23 people, a coin flip.
Put 23 random people in a room and the odds two of them share a birthday already top 50%. Intuition compares 23 people to 365 days — the right comparison is the 253 pairs of people in that room, and that count grows far faster than the group does.
P(n)
=
1−
n−1
Π
k=0
365−k
365
Group size n
pairs ∝ n²
2every extra person adds n−1 new pairs100
Chance of a shared birthday
%
Try
The birthday problem gives P(n) = 1 − the running product of (365−k)/365 for k = 0 up to n−1 — the chance no two of n people share a birthday, subtracted from 1. It isn't a real paradox, just a result that clashes with intuition: intuition compares n people to 365 days, when the quantity that matters is the number of pairs, n(n−1)/2, which grows quadratically. At n = 23 that's already 253 pairs, and P(23) ≈ 50.7%, just past P(22) ≈ 47.6%. This model ignores leap years (treats every year as 365 days) and assumes birthdays are spread uniformly across the year; real birth dates cluster slightly (for instance more births several months after year-end holidays), which in practice nudges the true probability of a match slightly higher, not lower — so 23 stays a safe, if approximate, threshold.