How many months with no birthday?

Question

A team of 20 people have birth months independent and uniform over 12 months. On average, how many months are nobody's birth month?

Accepted Answer

Consider one month at a time and add up with linearity of expectation. A given month stays empty only if all 20 people miss it: P(	ext{month empty}) = \left(\frac{11}{12}ight)^{20} \approx 0.176. Give each of the 12 months an indicator that equals 1 when it is empty. By linearity, the expected number of empty months is E[	ext{empty months}] = 12 	imes \left(\frac{11}{12}ight)^{20} \approx 12 	imes 0.176 \approx 2.11. So on average about 2 months have no birthdays. Equivalently, about 10 of the 12 months are covered, and the reason 20 people do not fill all 12 is precisely the clusteri…

How many months with no birthday?

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