Birthday Paradox Calculator
Discover how quickly the probability of shared birthdays grows as you add people to a room. Adjust the group size to see why only 23 people are needed for more than a 50% chance of a match.
Enter any group size between 0 and 200. The paradox becomes noticeable surprisingly quickly.
Chance of Shared Birthday
50.73%
At least two people share a birthday
Chance All Birthdays are Unique
49.27%
Every person has a distinct birthday
| Group Size | P(shared birthday) |
|---|---|
| 5 | 2.71% |
| 10 | 11.69% |
| 15 | 25.29% |
| 20 | 41.14% |
| 25 | 56.87% |
| 30 | 70.63% |
| 35 | 81.44% |
| 40 | 89.12% |
| 45 | 94.10% |
| 50 | 97.04% |
| 55 | 98.63% |
| 60 | 99.41% |
How to Use This Calculator
- Enter the number of people you want to analyze.
- Read the probability that at least two people share a birthday (and the complementary probability that all birthdays are unique).
- Use the summary table to compare common group sizes quickly.
- Experiment with larger groups to see how the probability approaches 100%.
Formula
- The product computes the probability that every new person avoids all previously used birthdays.
- We assume 365 equally likely birthdays and ignore leap years.
- The complement rule converts the probability of all unique birthdays into the probability of at least one match.
Full Description
The birthday paradox is not a true paradox; it simply illustrates that human intuition struggles with combinations. Each new person adds many opportunities for a match, so the probability rises much faster than linear. By the time you reach 23 people — a typical classroom — the chance of a shared birthday already exceeds 50%.
The calculation assumes a uniform distribution of birthdays across 365 days. Real-world data can deviate slightly due to seasonal birth patterns or leap days, but the paradox remains remarkably robust.
Frequently Asked Questions
Does the paradox consider leap years?
This calculator ignores leap years for simplicity. Including February 29 only changes the probabilities slightly and does not alter the key insight.
What if birthdays are not uniformly distributed?
Real data has modest seasonal variations, but the probability of a match remains high. The paradox survives because even small deviations still yield many pairing opportunities.
Why does the probability grow so fast?
Each person compares with every other person, creating n(n − 1)/2 potential pairs. The number of pairs grows quadratically, quickly outpacing the 365 available birthdays.
Can this be used for hash collisions?
Yes. The same math underlies the birthday attack in cryptography, where hash collisions become likely once you have about √(number of possible hashes) samples.