The Math and Myth of Coincidence


By Joseph Mazur

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A mathematical guide to understanding why life can seem to be one big coincidence-and why the odds of just about everything are better than we would think.

What are the chances? This is the question we ask ourselves when we encounter the strangest and most seemingly impossible coincidences, like the woman who won the lottery four times or the fact that Lincoln’s dreams foreshadowed his own assassination. But, when we look at coincidences mathematically, the odds are a lot better than any of us would have thought.

In Fluke, mathematician Joseph Mazur takes a second look at the seemingly improbable, sharing with us an entertaining guide to the most surprising moments in our lives. He takes us on a tour of the mathematical concepts of probability, such as the law of large numbers and the birthday paradox, and combines these concepts with lively anecdotes of flukes from around the world. How do you explain finding your college copy of Moby Dick in a used bookstore on the Seine on your first visit to Paris? How can a jury be convinced beyond a reasonable doubt that DNA found at the scene of a heinous crime did not get there by some fluke? Should we be surprised if strangers named Maria and Francisco, seeking each other in a hotel lobby, accidentally meet the wrong Francisco and the wrong Maria, another pair of strangers also looking for each other? As Mazur reveals, if there is any likelihood that something could happen, no matter how small, it is bound to happen to someone at some time.

In Fluke, Mazur offers us proof of the inevitability of the sublime and the unexpected. He has written a book that will appeal to anyone who has ever wondered how all of the tiny decisions that happen in our lives add up to improbable wholes. A must-read for math enthusiasts and storytellers alike, Fluke helps us to understand the true nature of chance.



The Stories


                   It starts as true story,

                   first wondrous, and rare,

                   then colossal collisions

                   of galactic affair

                   strikes with so much surprise,

                   we’re confused between thoughts

                   that it might be just fluke

                   we believe in a lot,

                   and what if it’s not.

J. M.

LIFE IS FILLED with expectancies, to-dos, and benign pleasures, but bewildering encounters and phantasmagorical stories give the blisses of being alive. Here is a browsing of just a few accounts of how our world is both enormous and small, and of how we come to distinguish flukes from coincidences. We shall return to these stories in Part 3, after we have some machinery to illuminate their hidden quantitative elements.

Chapter 1

Exceptional Moments

REMEMBER THAT TIME when you were leisurely strolling along a street in a foreign city, Paris or Mumbai perhaps, and bumped into an old friend you had not seen in a long time? That old friend that you bumped into: what was he doing there in your place and time? Or remember that moment when you wished for something and it happened just as you wished? Or the fluke of hard luck you had when everything went wrong during your vacation because of unfortunate timing? Or that time when you were astonished to meet someone who shared your birthdate? They were times when you must have had a sudden feeling of synchronicity that shrank the universe, an illuminating transformation that magnified your place in the cosmos. You felt part of an enlarged and centered circle of humanity with just a few persons—or perhaps just you—at that center.

Did you ever pick up a phone to call someone you hadn’t called in a year and, before dialing, hear the person on the line? It happened to me in 1969. Thinking about it, it seems more likely to happen than not. After all, a whole year had gone by—365 days when it didn’t happen. Add to that the number of days the year before, another year when it didn’t happen. And add to that the number of days from then to now. It never happened again. Now we are talking about a serious amount of time when the coincidence did not happen.

Imagine this story. You are sitting in a café in Agios Nikolaos on the island of Crete, when you hear a familiar laugh at a table in a nearby café. You turn to look at the person, a man. You cannot believe that he is your own brother. But there he is, unmistakably your own brother. He turns toward you and he is as surprised as you. It happened to me in 1968. Neither one of us knew that the other was not back home in New York or Boston.

