course of the book reading. The monograph is structured in eight topic chapters consisting of numerous sections and subsections. Each section describes a specific problem with its consideration in different aspects, explanation and solution, modifications, presents appendix with derivations and calculations, supplement exercises with dozens of questions, references to reading and multiple links to internet sources.

Chapter 1, “Probability, Evidence, and Reason,” is devoted to the famous Bayes’ Theorem, which is used to update the probability of an event, given some new evidence, so it combines prior beliefs with new data. It is presented in several possible formulations and applied to a dozen problems, including the prosecutor fallacy, the taxi color, the rare type beetle, the false positive virus test and vaccine efficacy, the tennis player, the broken window crime story, the detective story with five suspects, the bus waiting, William Shakespeare’s Otello theater play, Sally Clark and courtroom justice, and each one of those is completed with additional subproblems and variations.

Chapter 2, “Probability Paradoxes,” starts with the classic Joseph Bertrand’s box paradox, and continues with other counterintuitive conundrums on change the options, including the famous Monty Hall problem, its variations in the Three Prisoners and Deadly Doors problems, William Shakespeare’s Portia’s challenge in Merchant of Venice, the Boy-Girl paradox and its modification in the Girl Named Florida problem, the Two-Envelopes paradox and its Necktie variant, the famous Birthday problem, the Inspection paradox of a bus waiting, the J. Berkson’s paradox or a collider bias with some application to Covid-19 study, the well-known Simpson’s paradox, and the Will Rogers phenomenon (similar to the more known Lord’s paradox – S.L.).

Chapter 3, “Probability and Choice,” presents problems which do not have a unique agreed solution or with a surprising solution. It considers William Newcomb’s paradox discussed in decision theory as a choice between reason unsupported by evidence, and evidence unsupported by reason; the Sleeping Beauty problem; the God’s Coin Toss problem with the self-sampling and self-indication assumptions; the Doomsday Argument of the human race extinction, related also to the Lindy effect; the problem of when to stop looking and start choosing, aka the Secretary problem; why we more often are in the slower lane problem, related to the observer selection effect; Blaise Pascal’s Wager problem of preference to believe in God, and the related climate change and mugging problems; J.M. Keynes’ Beauty Contest, or money market problem; and the famous Benford’s Law, helping to identify the fraud series of numbers.

Chapter 4, “Probability, Games, and Gambling,” focuses on the classic probability estimations and their applications. It describes the Chevalier’s Dice problem, discussed by B. Pascal, P. Fermat, C. Huygens, and J. Bernoulli; the Pascal-Fermat “problem of points”; the Newton-Pepys problem about gambling odds; Staking to reach a target sum problem; the Favorite-Longshot Bias problem, related to the Gambler’s Fallacy in lottery play; Poisson distribution and its features; Card counting and a strategy for playing blackjack, aka “21”; the Martingale betting system for a profit, and its variant in the “Devil’s shooting room” paradox; a problem of how much to bet having an edge, decided by the Kelly’s criterion; the Expected Value paradox; and Options, Spreads, and Wagers problems, including estimations for buying or selling the call or put options on the example of the World Cup “Total Goals” market.

Chapter 5 “Probability, Truth, and Reason” explores some mind-bending ideas which could be testable or falsifiable. It includes C.G. Hempel’s paradox, aka the Raven paradox, due to which if something might or might not exist, and is not observed, it is more likely to exist if it is less observable than something else which is more observable; the Simulated World question on the probability that we live in a virtual reality created by an advanced civilization; the Quantum World thought experiments on the famous E. Schrödinger’s Cat simultaneously existing in different states of the life and death, with the related “quantum suicide” experiment about the alternative realities, or the “many-worlds interpretation” (MWI); the Fine-Tuned Universe puzzle, which discusses the perception and concepts of the cosmological constant and the universe expansion, dark energy existence and symmetry-asymmetry paradox, initial conditions for the Big Bang and alien civilizations; and William Occam’s Razor principle of simplicity, or parsimony in choice of theories with the fewest assumptions, which corresponds to escaping overfit in statistical estimations (also minted by Einstein—“Everything should be as simple as it can be, but not simpler”—S.L.).

Chapter 6, “Anomalies of Choice and Reason,” describes efficiency and inefficiency of markets with information and prices, and discusses curious and classic market anomalies related to the Halloween, Super Bowl, NYC weather and other indicators, including the effects of testosterone and cortisol levels in traders. The Ketchup Anomalies and financial puzzles, with their explanations by D. Kahneman and A. Tversky due to their Prospect Theory are considered. The Wisdom of Crowds, or a group estimation is known from F. Galton’s observation that averaging of many individual guesses yielded with a high precision the weight of an ox at an exhibition; a similar idea of aggregation of multiple pieces of information can be used for prediction of a market behavior at a macroeconomic and microeconomic level to yield characteristics valuable for government and commercial policymakers. In contrast to it, the opinions of experts are used in the so-called superforecasting for the outcome of national or international events. Anomalies of Taxation are also described, with the commodity or ad valorem tax systems levied on quantity or price, respectively.

Chapter 7 “Game Theory, Probability, and Practice” revises the main methods of the game theory, including John Nash equilibrium and a dominant strategy for the optimum outcome for a player accounting also for the actions of other players, with examples of the Prisoner’s Dilemma applicable to the poker card game in casino, the TV “golden balls” game show, and the Dollar Auction paradox; repeated strategies in many-stage games where players can learn from the other players’ previous moves; and the mixed strategies when there is no dominant one, considered on the example of a striker and goalkeeper decision making for a penalty situation in a European football game.

Chapter 8 “Further Ideas and Exercises” embraces a number of fun problems defying intuition. There are: the Four Card riddle, or else P.C. Wason’s selection task; the Bell Boy paradox with a returned and divided cash; a question on “can a line of finite length be equal to an infinitely long number?”; the “proof” that the sum of all natural numbers up to infinity equals -1/12, and other paradoxes with infinite series – as a famous mathematician of 19th century N.H. Abel put it, “Divergent series are the invention of the devil”; the famous Zeno’s paradox that Achilles can never catch a tortoise, and the related paradox of Thompson’s Lamp; and some other Cool Down exercises such as “is zero an even or odd number?”

The book is completed with an extensive list of Reading and References, Solutions to Exercises given in detail in 35 pages, and Index. In spite of its rather dry title and a rigorous investigation of each problem, the book is full of intellectual entertainment. It is useful to instructors, educational to students of all levels, and attractive for readers interested in tools of better reasoning on any topic. The book presents a unique collection of not so widely known problems with their meticulous consideration by meaning and interpretation, derivation and calculation. Some additional articles and books on the related subjects can be found in the references below.

DOI: 00401706.2021.2020523
Stan Lipovetsky
Minneapolis, MN
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