Cobra Effect · Statistics and probability
The birthday problem
Coincidences are far more likely than they feel.
6 cards, read aloud in 2:10, with a test and sources.
Twenty three people in a room. Bet on whether two share a birthday.
There are 365 days in a year. Twenty three people cannot even fill a month of them. So it feels like a long shot. Take the bet. You will win it more often than you lose.
The odds are just over even.
With 23 people, the chance of a shared birthday is 50.7%. With 30 it is 70%. With 50 it is 97%. By 70 people it is 99.9%, and you can stop being polite about it.
Why does it feel wrong? Because you counted the wrong thing.
You imagined someone sharing your birthday. That is 22 chances, and it is unlikely. But the bet was about any two people. Twenty three people make 253 pairs. Each pair is a small chance, and 253 small chances add up.
It is called the birthday problem, or the birthday paradox.
It is not a paradox at all. The sum is plain. The paradox is in us. We are bad at counting pairs, and pairs grow much faster than people. Double the people and you nearly quadruple the pairs.
Football squads have 23 players.
At the 2014 World Cup, someone checked all 32 squads. Half of them had two players with the same birthday. Exactly what the sum predicts. Coincidences are cheap when there are many ways for one to happen.
So when a coincidence stuns you, count the pairs.
How many people, how many days, how many ways could this have matched? Two friends with the same birthday is an event. Any two people in your office with the same birthday is Tuesday. Before you call something a sign, ask how many chances it had.
Sources
- Birthday problem, Wikipedia. The sum done properly, the table of probabilities for every group size, and the versions that matter in cryptography.
- Innumeracy, John Allen Paulos, 1988. A short, cross book about why people are bad with numbers, with the birthday sum and a dozen other coincidences that are not.
- Law of truly large numbers, Wikipedia. The bigger idea: with enough chances, any outrageous thing is likely to happen to someone. Diaconis and Mosteller wrote it up in 1989.
Nearby ideas
- Base rate neglect. A good test for a rare thing still gives mostly false alarms.
- Regression to the mean. Extreme results drift back towards normal, whatever you do.
- Survivorship bias. The failures you never see can reverse the lesson you draw.
- The gambler’s fallacy. Chance has no memory, so nothing is ever due.
- Berkson’s paradox. Filtering who you look at can create links that are not there.
- Simpson’s paradox. A trend in every group can reverse when the groups are combined.
- The law of small numbers. Small samples swing wildly, and the swings look like signals.
- The prosecutor’s fallacy. A rare match is not the same as a small chance of innocence.