What is the birthday problem and what does it reveal about probability intuition?
In a room of just 23 people, the probability that at least two share a birthday exceeds 50 %. The counterintuitive result comes from counting the large number of pairs rather than comparing each person to a fixed date — an example of how our intuition systematically underestimates collision probabilities.
How to think about it
The birthday problem is a benchmark for probabilistic reasoning under combinatorial counting. Walk interviewers through the complement approach — it is cleaner than computing shared-birthday probability directly.
Setting up the complement
Compute the probability that all n birthdays are distinct, then subtract from 1.
With 365 equally likely birthdays and n people:
P(all distinct) = (365/365) × (364/365) × (363/365) × ··· × ((365-n+1)/365)
= 365! / ((365-n)! × 365^n)P(at least one match) = 1 - P(all distinct)Key numbers
| n (people) | P(match) |
|---|---|
| 10 | 11.7 % |
| 23 | 50.7 % |
| 30 | 70.6 % |
| 57 | 99.0 % |
| 70 | 99.9 % |
At n = 23 the probability crosses 50 %. Most people guess you need roughly 183 people.
Why the intuition fails
You are not asking “does someone share my birthday?” (which would require ~253 people for 50 %). You are asking whether any of the C(23, 2) = 253 pairs share a birthday. Counting pairs, not individuals, is the key insight. The number of pairs grows as n(n-1)/2.
The approximation
Using ln(1-x) ≈ -x for small x:
P(match) ≈ 1 - e^(-n(n-1)/(2×365))This gives n ≈ √(2×365×ln 2) ≈ 23 — consistent with exact calculation.