Overview
The birthday paradox isn't a contradiction — just a collision between intuition and combinatorics. We think about people, but the probability is driven by pairs.
How to solve The Birthday Paradox
- Compute the chance everyone has a different birthday: (365/365)(364/365)(363/365)….
- Subtract from 1 to get the chance of at least one shared birthday.
- At 23 people this passes 50% (50.7%); by 50 it's 97%.
The key insight
n people form n(n−1)/2 pairs, and it's pairs that can collide. 23 people already make 253 pairs — far more chances to match than the count of people suggests.
Variations & echoes
- The 'birthday attack' exploits the same maths to break cryptographic hashes faster than brute force.
- The threshold grows like the square root of the number of possible days.
Frequently asked questions
Is 23 for someone matching MY birthday?
No — that's a different, much larger number (253). 23 is for any two people in the group matching each other.