🎂Birthday Paradox Calculator
Try 23 for the surprising result!
The Birthday Paradox is a famous probability result: in a group of just 23 people, there is a greater than 50% probability that two people share a birthday. This surprises most people because 23 seems small compared to 365 days. The probability grows rapidly — with 70 people it exceeds 99.9%. It is called a paradox not because it is logically contradictory, but because it strongly violates intuition.
- 1P(at least one shared birthday) = 1 − P(all different)
- 2P(all different) = (365/365) × (364/365) × (363/365) × ... × ((365−n+1)/365)
- 3P(match) = 1 − ∏ᵢ₌₀ⁿ⁻¹ (365−i)/365
- 4Assumes uniform birthday distribution (actual distribution varies slightly)
n = 23 people=P(shared birthday) ≈ 50.7%The famous threshold
n = 30 people=P ≈ 70.6%
n = 57 people=P ≈ 99.0%
n = 70 people=P ≈ 99.9%
| People (n) | P(shared birthday) | Approx. odds |
|---|---|---|
| 10 | 11.7% | 1 in 8.5 |
| 20 | 41.1% | roughly 2 in 5 |
| 23 | 50.7% | slightly more likely than not |
| 30 | 70.6% | about 2 in 3 |
| 40 | 89.1% | 9 in 10 |
| 57 | 99.0% | almost certain |
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