Birthday Paradox Calculator
The chance that two people in a group share a birthday — the result that reaches 50% at just 23 people and surprises almost everyone.
It is easier to compute the chance that everyone is *different* and subtract.
How the birthday paradox calculator works
It is easier to compute the chance that everyone is *different* and subtract. With n people and d equally likely days, that is (d/d)((d−1)/d)((d−2)/d)… for n terms; the answer is one minus it.
The count of *pairs* is what drives it: 23 people make 253 pairs, and each pair is a chance to match. This is also the mathematics behind hash collisions, which is why it matters well outside parties.
Formula: P(match) = 1 − d! / (d − n)! dⁿ
Worked examples
| Inputs | P(at least two share a birthday) | Note |
|---|---|---|
| 23 people | 50.7297% | just over 50% |
| 50 people | 97.0374% | 97% |
| 70 people | 99.916% | 99.9% |
FAQFrequently asked questions
Why does 23 people give 50%?
Because 23 people form 253 pairs, and each pair is an independent-ish chance to match. It is pairs that matter, not people.
Is it really a paradox?
No — just a strong intuition failure. People think about matching *their own* birthday, which needs about 253 people for 50%.
What is the chance someone shares MY birthday?
Much lower: about 6% in a group of 23. The calculator shows both.
How many people for near-certainty?
70 gives 99.9%; 100 gives 99.99997%. Only 366 makes it certain.
Where does this matter outside parties?
Hash collisions. A 64-bit hash has a 50% collision chance after about 5 billion items, not 18 quintillion — which is why cryptographic hashes are so much longer than they look like they need to be.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.