Question of the day
2026-09-27
The Birthday Problem: in a room of 23 people, what's roughly the probability that at least two share a birthday — and why is it surprisingly high?
Answer it out loud first — like you would in the room. Then check yourself:
Reveal the model answer
Model answer
About 50% (≈ 50.7%).
Use the complement — probability everyone is distinct: (365/365)(364/365)(363/365)…(343/365) for 23 people. That product is ≈ 0.493, so P(a shared birthday) ≈ 1 − 0.493 ≈ 0.507.
Why it feels wrong: you're not comparing to one fixed person — you're comparing all pairs. With 23 people there are C(23,2) = 253 pairs, and that many chances to match makes a collision likely. With 70 people it's already ~99.9%.
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