Concepts · 3 min read
The Birthday Paradox: Why 23 People Is Enough
Count the opposite
It is easier to find the chance that nobody shares a birthday and subtract from 1. Person 2 must avoid 1 date (364/365), person 3 must avoid 2 dates (363/365), and so on. Multiply those probabilities together.
For 23 people: P(no match) = (365/365) × (364/365) × … × (343/365) ≈ 0.4927, so P(match) ≈ 50.73%.
The table
| People | P(at least one shared birthday) |
|---|---|
| 10 | 11.69% |
| 15 | 25.29% |
| 20 | 41.14% |
| 23 | 50.73% |
| 30 | 70.63% |
| 40 | 89.12% |
| 50 | 97.04% |
| 60 | 99.41% |
| 70 | 99.92% |
Why it feels surprising
We think of matching our own birthday. But 23 people form 253 different pairs, and any pair could match.
Pairs among 23 people
- 23C2 = 253
Step-by-step working 3 steps
- 1Order does not matter, no repeats
nCr = n! / (r!(n−r)!) = nPr / r!
Use combinations when you only care which items are chosen (committees, lottery picks).
- 2Compute nPr and r!
23P2 = 506 2! = 2
- 3Divide
506 ÷ 2 = 253
Assumptions
The calculation treats all 365 days as equally likely and ignores 29 February; real birthdays are slightly uneven, which makes matches a bit more likely. More counting ideas: lottery odds.
Frequently asked questions
How many people for a 99% chance?
About 57 people give a probability above 99%.
Is it really a paradox?
It is a surprising result rather than a logical contradiction.
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