The Birthday Paradox Explained at a Wedding
The short version: with just 23 guests in a room, there's a better than 50% chance that two of them share a birthday. With 70 guests it's 99.9%. Your brain says that's impossible. Your brain is bad at counting pairs.
Cousin Kevin's wedding. Fifty guests, a free bar, and Kevin in an expansive mood. He turns to Uncle Gary and bets him $20 that two people in the room share a birthday.
"There are 365 days in a year, Kevin," says Gary, who has also visited the free bar. "Fifty people. Easy money."
They go table to table asking for birthdays. Two of the bridesmaids were born on 14 March. Gary pays. Kevin spends the $20 at the bar, which is free, and to this day nobody knows where the money went.
Why Kevin was always going to win
Gary was answering the wrong question. He was thinking: "What are the chances someone shares my birthday?" That really is unlikely. You'd need about 253 people for a 50% chance.
But Kevin's bet was about any two people. And the number of possible pairs grows fast. With 23 people there are already 253 different pairs, each with a small chance of a match. Add up all those small chances and a match becomes more likely than not.
| Guests in the room | Chance two share a birthday |
|---|---|
| 10 | 11.7% |
| 23 | 50.7% |
| 30 | 70.6% |
| 50 | 97.0% |
| 70 | 99.9% |
At Kevin's wedding, with 50 guests, Gary had about a 3% chance of keeping his $20.
Chance of a shared birthday
Where it costs you money
The birthday paradox is the reason amazing coincidences aren't amazing when there are lots of chances for them.
Thousands of people post market predictions online every day. Some of them will call five moves in a row by pure luck, the same way two bridesmaids share a birthday. When you meet that person, you're not looking at a genius. You're looking at the bridesmaids. This is also why we correct for luck when we test many strategies at once.
The moral
Before you call something an incredible coincidence, count how many chances it had to happen. And before you take Kevin's bet, count the pairs.
Assumes birthdays are spread evenly across 365 days and ignores 29 February, which is how everyone does this calculation. Educational content, not financial advice.