Birthday Paradox: Why 23 People Are Enough
How many people do you need in a room before two of them probably share a birthday? Most people guess around 180, half of the 365 days in a year. The real answer is just 23. With 23 people the chance of a shared birthday is 50.7%, a little better than a coin flip. This surprising result is called the birthday paradox, and once you see how it works you will never trust your gut about chance in quite the same way.
The Numbers
These figures assume 365 equally likely birthdays and ignore 29 February. They are exact calculations, not estimates.
| People in the room | Chance two share a birthday |
|---|---|
| 5 | 2.7% |
| 10 | 11.7% |
| 20 | 41.1% |
| 23 | 50.7% |
| 30 | 70.6% |
| 40 | 89.1% |
| 50 | 97.0% |
| 57 | 99.0% |
| 70 | 99.9% |
| 367 | 100% (guaranteed) |
By 70 people a shared birthday is almost certain. Only at 367 people is it truly guaranteed: with 366 possible birthdays including 29 February, one more person than that must share one with someone.
Why It Feels Wrong
The trick is that the question is not "does someone share my birthday?" It is "do any two people share a birthday?" Those are very different questions.
- In a room of 23, there are 253 different pairs of people (23 × 22 ÷ 2). Every one of those pairs is a chance for a match.
- The chance that someone shares your birthday among the other 22 people is only about 6%. You would need 253 other people to get to a 50% chance of matching you in particular.
- Our brains count people, not pairs. The number of pairs grows much faster than the number of people: 30 people already make 435 pairs.
How to Calculate It (the Easy Way)
Working out the chance of "at least one match" directly is messy. It is much easier to work out the chance that nobody matches, then subtract it from 100%.
- The first person can have any birthday.
- The second person must avoid that one day: 364 out of 365.
- The third must avoid two days: 363 out of 365.
- Keep going, and multiply all the fractions together. For 23 people that product is about 0.493, so there is a 49.3% chance that all birthdays are different.
- Subtract from 1: 1 − 0.493 = 0.507, a 50.7% chance that at least two people share a birthday.
This "work out the opposite" trick is one of the most useful ideas in probability. It works just as well for dice questions such as the chance of rolling at least one six in four throws: 1 − (5/6)4, about 52%. More on that in dice probability.
Do Real Birthdays Change the Answer?
In real life birthdays are not spread perfectly evenly: in many countries more babies are born in some months than others, and fewer on public holidays. Uneven birthdays only make a match more likely, never less. So the 23-person figure is, if anything, a slight underestimate.
Try It: a Classroom Experiment
- Ask everyone to write their birthday (day and month only) on a slip of paper.
- Before you check, ask the class to vote: do they think two people share a birthday?
- Read the dates out loud, or sort them on the board, and look for a match.
- Compare the result with the table above for your class size.
With a typical class of 25 to 30, there is a 57% to 71% chance of a match, so it works more often than not. If your group is small, simulate a big room instead: use the Random Number Generator to draw 23 numbers from 1 to 365 with No repeats unticked, one for each imaginary person, and try it a few times. You will see a repeat roughly half the time.
Where the Birthday Paradox Shows Up
- Football: a match has 23 people on the pitch (22 players and the referee), so you would expect two of them to share a birthday in about half of all matches.
- Computer security: the "birthday attack" uses the same maths to find two inputs that produce the same digital fingerprint far faster than you might expect.
- Coincidences: many "what are the odds?" moments, like bumping into someone from home on holiday, feel unlikely because we think about one specific match instead of all the possible ones.
It is the same lesson as why humans are bad at being random: our sense of what is likely is a poor guide. For decisions where you want true chance, not intuition, let the Yes or No Button answer, or read about real coin flip odds.
FAQ
What is the birthday paradox?
It is the surprising fact that in a group of just 23 people there is a 50.7% chance that at least two of them share a birthday. It is not a true paradox, only a result that goes against intuition.
Why are only 23 people needed?
Because the question is about any two people, not about you. 23 people make 253 different pairs, and each pair is a chance for a shared birthday.
How many people are needed for a 99% chance of a shared birthday?
57 people give about a 99.0% chance, and 70 people give about 99.9%. A match is only completely guaranteed with 367 people.
What is the chance that someone shares my birthday?
Much lower. Among 22 other people the chance that one of them has your exact birthday is only about 6%. You need 253 other people to reach a 50% chance.
How do you calculate the birthday paradox?
Multiply the chances that each new person avoids all the earlier birthdays (364/365, 363/365 and so on) to get the chance of no match, then subtract that from 1.