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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  
\[n=20\]
.

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