Thank you for visiting this site. This article covers the “Birthday Paradox.”
A year has 365 days, so sharing a birthday with someone seems like it should be quite rare. Yet in practice, in a group of just 23 people, the probability that at least one pair shares a birthday exceeds 50%. With 70 people, it surpasses 99.9%.
Doesn’t that seem like too few? That feeling of disbelief is the very heart of this paradox.
Why 23 People Give More Than 50%
Most people get tripped up by confusing “the probability that someone shares your birthday” with “the probability that any two people in the group share a birthday.”
The probability that someone among 23 people shares your specific birthday is indeed low (about 6%). But the question asks for “the probability that at least one pair of any two people — regardless of who — shares a birthday.”
In a group of 23 people, the number of two-person combinations is 23 × 22 ÷ 2 = 253 pairs. We’re asking whether at least one of those 253 pairs shares a birthday — and suddenly exceeding 50% doesn’t seem so surprising.
The Calculation
Rather than computing the probability of a shared birthday directly, it’s easier to compute the probability that “everyone has a different birthday” and subtract from 1.
Person 1 can be born on any day: 365/365.
Person 2 must be born on a different day: 364/365.
Person 3 must differ from both: 363/365.
Continue to person 23…
Probability all birthdays differ = 365/365 × 364/365 × 363/365 × … × 343/365 ≈ 0.4927
So the probability that all 23 birthdays are different is about 49.3%. That means the probability that at least one pair matches is 1 − 0.4927 = about 50.7%.
With 50 people the probability rises to 97%; with 70 people it exceeds 99.9%.
Why Intuition Fails
Humans systematically underestimate how quickly the number of “pairs” grows.
Each time one more person joins, the number of new pairs equals the total number of people already present. When person 23 joins, 22 new pairs are created. Because the number of pairs grows roughly with the square of the group size, the probability shoots up much faster than intuition suggests.
The same dynamic applies in everyday life. In a meeting of 10 people, it is hardly unusual for two attendees to share a hobby — because those 10 people form 45 distinct pairs.
Verified by Real Data
The Birthday Paradox has been confirmed not just theoretically but in actual data.
For instance, examining the rosters of teams at the FIFA World Cup (23 players per team), roughly half of all teams contain at least one pair of players sharing a birthday — exactly as theory predicts.
A school class of 30–40 students has a 70–90% chance of containing at least one shared birthday pair. Looking back at your own school days, you may well remember classmates who shared a birthday.
Applications in Cryptography
The Birthday Paradox plays an important role in cryptography and security. An attack method called a “birthday attack” exploits this principle.
If a hash function produces an n-bit output, intuition suggests that generating a collision (two different inputs with the same hash) would require approximately 2^n attempts. In reality, however, roughly 2^(n/2) attempts suffice — the same mathematics as the Birthday Paradox. This is a critical factor in evaluating the security of cryptographic systems and cannot be ignored.
Summary
This article covered the “Birthday Paradox.”
Strictly speaking, this is not a contradiction but a superb example of how inaccurate human probabilistic intuition can be. Intuitively grasping the explosive growth of combinations is genuinely difficult for the human mind.
To return to the full list of paradoxes, follow the link below.
Thank you for reading. We hope to see you in the next article.
