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What are the Odds?

Looking at Coincidences Mathematically

By James Case

Fluke: The Math and Myth of Coincidence. By Joseph Mazur. Basic Books, New York, NY, March 2016. 288 pages, $26.99.

Fluke: The Math and Myth of Coincidence. By Joseph Mazur. Courtesy of Basic Books.
The subject of Joseph Mazur’s fifth popular math book is “seemingly unlikely events.” He recognizes three main types—coincidences, flukes, and serendipities—and offers relatively concise definitions for each. Sandwiched between a brief “Introduction” and an equally-brief “Epilogue” are four main parts: “The Stories,” “The Mathematics,” “The Analysis,” and “The Head Scratchers.”

Part I, consisting of three chapters, introduces the ten “real-life stories” on which the greater part of the book is based. They range in complexity from that of a Texas woman who won four multimillion dollar lotteries in less than twenty years to the tale of two men named Francesco who arrived in the same hotel lobby at the same time, expecting to meet and interview a stranger named Manuela for possible employment. Both met a woman of that name, retired to a nearby conference room, and began a somewhat bewildering conversation before returning to the lobby and discovering that each Francesco had been talking to the wrong Manuela!

Part II contains five chapters on the computation of probabilities of compound events. Mazur offers many examples, including the following: diffusion processes, laws of large numbers, Bernoulli trials, the birthday problem, Pascal’s triangle, Galton’s peg board simulation of an unbiased one-dimensional random walk, Huxley’s typewriting monkeys, and the proverbial flap of a butterfly’s wing. He addresses all this and more without invoking anything beyond grade-school arithmetic and elementary algebra.

Part III, which contains only two chapters, applies the lessons learned in Part II to the analysis of the stories from Part I to demonstrate that, whereas some seemingly unlikely events are indeed highly unlikely, others should not even be considered surprising. In particular, given the large number of high-value lotteries in the world and the vast number of people who participate religiously, it is all but inevitable that someone somewhere will win more than once, and by no means unlikely that there will be a four-time winner!

Part IV of Fluke is perhaps the most interesting. It consists of five essays concerning coincidences that, at least for the moment, completely escape analysis. The first investigates coincidences in DNA evidence gathered at crime scenes and lawyerly attempts to mislead jurors regarding the likelihood of consequent mistakes. The second describes accidental findings by scientists studying seemingly unrelated phenomena, such as the discovery of penicillin in Alexander Fleming’s untidy laboratory, where fungus from a separate study inadvertently contaminated a staphylococcus culture, and Röntgen’s discovery of X-rays while investigating electrical currents in a partially evacuated glass tube. According to Mazur, any number of earlier investigators also produced such rays, but failed to notice them due to weaker currents and/or less complete vacuums. The third essay involves a rogue trader who wagered recklessly while failing to anticipate two massive flukes. The first one made him rich, while the second left him bankrupt. The fourth considers attempts to evaluate psychic powers and extrasensory perception (ESP) statistically, while the fifth essay compares some elaborate coincidences in literary fiction with their (typically less-elaborate) counterparts in real life.

Books of the present sort can serve as texts for the “quantitative reasoning” courses colleges and universities are increasingly obliged to offer victims of “math anxiety.” Intended to enhance a student’s ability to assess the risks and uncertainties encountered in daily life, such courses can prove immediately useful to novices faced for the first time with the need to purchase healthcare insurance, schedule a holiday picnic, or acquire additional student loans.

James Case writes from Baltimore, Maryland.

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