On the very day you think of a friend you haven’t heard from in years, that friend calls you. We tend to call moments like this fate or a miracle. But one mathematician said that such things happen far more often than we think, and that it is actually natural for them to occur roughly once a month. This is Littlewood’s Law.
Put simply, Littlewood’s Law is the rule of thumb that “even an event so rare it happens once in a million times will occur about once a month in an ordinary person’s life.” It is named after the British mathematician John Edensor Littlewood and is often cited as a way to explain seemingly miraculous coincidences without any supernatural force.
In this article, we will look in turn at how Littlewood’s Law came about, the reasoning behind the once-a-month calculation, everyday examples, and the lessons the law offers us.
Littlewood’s Law: Looking at Miracles Through Numbers
Where Did Littlewood’s Law Come From?
This law was not completed by one person at one time; it grew out of the ideas of two mathematicians.
(1) Littlewood’s observation
In the “large numbers” discussion of his 1953 book A Mathematician’s Miscellany, Littlewood pointed out that people tend to greatly exaggerate how improbable unlikely events are. He added that, over a whole lifetime, a coincidence of about one in a million is trivial.
(2) Dyson gives it a name
The person who gave this idea the name “Littlewood’s Law of Miracles” and made it widely known was the physicist Freeman Dyson. In a 2004 review of a book criticizing psychic powers and pseudoscience, he set out the law neatly.
(3) Who was Littlewood?
Littlewood (1885–1977) was a mathematician at Trinity College, University of Cambridge, famous for his long collaboration with his colleague Godfrey Harold Hardy in analysis and number theory. His habit of weighing probabilities with numbers rather than intuition is reflected in this law.
The Calculation Behind a Monthly Miracle
The calculation Dyson presented is surprisingly simple.
(1) Defining a miracle
First, a miracle is defined as “an event that has special significance when it occurs, but whose probability of occurring is one in a million.” This also fits well with the feeling we usually associate with the word miracle.
(2) The number of events in a day
Assume that a person is awake and paying attention to what is happening around them for about eight hours a day, and that during that time they see or hear something at a rate of roughly one per second. That comes to about 30,000 events a day, or about one million a month.
(3) The conclusion
It is not strange for one event with a one-in-a-million probability to be mixed in among a million events. So the conclusion is that, on average, encountering something miraculous about once a month is to be expected.
Of course, these numbers are only rough estimates, and what counts as a miracle differs from person to person. Still, they are enough to show the principle that rare events, as they pile up, eventually appear right in front of us.
Understanding Littlewood’s Law Through Everyday Examples
The law shows up not only in individual experience but even more clearly where many people are gathered.
- Miracle at the card table (Bridge Hand): In the card game bridge, the probability that the 13 cards dealt to one player are all of the same suit is extremely low, on the order of one in hundreds of billions. But if we assume that 2 million people in Britain play 30 hands a week all year round, the number of hands dealt in a year reaches several billion, so even such a hand can be expected to actually turn up roughly once every few decades.
- Big-city coincidences (Big City Odds): The magician Penn Jillette said that in New York, with a population of about 8 million, one-in-a-million events happen eight times a day.
- Dreams that come true (Prophetic Dreams): Because so many people dream every night, it is entirely possible for someone’s dream to coincide by chance with something that happens the next day.
The idea that, given enough attempts, even the rarest event will eventually happen is called the “Law of Truly Large Numbers.” The phrase comes from observations made by the statisticians Persi Diaconis and Frederick Mosteller in their study of coincidences, and Littlewood’s Law can be seen as applying it to a single person’s day.
A Balanced View in the Face of Coincidence
Littlewood’s Law is less a claim that miracles do not exist than a story about how we look at coincidences.
(1) Surprise is not the same as meaning
Strange coincidences stay in our memory for a long time, while the countless moments when nothing happened are quickly forgotten. That is why we easily feel that coincidences are much rarer and more special than they really are.
(2) Be wary of hasty interpretations
Taking a chance coincidence straight away as evidence of prophecy or psychic power can cloud our judgment. The law is a good yardstick when confronting pseudoscience and unfounded claims.
(3) Still, enjoy the moment
That does not mean we need to belittle a happy coincidence. Enjoying the moment while knowing it can be explained by probability is the healthiest attitude.
Littlewood’s Law reminds us that even something that looks like a miracle can become an ordinary event in the face of large numbers. Getting into the habit of thinking of the numbers hidden behind an astonishing coincidence helps us see the world more calmly.
Telling surprise apart from meaning when faced with an uncanny coincidence: that is the first step toward probabilistic thinking.
