how many people must be in a room so that the probability of two of them being born in the same month is 0.9?

Answer 1

Assuming the birth months are uniformly distributed (1/12 each), then the probability of n people all being in different months is
1 (11/12) (10/12) .. ((13-n)/12)
and the probability that there are two with the same month is therefore
1 - 1 (11/12) (10/12) .. ((13-n)/12)
Tabulating this against n we get, to 5 d.p.:
n P
01 0
02 0.08333
03 0.23611
04 0.42708
05 0.61806
06 0.77720
07 0.88860
08 0.95358
09 0.98453
10 0.99613
11 0.99936
12 0.99995
13 1

So for the probability to be at least 0.9, we need 8 people in the room - in which case the probability is in fact over 95%.