The birthday paradox
In a room of just 23 people, there is a better than 50% chance that two of them share a birthday. With 70 people it is almost certain. Try it yourself below.
chance that at least two of 23 people share a birthday
23 people make 253 possible pairs.
Number of people in the group. Hover or drag across the chart to explore.
Try it: simulate random groups
Create random groups of people and count how often two share a birthday.
Why it works
It is easier to calculate the chance that nobody shares a birthday. The second person has a 364/365 chance of missing the first person's birthday, the third a 363/365 chance of missing both, and so on. Multiply those together and subtract from 1:
P(shared) = 1 − (365 × 364 × … × (365 − n + 1)) / 365n
The key is the number of pairs. With 23 people there are 23 × 22 ÷ 2 = 253 pairs, and each pair is a chance for a match. That is why the probability climbs so fast.
Birthday paradox probability table
| People | Pairs | Chance of a shared birthday |
|---|---|---|
| 5 | 10 | 2.7% |
| 10 | 45 | 11.7% |
| 15 | 105 | 25.3% |
| 20 | 190 | 41.1% |
| 23 | 253 | 50.7% |
| 30 | 435 | 70.6% |
| 40 | 780 | 89.1% |
| 50 | 1,225 | 97.0% |
| 57 | 1,596 | 99.0% |
| 60 | 1,770 | 99.4% |
| 70 | 2,415 | 99.9% |
| 75 | 2,775 | >99.9% |
| 100 | 4,950 | >99.9% |
Assumes 365 equally likely birthdays and ignores February 29. See how birthdays are really spread across the year on our most common birthdays page.
Frequently asked questions
What is the birthday paradox?
The birthday paradox is the surprising fact that in a group of just 23 people, there is a better than 50% chance that at least two of them share a birthday. With 70 people, the chance is over 99.9%.
Why is it called a paradox?
It is not a true paradox, just a result that feels wrong. People tend to think about the chance of someone sharing their own birthday, but the question is whether any two people in the group match, and the number of possible pairs grows very quickly.
How many people do you need for a 50% chance of a shared birthday?
Only 23. A group of 23 contains 253 different pairs of people, and each pair is a chance for a match.
Does it matter that some birthdays are more common?
Slightly. Real birthdays are not spread evenly across the year, which makes shared birthdays a little more likely than the classic calculation suggests. The simulator on this page can use real U.S. birth frequencies.