In a room of 23 people, two probably share a birthday
It sounds impossible with 365 days to choose from, but the odds pass 50% with just 23 people and reach 99.9% with 70.
Ask most people how many are needed for a 50% chance of a shared birthday and they guess around 180. The answer is 23.
Why intuition fails
We imagine ourselves finding a match, which really is unlikely. But the question is about any two people. With 23 people there are
23 × 22 ÷ 2 = 253 pairs,
and each pair is a separate chance. The easiest way to calculate it is through the opposite: the chance that everyone is different.
365/365 × 364/365 × 363/365 × … × 343/365 ≈ 0.493
So the chance of at least one match is about 50.7%. With 70 people it reaches 99.9%.
| People | Chance of a shared birthday |
|---|---|
| 10 | 12% |
| 23 | 51% |
| 50 | 97% |
| 70 | 99.9% |
Why security engineers care
Cryptographic hash functions give data a fingerprint. If two different documents ever share a fingerprint, called a collision, forgeries become possible. The birthday effect means an attacker needs only about the square root of the number of possible fingerprints to find one. That is why modern hashes have 256-bit outputs.
A useful takeaway: when something seems like an amazing coincidence, count how many chances there were for some coincidence to occur.
Bring it up
“In a group of just 23 people, it’s more likely than not that two of them share a birthday. Want to test it?”
Follow the thread
- RelatedOnce you notice something, it’s everywhere Coincidences feel meaningful because intuition is bad at counting how many chances there are.
- Leads toJulius Caesar’s secret code (and why it failed) Modern cryptography relies on probability, and the birthday paradox is one of its most important attacks.
