probability
did you know 23 people can be enough for a birthday match?
a small room hides a surprisingly large number of pairs.
23 → 253
people → possible pairs
under the usual birthday model, those pairs produce a 50.7% chance of at least one match.
- 23 people
- 253 pairs
- 50.7% chance
the idea
in a group of 23 people, the chance that at least two share a birthday is just over half in a simple model. the surprise is that you are comparing everyone with everyone, not just everyone with you.
how it works
start with the opposite outcome: everybody has a different birthday. the second person must avoid one date, the third must avoid two, and so on. multiply those chances, then subtract from one.
go deeper
how many chances are hiding in the room?
in a group of 23 people, the chance that at least two share a birthday is just over half in a simple model. the surprise is that you are comparing everyone with everyone, not just everyone with you.
23 people make 253 distinct pairs. no individual pair is particularly likely to match, but there are plenty of opportunities. a match involving any two people counts.
why count the people who do not match?
start with the opposite outcome: everybody has a different birthday. the second person must avoid one date, the third must avoid two, and so on. multiply those chances, then subtract from one.
the calculation gives about 50.7% for 23 people and 97.0% for 50. these are probabilities across many possible groups, not promises about a particular party.
where does 50.7% come from?
assume 365 equally likely birthdays, independent of one another, and ignore leap days. multiply the 23 fractions from 365/365 down to 343/365.
P(at least one match) = 1 − (365 × 364 × … × 343) / 365²³ ≈ 50.7%
what changes outside the model?
- real birthdays are seasonal, and relatives can have linked birthdays. the tidy assumptions are an approximation.
- matching your particular birthday is a different question: among 22 other people, its probability is only about 5.9% under the same assumptions.
sources & further reading
concepts: complements · combinatorics · birthday paradox
2 minute read