How many people should be invited to a party in order to make it likely that there are three people with the same birthday?

Let \(X_{ijk}\) be the indicator random variable for the event that the birthdays represented by \(i\), \(j\) and \(k\) are the same. It follows that \(X_{ijk} = \frac{1}{n^2}\). To calculate the number of people to invite:

\[\begin{split} E[X] &= E \left[ \sum\limits_{i=1}^m \sum\limits_{j=i+1}^m \sum\limits_{k=j+1}^m X_{ijk} \right] \\ &= \sum\limits_{i=1}^m \sum\limits_{j=i+1}^m \sum\limits_{k=j+1}^m \frac{1}{n^2} \\ &= {m \choose 3} \frac{1}{n^2} \\ &= \frac{m(m-1)(m-2)}{6n^2} \\ \end{split}\]

And then to solve for \(m\):

\[\begin{split} &= \frac{m(m-1)(m-2)}{6(365)^2} \\ 6(365)^2 & \leq m(m-1)(m-2) \\ 6(133,225) &\leq (m^2 -m)(m-2) \\ 799,350 &\leq m^3 - 3m^2 + 2m \\ \end{split}\]

The smallest integer for which this equation holds is \(m = 94\).