Or imagine this. You are browsing used books in a bookstore far from home when you come across a book you remember from your childhood. You open it and find your own inscription. It is a copy of Moby Dick with your own name on the inside cover and your own marginal markings throughout the book. It was a book you had in college. It happened to a friend who told me that he was browsing the shelves of a used bookstore in Dubuque, Iowa, a city he had never been to before.1

In 1976 my wife, my two children, and I had been touring through Scotland, when on one snowy day our Vauxhall car broke down in the small town of Penicuik. A mechanic at the only garage in town told us that the problem was our alternator, and that it would be three days before he could replace it. We headed to the nearest pub, hoping to spend the night. The publican was a man of few words, but when we told him that we were from America, he livened up to proudly say, “Next week we will have someone from America coming to sing. You probably know her. I don’t know her name, but there is a poster downstairs.” He brought us to a large poster announcing a stovies night2 concert by Margaret MacArthur.

“Margaret MacArthur!” my wife and I exclaimed simultaneously. “She is our neighbor. We know her very well!”

The publican nodded, and with blank countenance muttered, “Thought you would.”

America is truly a small country.

There are moments when we are struck by magnificent coincidences. They are the foci of nature’s web of associations because, especially in the solitude of this digital age, we want to fit into the intimidating world with a sense of self, an identity, a purpose, and a feeling that some parts of our lives have destinies. Daunted by the chilling vastness of the forever-expanding universe in an endless space and time, it is reassuring to know that we are more connected than we think, or that the universe lines up for us.

With any coincidence story is the question of whether there is something in the universe that perturbed time and place enough to trigger the coincidence and conceal its cause. Some people have questioned whether there are metaphysical connections. Some say that there is oneness in this universe, an energy that we cannot be aware of, a force that changes our patterns of behavior, a knowing something that we do not know.

Causality is the Western way of interpreting the meaning of events. Nineteenth-century Western causality had a strict classical physics view that the laws of nature govern the movement and interaction of all observable objects. If the variables of the present state are precisely known, then the future is completely predictable. In other words, any predictions of the future are tied to whatever we can know of the past and present. However, by the early twentieth century, with the invention of quantum mechanics, Western philosophy took a radical shift of viewpoint: observable objects are driven by non-observable events of the quantum world, governed by simple, wondrous rules. One such rule claims there are no roads not taken. Every particle is ordered to follow not just one path, but also every possible path with a probability that depends on the path. Predictability, in that quantum mechanics point of view, is limited to probabilities that an object will be somewhere on each path and in a particular state. In other words, careful observation of exactly what happened in the past only gives us uncertain probabilities of what might happen in the future.

Of course, there is always the question of what causes one person to choose a path forward. We are not talking about the mechanical path of an object. Why did you, dear reader, choose to read this far into this book? You have free will that has almost nothing to do with classical physics, or the path of observable objects, or the new physics. The coincidences of this book are related to decisions that people make, roads taken and those not. Human decisions are a matter of free will, where neither relativity nor quantum mechanics come into play, although there are always other strong external influences. We decide on a path. Someone else decides on a path. Then, bam! The paths meet, and we have no apparent cause. The problem with apparentness is that it requires an observable object traveling on an observable path. So, unless there are brain-wave connections between distinct individuals, free will trumps all quantum influences.

There is also the Eastern way, however. The Chinese, for instance, have their Tao, in which opposites cancel out to make the whole and total picture. In it the nothingness is also part of the whole. A block of stone can become a sculpture defined by the remaining stone and the stone that has been carved away. It is surely a different way of thinking. And still the Tao belief is certainly different than any theology that looks at the world as if everything in the world, from cells of organisms to subatomic particles of minerals, is prearranged from the time of creation, and laws governing causality can be broken only if ratified by God’s will. The Taoist believes that coincidences are in the sympathy of all things, and for that reason all events in the world stand in one relationship beyond any causality and any appearances. In other words, there are no flukes. But that same Taoist also believes that underneath there is a hidden rationality. The revered Tao Te Ching, now some 2,500 years old, tells us:

Heaven’s net is wonderfully vast and enveloping;

Though wide-meshed, nothing slips through.3

Just as all parts of a whole work in harmony to complement one another, so, too, all events in the world stand in one meaningful relationship with the whole, which is in central “meaningful” control.

