If something is so unlikely, why does it keep happening?
A tiny chance. Many opportunities.
Something has a 1 in 100 chance of happening. You give it 100 chances. Most people guess it will almost certainly happen, or that it should happen exactly once. Neither is quite right. Move the controls and see.
One run
Each square is one try. Filled squares are the times it happened.
Turn on JavaScript to see the simulation and move the controls live.
What to notice
When the tries match the odds (1 in 100, 100 tries), the answer is about 63%, not 100%. That number keeps turning up: it is 1 − 1/e.
Double the tries and it climbs to about 87%. Rare things become likely when they get enough chances.
This is why coincidences are everywhere. With billions of people living billions of days, a one-in-a-million event happens to someone all the time.
Assumptions
- Every try is independent: one result does not change the next.
- The chance is the same on every try.
- The simulation uses a seed, so a shared link shows exactly the same run.
Show me the reasoning
On one try, the chance it does not happen is 1 − 1/100 = 0.99.
For it never to happen, it has to fail on every one of the 100 tries: 0.99^100 ≈ 36.6%.
So the chance it happens at least once is 1 − (1 − 1/100)^100 ≈ 63.4%.
When the number of tries equals the odds, this gets close to 1 − 1/e ≈ 63.2%, however large the numbers are.
Sources
- Wikipedia, Law of truly large numbers · en.wikipedia.org
- OpenStax, Introductory Statistics, independent events · openstax.org