Birthday problem given the first birthday The famous 'birthday problem' finds that it takes 23 people to have a 50% chance of some two people having the same birthday.
However, how does the question change if given that person 1's birthday is January 1st, how many people would it take to have 50% chance of another January 1st birthday?
I think that the second question would require more people because we are looking for a specific pair while the original is looking for any pair. But exactly how are the numbers affected?

If you have $n$ people, number one has a birthday on the first day of the year and the remaining $n-1$ of their birthdays are independent and uniformly distributed across a 365 day year, then the chances that someone else also has a birthday on the first day of the year equals one less the chance that none of the other $n-1$ have a birthday on the first day of the year.
So, you want to figure out the smallest $n$ such that $1-(\frac{364}{365})^{n-1} \ge \frac{1}{2}$. This means we need $n \geq 1 + \frac{\ln \frac{1}{2}}{\ln \frac{364}{365}} \approx 253.65$, so we need $n = 254$.