Walt Whitman also told us that we have some connection to the All, and that there is a moral purpose and intention that we are forced to follow unconsciously. He put it this way:

            As within the purposes of the Cosmos, and vivifying all meteorology, and all the congeries of the mineral, vegetable and animal worlds—all the physical growth and development of man, and all the history of the race in politics, religions, wars, &c., there is a moral purpose, a visible or invisible intention, certainly underlying all . . . something that fully satisfies . . . That something is the All, and the idea of All, with the accompanying idea of eternity, and of itself, the soul, buoyant, indestructible, sailing space forever, visiting every region, as a ship the sea.4

Chapter 2

The Girl from Petrovka and Other Benign Coincidences

            What connection can there have been between many people in the innumerable histories of this world, who, from opposite sides of great gulfs, have, nevertheless, been very curiously brought together!


IF YOU LEAVE your house, a great many encounters and happenings are possible. The probability of each may be small, but when we group them together and ask for the probability that at least one of them will happen, the likelihood goes up. These stories are just ten of many that effectively represent ten characteristic classes. They will be analyzed in Part 3.

Story 1: The Girl from Petrovka

Class: Lost, unlikely to be found, objects accidentally found by someone deliberately looking for them

One of the most celebrated coincidence stories involves actor Anthony Hopkins. After being cast to play the part of Kostya in a movie version of The Girl from Petrovka, Hopkins spent some time searching for the novel in bookstores near London’s Leicester Square Underground station. Unsuccessful in his search and about to return home, he noticed a book lying on one of the benches of the same Underground stop. It was not just a copy of The Girl from Petrovka, but the lost copy belonging to its author, George Feifer.

It is a truly bewildered story. I should be forced to concede that this one would be so foreign to any reasonable theory of the frequency of coincidences that I should have to congratulate the story for escaping any explanation. But in truth it does not escape analysis. George Feifer told me the true story himself: He had used a copy of the American edition of The Girl from Petrovka to highlight words that required British translations for the UK publication of his book. He submitted his translations to the British publisher and checked them on the copy plates. One day he met a friend in Hyde Park Square and gave the friend his marked-up American edition. In a daze of the moment, the friend put the book on top of his car and, late for an afternoon meeting with a girl, speedily drove off. On seeing Feifer on the movie set, Hopkins told Feifer that he found the book at an Underground stop. I wrote to Hopkins for his side of the story. Predictably, he never replied.

Story 2: Jack Frost and Other Stories

Class: Unexpectedly found familiar personal objects not searched for

A comparable story involves writer Anne Parrish. According to the original telling (a story very different from the many floating through cyberspace), while in Paris, after attending Mass at Notre-Dame and visiting the bird market on a sunny June Sunday in 1929, Anne and her husband, industrialist Charles Albert Corliss, stopped at Les Deux Magots for lunch. Leaving Charles alone with his wine, she strolled the bookstalls along the walls of the Seine. It was not unusual for her to spend hours rummaging rows of books on long tables. On that day, she found Helen Wood’s Jack Frost and Other Stories. After a short haggle with the bookseller, she paid one franc, ran to her husband who was still sitting with his wine, excitedly put the book in his hands, and told him that it was one of her favorite books when she was a child. He slowly turned the pages. After a few moments of silence, he handed the book back to her opened at the flyleaf where penciled “in an ungainly childish scrawl, was: ‘Anne Parrish, 209 North Weber Street, Colorado Springs, Colorado.’”2 It had been her book when she was a child.3

Story 3: The Rocking Chair

Class: Requiring reasonably precise time and space and not human chance meetings

A coincidence has to be more than a story forced to give a surprise or to hide its cause. Here’s one that happened to me some years ago. My wife was pregnant and her aunt told her that she must have a comfortable rocking chair for nursing the newborn baby. She sent a check to cover the purchase of a new rocking chair. My brother had the perfect rocking chair, and my wife and I found that same chair at a furniture store in Cambridge. It was uncommonly wide, a Shaker design with thin black spindles and a high back. But the chair was not in stock, and so we asked that it be delivered to my brother’s house in Cambridge when it was available for delivery. We would pick it up from there and bring it home to Vermont at our next visit. Some weeks later my brother and his wife were hosting a small gathering. Someone sat on their rocking chair, and the entire chair collapsed underneath him, breaking to pieces. Embarrassed, my brother courteously told his guest not to worry. At that precise moment the doorbell rang and our new chair was delivered. One can only imagine the surprise at the party, when my brother took the golden opportunity to console the guest by saying, “Oh, that’s fine; we just called for a replacement.”

