## How Many People in a Group Such That The Probability That Two People Share Same Birthday is More Than a Half

How large a group of people must we have before the probability of two people having the same birthday is more than a half?
Suppose we have a group of n people.There are 364 days in every 3 out of 4 years, and 365 days in 1 out of 4 years. The probability of any two randomly selected people having the same birthday is then
$\frac{3}{4} \times \frac{1}{364} + \frac{1}{4} \times \frac{1}{365}$

In a group of
$n$
people there are
$\frac{n(n-1)}{2}$
pairings so the probability of any two people from this group having the same birthday is
$\frac{n(n-1)}{2}( \frac{3}{4} \times \frac{1}{364} + \frac{1}{4} \times \frac{1}{365})$

We require that this be more than a half, so solve
$\frac{n(n-1)}{2}( \frac{3}{4} \times \frac{1}{364} + \frac{1}{4} \times \frac{1}{365}) \gt \frac{1}{2}$

This simplifies to
$5836n^2-5836n-2125760=0$

The solution to this equation are
$n=19.59, \: -18.59$
to 2 decimal places.
Obviously we take the first of these and round it up, taking
$n=20$
.