Birthday Paradox Calculator
Shared birthday odds for any group size, exact math.
What it does
The birthday paradox surprises because pairs multiply quadratically: 23 people make 253 pairs, and each pair rolls year long odds against matching. Half the groups that size hold a shared birthday.
The tool above takes the headcount from 2 to 500. It returns the exact match chance with pair count and expected matches beside them.
The default 23 reads 50.7% across 253 pairs (0.69 expected). Thirty people reach 70.6% while fifty hit 97%, and past 365 certainty rules.
Counting follows Khan Academy probability units with the same complement method in LibreTexts statistics references. Both assume uniform birthdays, ignoring seasonal bumps.
How to use
- Enter the group headcount.
- Read match chance first for the headline.
- Read pairs to see why it climbs fast.
- Use expected matches for the count feel.
- Test classrooms live for the wow.
Formula
- complement
- One minus the all different chance.
- pairs
- Quadratic pair count driving it.
- 365
- Days ignoring Feb 29.
Uniform birthdays assumed; real seasonality nudges numbers slightly. Past 365 the page returns certain by pigeonhole instead of computing.
Worked example
Example: 23 people
Default headcount 23.
- Pairs: 23 x 22 / 2 = 253.
- All different: 365/365 x 364/365 x ... x 343/365 = 49.3%.
- Match: 100 - 49.3 = 50.7%.
- Expected: 253 / 365 = 0.69 matches.
- Tool shows 50.7%, 253 pairs, 0.69 shared.
Reference table
| People | Chance | Pairs |
|---|---|---|
| 10 | 11.7% | 45 |
| 23 | 50.7% | 253 |
| 30 | 70.6% | 435 |
| 50 | 97% | 1,225 |
| 70 | 99.9% | 2,415 |
Limits
- Uniform birthdays assumed; seasons bump reality.
- Feb 29 ignored in the 365 model.
- Twins break independence assumptions.
- Groups past 365 read certain, not computed.
Common mistakes
- Reading 23/365 (6%) instead of pair math.
- Expecting your birthday matched, not any pair.
- Trusting small samples over exact odds.
- Forgetting twins in classroom demos.
- Assuming uniform births exactly.
Try it yourself
Quick checks. Answers sit tucked under each question.
30 people?
70.6% across 435 pairs.
50 people?
97% across 1,225 pairs.
Pairs in 23?
23 x 22 / 2 = 253.
Questions
Why 50% at only 23?
253 pairs each roll the odds; pairs grow quadratically with heads.
Is it really a paradox?
Only psychologically. The math counts pairs, not your birthday.
What about Feb 29?
Ignored here; including it shifts decimals slightly.
Expected vs chance?
Chance is any match; expected counts how many (0.69 at 23).
70 people?
99.9%: near certain with 2,415 pairs.
Classroom demo?
Groups over 30 surprise more often than intuition says.