Story 4: The Golden Scarab

Class: Dream coincidences in somewhat generous time and space

A young woman patient was telling Swiss psychiatrist Carl Jung of her dream about a golden scarab. We have Jung’s version: “While she was telling me of this dream I sat with my back to the closed window. Suddenly I heard a noise behind me, like a gentle tapping. I turned round and saw a flying insect knocking against the windowpane from outside. I opened the window and caught the creature in the air as it few in. It was the nearest analogy to a golden scarab that one finds in our latitudes, a scarabaeid beetle.”4 Jung goes on to say, “We often dream about people from whom we receive a letter by the next post. I have ascertained on several occasions that at the moment when the dream occurred the letter was already lying in the post-office of the addressee.”5

Story 5: Francesco and Manuela

Class: Chance meetings of humans in precise timing and space

My wife and I were in a van riding along the hairpin road that cut through la Costa Smeralda, the eastern coastal hills of Sardinia, high above the clear emerald waters of the Tyrrhenian Sea. They were breath-halting moments when our Italian driver was gesticulatively pointing out historical sites while turning his head back and forth for quick peeps of the dangerous curves ahead and glances at passengers in the backseats. We were spending some time at Studitalia, an Italian language school in Olbia, a picturesque small port city on the northeast coast of Sardinia. It was a weekend. And—as it happened every weekend—the school had offered its students an excursion immersed in Sardinian culture and beauty. The driver was Francesco Marras, the school’s director.

A student sitting in the front passenger seat asked him when and how the school started.

“Well—” he began his answer, thinking about the story he was about to tell as the van wigwagged from one side of the road to the other seconds before the next curve. “When the school opened just three years ago in 2010 there was only one student.” In a typical Italian way, he used his right hand to illustrate his story, and his left to blithely steer the van.

And so, we learned how on that opening day Francesco went to the lobby of the Hotel de Plam to meet the school’s first student, Manuela from Madrid, for an orientation excursion, which involved a boat trip to the magnificent Isola Tavolara, an enormous flat-topped rock island some 3 miles offshore. Since Francesco and Manuela were early and their boat was late, they went off to a café for a drink. They sat for an hour, chatting in Italian. Manuela talked about where she lived in Spain, about her work, her boyfriend, and her interests. Francesco talked about the school. Soon Francesco began to wonder why Manuela would want to take Italian when her command of the language was quite excellent.6 When he finally asked about the level of Italian she expected to learn at the school, the mix-up became clear.

“Learn Italian? Why, do you think I need Italian lessons?” she asked.

The confusion lasted several more minutes before Francesco realized that Manuela was the wrong Manuela, who had expected to meet in the hotel lobby someone by the name of Francesco!

They both returned to the hotel lobby to find the other Francesco interviewing the other Manuela for a job she neither expected nor wanted.

Why is this story so surprising? Because it has been humanized as a story with a place and time, specific names, a colorful character who seems to be telling the truth. Intellectually, we are not fooled. We know that with large numbers of possibilities these encounters happen, and that they are not so unusual.

Story 6: Albino Taxi Driver

Class: Chance meetings of humans under generous timing and generous space

These stories and others like them are more common than we generally think. We are told such stories, and many of us have experienced them. Just the other day I met a woman who told me a wonderful story: In Chicago one day she got into a taxi driven by a man with albinism. Three years later she got into the same man’s taxi in Miami. “Now, what are the chances of that?” she asked me. Yes, this is a wonderful story, but let us deconstruct it. Taxis frequent particular neighborhoods. The woman is an executive of a private equity firm, someone who often takes taxis in different major cities. Taxi drivers that do not have albinism are not as distinguishable; so, a person who uses taxis often, might expect to hail a taxi without noticing that the driver is familiar, unless that driver happened to be a person with albinism. Still, I would agree that we should indulge in some sort of fascination with the fact that Miami and Chicago are 1,200 miles apart.

