It takes a room of just 23 people for two of them to probably share a birthday — far fewer than 365 days in the year would suggest. That's the birthday paradoxThe counterintuitive result that a small group very likely contains a shared birthday — because it's the number of pairs, not people, that drives the odds.: a match between any two people piles up far faster than a match to one particular date. Pick a group size and a question.
that two of the 23 share a birthday.
people in the room →
| Reach this chance | People needed |
|---|---|
10%one in ten | 10 people |
50%even odds | 23 people |
90%nine in ten | 41 people |
99%ninety-nine in a hundred | 57 people |
99.9%all but certain | 70 people |
The gap between the two questions is the whole paradox. With 23 people there are 253 different pairs, and any one of them can be the match — so "do any two share?" climbs with the square of the group. Matching your birthday has only 23 chances, against a single fixed date, so it crawls: even odds arrive at 23 people for any pair, but 253 for yours. (Birthdays assumed uniform across 365 days; leap days ignored.)