Answer to Question #166469 in Statistics and Probability for phemelo mongae

Question

 Suppose there are n people in a room. (a) What is the probability that at least two are born in the same month? Assume all the months are equal. (b) What is the probability if n = 13?

Answer 1

a)

Events:

A - Аt least two are born in the same month.

$\lnot$ A - Nobody are born in the same month.

P(A) = 11/12 * 10/12 * … * (12-n+1)/12 = (12-n+1)/(12^n-1)  where n - number of people in the room.

P($\lnot$A ) = 1-(12-n+1)/(12^n-1)

b)

If n = 13 then according to the Dirichlet principle, at least two people will have a birthday in one month (n = 13, the number of months in a year is 12 and 13>12). Hence P($\lnot$A ) = 1. We can check this value using the formula from “a)”:

P($\lnot$A ) = 1-(12-13+1)/(12^13-1) = 1- 0/12¹² = 1