Sharing a birth month

Question

Assume birth months are spread evenly over the 12 months of the year. What is the smallest group size for which two people are more likely than not to share a birth month?

Accepted Answer

Same idea as the birthday problem, but with only 12 "slots" instead of 365. Track the probability that all birth months are different. P(	ext{all different}) = \frac{12}{12}\cdot\frac{11}{12}\cdot\frac{10}{12}\cdots Compute step by step: - n = 4: \dfrac{12\cdot 11\cdot 10\cdot 9}{12^4} = \dfrac{11880}{20736} \approx 0.573, so a shared month has probability about 0.427, under a half. - n = 5: multiply by \dfrac{8}{12} to get \approx 0.382, so a shared month has probability about 0.618, over a half. So it first becomes more likely than not at n = 5 people. The answer is 5. With only twelve slo…

Sharing a birth month

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