The Birthday Problem: Shared Birthdays in a Room

You are at a party and wondering about birthday coincidences. **Part 1:** There are $N \leq 365$ people in the room. What is the probability that at least two of them share the same birthday? Assume birthdays are uniformly distributed across 365 days and independent. **Part 2:** Now focus on yourself. Given $N$ people in the room (including you), what is the probability that at least one other person shares your birthday?