Math Calculators
Birthday Paradox Calculator - Group Size & Collision Probability
Calculate the probability that at least two people in a group share a birthday. See why just 23 people gives a 50% chance and visualize the probability curve.
50.73%
probability that at least two people share a birthday in a group of 23 (with 365 day year)
50% crossover: 23 people
Why Is It a Paradox?
Most people intuitively guess you need ~180 people to have a 50% chance of a shared birthday. The real answer - just 23 - feels deeply wrong. The key insight is that every pair of people is checked, not just "does someone share my birthday." With 23 people there are 253 unique pairs, which quickly accumulates probability.
The probability formula
It is easier to calculate the complement - the probability that no two people share a birthday:
P(no match) = 365/365 × 364/365 × 363/365 × … × (365 − n + 1)/365
P(at least one match) = 1 − P(no match)
Each fraction represents the probability that a new person's birthday is different from all previously checked people. As more people are added, the product shrinks rapidly.
Probability table
| People (n) | Probability of a shared birthday |
|---|---|
| 5 | 2.7% |
| 10 | 11.7% |
| 15 | 25.3% |
| 20 | 41.1% |
| 23 | 50.7% - crosses 50% |
| 30 | 70.6% |
| 40 | 89.1% |
| 50 | 97.0% |
| 57 | 99.0% |
| 70 | 99.9% |
Generalizations
The paradox generalizes to any scenario with a fixed number of equally likely outcomes:
- Days of the week (7 options): only 4 people are needed for a >50% chance of two sharing a weekday birthday.
- Months of the year (12 options): only 5 people needed to exceed 50%.
- Hash collisions: the same math applies to cryptographic hash functions. A 256-bit hash needs approximately 2¹²⁸ hashes before a collision is likely - an astronomically large number, which is why SHA-256 is considered collision-resistant in practice.