Story 7: Plum Pudding

Class: Associations with familiar objects

And here is another story about a doorbell ringing to announce a coincidentally surprise visitor. I learned this one along with several others in L’Inconnu: The Unknown by early twentieth-century astronomer Nicolas Camille Flammarion.7 It’s one of those double coincidences, the kind that brings some astonishment, and then a new surprise happens to top it, turning it into a triple coincidence.

Flammarion writes that Emile Deschamps, a celebrated nineteenth-century poet, told this tale. Deschamps was a young boy at boarding school in Orléans, France, when he met an English émigré by the curiously non-English name of M. de Fortgibu. Dining at the same table, M. de Fortgibu suggested that the young Deschamps taste a dish that was almost unheard of in France, plum pudding.

For ten years Deschamps, having not seen or heard of it again, forgot about his discovery of a plum pudding that oddly contained no plums. Ten years later, passing a restaurant on the boulevard Poissonière that displayed the strange pudding on its menu, Deschamps was reminded of M. de Fortgibu. He ordered a slice, but was told by the counter ladies that a certain gentleman had ordered the whole pudding. One of the women turned toward a man in a colonel’s uniform who was eating at one of the tables.

“M. de Fortgibu,” she called out, “would you have the goodness to share your plum pudding with this gentleman?”

Deschamps didn’t recognize M. de Fortgibu.

“Of course,” M. de Fortgibu answered. “It would give me great pleasure to share a part of this pudding with the gentleman.”

It wasn’t likely that he recognized Deschamps, either.

Now, that should have been the full coincidence, but it was not. Some years passed. Deschamps had not seen or thought about plum pudding. And then one day he was invited to a dinner at the home of a lady who announced that an unusual dish was to be served: a real English plum pudding.

“I expect a M. de Fortgibu will be there,” he joked.

The evening of the dinner arrived. A magnificent plum pudding was served to the ten guests seated while Deschamps told the coincidence story of M. de Fortgibu and the plum pudding. Just as Deschamps completed his story, everyone heard the doorbell ring and M. de Fortgibu was announced.

You and I would think this was all planned. Deschamps thought so. Perhaps the dinner host had used Deschamps’s little joke to build a joke of her own. But no! It gets even more interesting. By this time M. de Fortgibu was an old man who walked with a cane. He walked slowly around the table looking, for someone in particular. As he came close, Deschamps recognized him. Surely it was he.

“My hair stood up on my head,” Deschamps said on relaying this story sometime later. “Don Juan, in the chef d’oeuvre of Mozart, was not more terrified by his guest of stone.”

But Deschamps was not the person the newcomer sought, It turned out that M. de Fortgibu (the same) was also invited to dinner, but not that dinner. He had mistaken the address and rung the wrong doorbell. It was a triple coincidence that must be so rare that you would think that the chances of its happening in one’s lifetime must be staggeringly close to zero. Yet it did happen, if we can trust M. Flammarion.8

“Three times in my life have I eaten plum pudding,” Deschamps reflected on his confounded experience, “and three times have I seen M. de Fortgibu! A fourth time I should feel capable of anything . . . or capable of nothing!”

Flammarion, the respected astronomer who has lunar craters, Martian craters, and asteroids honoring his name, was a collector of coincidences. As he was known for being a collector, people sent him their stories. He collected hundreds. Some quite astounding! Many were sent to him anonymously from all parts of the world, so it is very hard to trust their truthfulness, even though he does say that some had multiple witnesses, that some had a sincerity he vouches for, and that others have “all the marks of good faith.”

