## a simpler (?) birthday problem

A monthly birthday problem from the Riddler:

What was the probability that none of the 40 people had birthdays this month? What is the probability that there is at least one month in the year during which none of the 40 people had birthdays (not necessarily this month)?

Assuming the same number of days in all months, the probability that one individual is not born in March is 1/12 and hence the probability that none of 40 (independent!) persons are not born in March is (11/12)⁴⁰, about 3%. The second question can be solved by reading Feller’s chapter on the combination of events (1970, Chapter IV, p.102). The probability that all months are seeing at least one birthday is

$\sum_{i=0}^{12} (-1)^i {12\choose i}(1-i/12)^{40}=0.6732162$

which can be checked by a quick R simulation. The complement 0.326 is thus close to 11 x 0.03!

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