Occasionally, we are faced with events that seem so unlikely that we feel like the universe is sending us a message. That’s why it’s common to say “I don’t believe in coincidences”.
Say that the “coincidences don’t exist” It is one of the most common mathematical errors.
Maybe it’s actually more fun to think that the universe is sending us messages than to think that our life is full of mere coincidences… but the point is, it really is.
To math Sarah Hart explains what coincidences are often much more likely to occur than we think.
The 0.7 Rule
Even very unlikely events are likely to occur if given enough opportunity. We can express this mathematically as follows:
- Com k opportunities for an event to occur, in each of which there is a hypothesis in n that it actually happensthen (assuming that they are independent events) the probability of occurring at least once is: 1 – (1 – 1/n)k
“But don’t let algebra discourage you,” urges Sarah Har.
The message to remember is that the probability of a coincidence increases rapidly as the number of opportunities increases.
From this formula, we can deduce the following fact: the probability of a “one in n” event occurring at least once exceeds 50% whenever there are more than 0.7n independent opportunities for it to occur.
This is a phenomenon that the author calls “0.7 rule”.
Let’s look at some examples
Suppose everyone in Birmingham, England, has a one in a million chance of dreaming every night about the next day’s news. As the population is over 700,000, it is more likely that every morning someone in Birmingham will convince themselves that they are psychic – they will note maths.
Every day, there are many ways for a strange coincidence to happen to you. If we consider that we have one opportunity per hour to experience such an event and that the average life expectancy is 706,000 hours, there is more than one chance of experiencing a one in a million coincidence at least once.
These analyzes can help explain some of history’s most astonishing coincidences, such as the curious case of . In September 2009, the numbers – 4, 15, 23, 24, 35 and 42 – came in two consecutive draws.
It seems like an isolated case, but statistician David Hand calculated that it was enough for the game to run for 43 years for it to become more likely that the same numbers would be drawn at least twice (the lottery had, at this time, been running for 52 years).
Another example given by Sara Har was that of Roy Sullivanone ranger of Virginia, USA, was struck by lightning seven times between 1942 and 1977.
You could say that Roy is an unlucky man, and in fact, he might be. However, the guard had more “opportunities” than most people: There is more lightning in the hottest, most humid regions of the US, and people who work outdoors are more exposed than those sheltered indoors.
Plagiarism may not exist?
The distinction between coincidences and deeper causes is fundamental to the scientific method and can also resolve accusations of plagiarism.
The success of , for example, was the subject of a lawsuit in 2022 that alleged that a sequence of four notes had been removed from the song Oh Why? by Sami Chokri, from 2015.
“It’s natural for coincidences to happen if 60,000 songs are released every day on Spotify,” Sheeran told his fans after winning the case. “There are 22 million songs a year and there are only 12 notes available“.
Ed Sheeran is right. It is a mathematical certainty that many of these 22 million songs will contain eerily similar passages.
In academia, distinguishing between coincidental similarities and plagiarism or AI assistance in student work is also a growing challenge.
A plagiarism detection tool called Turnitin has an estimated 1% false positive rate for identifying documents with more than 20% AI content. If we are among that 1%, how do we prove we didn’t cheat?
Sarah Hart gives the example of a University of California student, Louise Stivers, who was only able to convince the university of her innocence by providing time-stamped drafts of her work.
Destiny vs. Mathematics
It is estimated that each of us knows 1000 people (acquaintances, colleagues, neighbors, etc.). If these connections are randomly distributed, there is a greater than 99% probability of having a mutual friend or a friend of yours meeting a friend of mine.
Mathematicians Steven Strogatz and Duncan Watts discovered that the presence of a few highly connected individuals can cause the overall distribution of the network to approach random distribution. This means that even if people only have 100 friends on average, any two people in the world are probably connected by a chain of less than five mutual friends.
The cliché is true: the world is really small (and destiny doesn’t exist).
Thanks to the laws of mathematics, coincidences abound. As Sarah Hart explains, events that appear to be a matter of “fate” are little more than a numbers game.
