Table of Contents
The Birthday Paradox
The birthday paradox is the counterintuitive result that in a group of just 23 people, there is a greater than 50% chance that at least two share the same birthday. With 70 people, the probability exceeds 99.9%. This surprises most people because they confuse it with the probability of someone sharing THEIR specific birthday.
The key insight is that with n people, there are n(n-1)/2 pairs to compare. With 23 people, that's 253 pairs, each with a small chance of matching, but collectively reaching over 50%.
Formula
Probability Table
| People | P(Match) |
|---|---|
| 10 | 11.7% |
| 20 | 41.1% |
| 23 | 50.7% |
| 30 | 70.6% |
| 50 | 97.0% |
| 70 | 99.9% |
FAQ
Why is it called a paradox?
It is not a true logical paradox, but a veridical paradox -- a result that is provably true but seems false. Human intuition struggles with combinatorial growth.
What are real-world applications?
Hash collisions in computer science, cryptographic attack planning (birthday attack), and DNA database matching all rely on birthday paradox mathematics.