Put 23 people in a room, and the odds already favor a shared birthday
Most people guess you'd need close to 183 — half of 365 — before it's a coin flip. The real number is 23, because you're not checking one date against one person. You're checking every pair of people against each other.
| Odds of a shared birthday | People needed |
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It's about pairs, not dates. With 23 people there are 23×22÷2 = 253 different pairs of people, and each pair has a small independent chance of matching. Add up 253 small chances and you cross 50% much faster than intuition expects — the number of pairs grows roughly with the square of the group size, not the group size itself.
This assumes birthdays are spread evenly across 365 days (no Feb 29, no accounting for real-world clustering like September birth spikes) — the standard simplification used in the classic version of this problem.
P(shared) = 1 − (365 × 364 × … × (365−n+1)) / 365ⁿ, computed exactly for each n and cross-checked against the published birthday-paradox values (23 → 50.7%, 57 → 99.0%, 70 → 99.9%).