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?

You are correct that the number will be higher. In fact, it will be much higher.
Ignoring leap years, each person has a $1\over365$ chance of having a January 1 birthday, and a $364\over365$ chance of having some other birthday.
Given a group of $n$ people, the chance that none of them have a January 1 birthday, then, is $\frac{364}{365}^n$, so the chance that at least one has a January 1 birthday is $1-\frac{364}{365}^n$.
You asked how large $n$ must be to make this chance over $\frac12$, or equivalently how large $n$ must be to make $\frac{364}{365}^n < \frac12$.
Since $\frac{364}{365}^{252} = 0.5008\dots$, and $\frac{364}{365}^{253} = 0.4995\dots$, the minimum number of people to get a $50\%$ chance of a January 1 birthday is $253$.
