How many people to share a birth month?

Question

Assume birth months are independent and uniform over 12 months. How many people do you need before two of them sharing a birth month is more likely than not?

Accepted Answer

Seat people one at a time; each must avoid the months already used: P(	ext{all distinct}) = \frac{12}{12}\cdot\frac{11}{12}\cdot\frac{10}{12}\cdots Track it: - n = 4: \frac{12 \cdot 11 \cdot 10 \cdot 9}{12^4} = \frac{11880}{20736} \approx 0.573, so match probability \approx 0.427. - n = 5: multiply by \frac{8}{12} to get \approx 0.382, so match probability \approx 0.618. The match probability first exceeds 	frac12 at n = 5 people. A caution on the approximation. The rule of thumb n \approx 1.18\sqrt{d} = 1.18\sqrt{12} \approx 4.1 suggests 4, but the exact count is 5. The \sqrt d formula is …

How many people to share a birth month?

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