Coincidence and the Law of Very Large Numbers

This article is an excerpt from the Shortform book guide to "The Art of Thinking Clearly" by Rolf Dobelli. Shortform has the world's best summaries and analyses of books you should be reading.

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What is the law of very large numbers? How does the law of very large numbers help explain coincidences?

The law of very large numbers is one of the key concepts in probability and statistics. The concept is quite simple: with a sample that’s large enough, even the most improbable events could happen.

Keep reading to learn about the law of very large numbers and how it explains coincidences and improbable events.

Hardly Anything Is Impossible

People attribute strange events to the supernatural—for example, believing someone getting struck by lightning twice is a message from God or the universe—because they think such events are impossible and can’t be the result of coincidence.

In reality, these strange events are possible, though unlikely. People can’t accurately estimate probability, so they think strange events are less likely than they actually are. Continuing our example, someone getting struck by lightning twice seems impossible, but with 7.7 billion people on earth and 1.4 billion lightning strikes per year, it’s bound to happen eventually.

Thus, don’t get too excited when something unusual happens, Dobelli suggests. Hardly anything is impossible, and the probability of any event occurring is likely higher than you think.

The Law of Very Large Numbers

Dobelli’s explanation that coincidences are more likely than you think is supported by the law of very large numbers: When your sample size is very large, “impossible” events are inevitable. For example, if the chance of being struck by lightning twice is one in 3 million and there are 7.7 billion people on earth, 2,000 people will be struck twice by coincidence.

So why don’t people take the law of very large numbers into account? It’s because humans can’t visualize very large numbers. Since early humans didn’t need to understand such large numbers, your brain didn’t evolve the instinctive ability to do so. Humans’ ability to understand just how large a number is starts to degrade after 100, and it breaks down completely for numbers above a million.

Thus, when people try to estimate probabilities that involve very large numbers—for instance, the size of the human race—they underestimate their sample size and don’t take the law of very large numbers into account. For example, you can’t visualize how big 7.7 billion or 3 million are. You think those numbers are closer in size than they really are, so you assume the chance of being struck by lightning twice is much lower than 2,000. Thus, when it happens, you search for another explanation, attributing it to supernatural intervention.

Coincidence and the Law of Very Large Numbers

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Darya Sinusoid

Darya’s love for reading started with fantasy novels (The LOTR trilogy is still her all-time-favorite). Growing up, however, she found herself transitioning to non-fiction, psychological, and self-help books. She has a degree in Psychology and a deep passion for the subject. She likes reading research-informed books that distill the workings of the human brain/mind/consciousness and thinking of ways to apply the insights to her own life. Some of her favorites include Thinking, Fast and Slow, How We Decide, and The Wisdom of the Enneagram.

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