Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 3 / b

Under the independent uniform-birthday model, condition on the first person's birthday. Exactly one of the second person's $365$ equally likely dates matches, soThis concerns one selected pair; the birthday problem allowing any pair among all guests is a different event.

For matching one specified birthday, each of the $N-1$ other guests avoids the host's date with probability $364/365$. Independence givesFor this to exceed $1/2$, the negative logarithm of $364/365$ reverses the inequality on division, giving $N-1>\ln (1/2)/\ln (364/365)\simeq 252.652$. Thus the smallest total attendance is $N=254$, including the host.