Coincidences.

Somebody tells you the story, and it always ends with the same four words. What are the odds.

Nobody who says it wants a figure. But take the question literally and there is a confusion inside it, and that confusion is the subject of this book.

Every coincidence carries two probabilities, not one. There is the chance that the event happened at all. And there is the chance that anybody ever found out. They are separate quantities, they can differ by orders of magnitude, and almost every coincidence you will ever hear reports the first while feeling the second. The second is never asked about, for a simple reason: when a coincidence happens and nobody notices, the world does not record a near miss. It records nothing at all.

Seven cases are taken apart here, in the same order, under the same pressure.

Four belong to the author. A former student rising from behind a director's desk in a Tunisian public office. A horn sounded on a Tunis highway by a driver who turned out to be his wife, and who had no idea. A flatmate not seen for decades, standing in a shopping mall three thousand kilometres away in Oman. And a fish, given away by two strangers on a street in New York, whose two ends both turned out to lie inside one man's small circle of acquaintance.

Three belong to everybody, and they can be checked. The D-Day codenames that surfaced in a national newspaper crossword in the weeks before the invasion of 1944. The balloon that carried a girl's name a hundred and forty miles across England and was found by a farmer who happened to know a girl with the same name. The Bulgarian lottery that drew the same six numbers twice in four days, and the commission that was sent to find out why.

Six of the seven empty out. One does not.

And the case that survives is the worst documented in the book, while the three that are best attested are the three emptiest. The author sets that out rather than tidying it away, because he cannot decide what it means.

This is not an argument that the universe is signalling to you. It is not a debunking manual either, and it makes the case that the man at the dinner table who explains that nothing is remarkable is worse than the credulous man he is correcting. One rule governed the writing throughout: nothing is invented. Where a fact was missing it was left missing, and the gaps are visible in the text. A city goes unnamed because the author is no longer certain of it. A figure is refused rather than estimated. Two sources disagree and the disagreement is printed.

Drawing on probability, cognitive psychology, the mathematics of human movement, migration, and the psychology of rumour, and written by a Tunisian educator who told these stories at dinner tables for forty years before he did the arithmetic on them, the book leaves the reader with three questions that take roughly a minute and will empty most of what they are applied to.

What the arithmetic never touches is the thing anybody actually remembers. Improbability and significance are separate quantities, they were never correlated, and nearly all the confusion in this subject comes from people who have felt the second and reported the first.

Nobody has ever gone home moved by a denominator.