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10. Birthday problem
Chance that at least two people in a room share a birthday. Type the head count.
| People | P(shared birthday) |
|---|---|
| 2 | 0.0027 |
| 10 | 0.1169 |
| 20 | 0.4114 |
| 22 | 0.4757 |
| 23 | 0.5073 |
| 30 | 0.7063 |
| 40 (yours) | 0.8912 |
| 50 | 0.9704 |
| 57 | 0.9901 |
- P(at least two share a birthday), n = 400.89121 − P(all different). Complement, the same move as at least one.
- P(all different birthdays)0.1088365/365 × 364/365 × 363/365 × ... × 326/365 (40 fractions)
- P(someone shares YOUR birthday)0.10151 − (364/365)^39. Different question: only pairs that include you count.
- Number of pairs78040 × 39 / 2. Why it climbs so fast: every pair is a chance to match.
- Expected matching pairs2.137780 / 365
- People needed for a 50% chance23The famous answer: 23 people
- People needed for a 99% chance57
- Assumption to say out loudEqual odds for every day, no leap day, no twinsReal birthdays cluster a little, which makes a match slightly more likely.