What is Littlewood’s law? Littlewood’s law is a law made by a mathematician called John Littlewood that says we should expect to see a miracle, or something improbable, once every 35 days.
John Littlewood was a British mathematician. He was born in 1885 and he spent most of his life working on analysis, number theory, differential equations, and the distribution of prime numbers. This isn’t really connected to today’s topic, but the theory is that there is a logical distribution to prime numbers. As numbers get larger, the probability of a random number being a prime number becomes less common and this is inversely proportional to its number of digits. Prime numbers are obviously not random, but they do appear to be random. If anybody can work out the pattern behind prime numbers, they will win a lot of awards.
Amongst all of his research, papers, and books, John Littlewood came up with Littlewood’s law. His conjecture was that so many events happen to us every day, that the probability of us seeing an improbable event, or something akin to a miracle, was likely once every 35 days. He had statistics to back this up, which we will look at in a minute, but he wasn’t being overly serious. He was merely trying to make a point that miracles and things we think of as huge coincidences, are actually not that unusual and we should be more surprised if we don’t see any. This is connected to the law of truly large numbers.
The law of truly large numbers says that if you do something enough times, even things that are exceedingly improbable will happen. That makes logical sense. In a regular lottery, where you pick six numbers out of 49, the chances of winning are 1 in 13,983,816. It is incredibly unlikely that I will win. However, it is not unlikely that the lottery will be won in a single week because more than 14 million people buy tickets. We find this hard to believe sometimes because we get confused with my chance of winning the lottery and the chance that the lottery will be won by someone that week. There are people who win the lottery twice, the odds of which are quite high. Still, even if it is billions to one, there are probably more than a billion lottery tickets sold each week around the world. The odds are high that someone will win the lottery more than once. If you think about it, there are over 8 billion people on Earth, so, if something has a one in a billion chance of happening, it should happen somewhere on Earth eight times a day.
This is what John Littlewood said with his law. The law of truly large numbers can be applied to our lives. Littlewood used some numbers to explain this. He assumed that people are actively about and seeing things for about eight hours a day. This takes out sleeping time, eating time, and watching TV time. He also assumes that we see something about once a second. That means we see roughly 28,800 things a day. Over 35 days, that multiplies up to 1,08,000 things that we see. Littlewood’s assumption is that if something has a one in a million chance of happening and we see over a million things every thirty-five days, we will probably see a “miracle” every month. However, Littlewood was merely trying to demonstrate that the things we call “miracles” are not actually that much of a miracle and probability means that if enough people are looking at the world, these things should be observed.
All of this is true and the law of truly large numbers explains away any coincidence. It doesn’t make it any less special for the people that the coincidence happens to, but it does show the likelihood of something amazing happening. More interesting, perhaps, is that Littlewood never actually appears to have made this law. He never mentioned it in any of his writings or teachings. It was ascribed to him by a student of Littlewood’s, called Freeman Dyson, writing in 2004. Whether Littlewood mentioned it in a class and Dyson overheard it and remembered it many decades later, or whether Dyson just thought he had, is unknown. Littlewood might be amused to know that his name is more commonly known for this law than for the mathematics he worked hard at. Still, what are the odds of something like that happening? And this is what I learned today.
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Sources
https://www.ucl.ac.uk/~ucahjmt/itwontbeyou.html
https://www.geeksforgeeks.org/how-to-calculate-probability-of-winning-the-lottery
https://brilliant.org/wiki/distribution-of-primes
https://www.imperial.ac.uk/be-inspired/magazine/issue-41/improbable-probability
https://en.wikipedia.org/wiki/Littlewood%27s_law
https://en.wikipedia.org/wiki/John_Edensor_Littlewood
https://gwern.net/littlewood-origin
https://rationalwiki.org/wiki/Littlewood%27s_law
https://science.howstuffworks.com/science-vs-myth/unexplained-phenomena/littlewood-law-miracles.htm
Image By no conegut – Alchetron: https://alchetron.com/John-Edensor-Littlewood, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=91191788