Birthday Paradox Calculator (Shared Birthday Probability)
By Santaji GadepatilinUpdated November 4, 2026
In a room of just 23 people there is a better than even chance that two share a birthday, which surprises most people because there are 365 days. This birthday paradox calculator gives the exact probability of a shared birthday for any group size, and the much smaller chance that someone shares your own birthday.
How to use the birthday paradox calculator
Enter the number of people in the group. The calculator returns the probability that at least two of them share a birthday, the probability that all birthdays are different, the chance that at least one other person shares your birthday, and the number of pairs. Change the number of days to apply the same maths to other “collisions”, such as hash values or ID numbers.
Birthday problem formula
P(all different) = d × (d − 1) × … × (d − n + 1) ÷ dⁿ = ₙP(d, n) ÷ dⁿ
P(shared) = 1 − P(all different) · P(shares yours) = 1 − ((d − 1) ÷ d)n−1
Example
A group of 23 people:
- P(all different) = 365 × 364 × … × 343 ÷ 365²³ = 49.27%
- P(at least one shared birthday) = 1 − 0.4927 = 50.73%
- P(someone shares your birthday) = 1 − (364/365)²² = 5.86%
- Pairs = 23 × 22 ÷ 2 = 253
Shared-birthday probability by group size
| People | P(any two share) | P(someone shares yours) |
|---|---|---|
| 2 | 0.27% | 0.27% |
| 5 | 2.71% | 1.09% |
| 10 | 11.69% | 2.44% |
| 15 | 25.29% | 3.77% |
| 20 | 41.14% | 5.08% |
| 23 | 50.73% | 5.86% |
| 30 | 70.63% | 7.65% |
| 40 | 89.12% | 10.15% |
| 50 | 97.04% | 12.58% |
| 57 | 99.01% | 14.24% |
| 60 | 99.41% | 14.94% |
| 70 | 99.92% | 17.25% |
| 80 | 99.99% | 19.49% |
| 100 | 100.00% | 23.78% |
Notes
- Real birthdays are not perfectly uniform, which makes a match slightly more likely than these figures.
- The counting uses permutations — see the permutations calculator.
Frequently asked questions
Why do only 23 people give a 50% chance?
Because every pair of people is a chance for a match, and 23 people form 253 pairs. The probability that all birthdays differ is 365/365 × 364/365 × … × 343/365 = 49.3%, so a shared birthday has a 50.7% chance.
What is the formula for the birthday problem?
P(shared) = 1 − 365! ÷ ((365 − n)! × 365ⁿ) = 1 − (365 × 364 × … × (366 − n)) ÷ 365ⁿ.
How many people do you need for a 99% chance?
57 people give about 99.0%, and 70 people about 99.9%. With 366 people a match is certain (ignoring 29 February).
How likely is it that someone shares my birthday?
Much less likely — 1 − (364/365)^(n − 1). In a group of 23 (you plus 22 others) it is only about 5.9%, and you need 253 other people for a 50% chance.