Story 8: The Windblown Manuscript

Class: Coincidences dictated by natural causes

The most remarkable coincidences are those of Flammarion’s personal experience. One is a captivating story that suggests that there are some miraculous forces looking out for us, chance perhaps, or unknown forces that parallel those of nature. He was writing his eight-hundred-page popular treatise on the atmosphere.9 It was to become his definitive work. At the end of the nineteenth century it was very celebrated both for its detail and its accessibility. Just at the point where he was busy writing the third chapter of the fourth section, a chapter about the force of wind, the most extraordinary thing happened. It was a cloudy midsummer day. He was in his study. One window, facing east and overlooking some chestnut trees and avenue de l’Observatoire, was open. Another window faced southeast with a magnificent view of the Paris Observatory, and a third faced south onto rue Cassini. He had just finished writing: “


  • "Always entertaining and frequently insightful, Fluke is never less than thought-provoking."
    --Amir Alexander, Wall Street Journal

    "Mazur gently dashes icy water on our sense of wonder, patiently doing the math to explain multiple lottery winners, 'remarkable' accidental scientific discoveries and wrongheaded government policy."
    --Keith Blanchard, Wall Street Journal

    "Mazur takes what could be difficult, abstruse subjects--probability and statistics--and makes them entertaining. The author draws examples and illustrations from a variety of fields--law enforcement, economics, the sciences--and, when he unavoidably gets into some fairly complicated mathematical discussions, he explains his terms and remembers that, for the most part, his readers aren't mathematicians. An ideal book, then, for the lay reader who is curious about the nature of coincidence."
    --Booklist Online

    "Well written, entertaining... an understandable introduction to probability for the layman."

    "Mazur's thoughtful tour reveals the explanatory power of probability theory in the larger world."
    --Publishers Weekly
  • "By studying anecdotes, probability, the past, the present, and the connectivity of everything, Mazur finds proof of the inevitability of the sublime and inexplicable. If you've ever wondered how the decisions of seven billion people can add up to moments of coincidence, Mazur's careful and clear explanations are for you."
    -UTNE Reader
  • "In Fluke, the author takes us on a marvelous guided tour of the world of the unlikely and the improbable. After reading Fluke, you will definitely come away with a deeper understanding of why wildly improbable coincidences may not be so improbable after all."
    --Ronald Graham, Chief Scientist at the California Institute for Telecommunications and Information Technology

    "A tour de force of masterful writing that weaves together simple and not-so-simple mathematical notions of probability and statistics into various intriguing coincidences from fact and fiction, explaining with nuance various strange phenomena. Mazur's book will teach you some of this mathematics, leaving you quite equipped to understand the role of chance in your life without resorting to magical thinking."
    --Gizem Karaali, Editor, Journal of Humanistic Mathematics

    "The chances are very slim that you'd ever read this blurb. A simple-minded calculation puts the odds at about 50,000 to one against. Yet... here you are. How weird is this seemingly far-fetched coincidence? Well, dear reader, you've picked up the right book to answer that question."
    --Charles Seife, author of Zero: The Biography of a Dangerous Idea
  • "With charm and clarity, Joe Mazur leads us through the strange terrain of chance and surprise. He explains why apparently remarkable coincidences are usually more likely than we imagine, because we underestimate how large our world really is. Not so much probability theory, as improbability theory! A terrific read, and a welcome antidote to superstition and gullibility."
    --Ian Stewart, author of Professor Stewart's Incredible Numbers

    "Mazur has written a wonderfully insightful book. He shows how it is that our purely psychological expectations about what might happen in the real world, and our culturally acquired notions of order and disorder, often give us a completely false sense of the chance that something will, in fact, occur in the world outside."
    Richard Lewontin, Professor of Biology Emeritus at Harvard University and author of The Genetic Basis of Evolutionary Change

    "Clear, humorous, and grounded in history and culture, Fluke shows you why anything that can happen is bound to happen, sometime. But just as rainbows still thrill us when we parsed the physics, dissecting bizarre coincidences doesn't dilute our amazement. Mazur has accomplished the seemingly impossible feat of writing a book for everyone."
    --Marjorie Senechal, Editor-in-Chief, The Mathematical Intelligencer

On Sale
Mar 29, 2016
Page Count
288 pages
Basic Books

Joseph Mazur

About the Author

Joseph Mazur is an emeritus professor of mathematics at Marlboro College, and the author of four other popular mathematics books, the most recent of which is the highly acclaimed Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Among his many honors is a Guggenheim fellowship. Mazur lives with his wife, Jennifer, in Marlboro, Vermont.

Learn more about this author