Progressive Birthday problem

Question

The Birthday Problem is very interesting to me. The more dates you fill up, the lower the chances a date will be outside what has already been seen.

But I do not seem to understand at all how statistics work. Take this small twist to the situation:

After asking people one after one, what is the probability that the first time two people share the same date will be exactly at the $10^{th}$ person

MY UNDERSTANDING

After asking 2 people: The chance of two people having the same Bday $P(S)=1- P(\overline{S}) \Rightarrow P(\overline{S})=\frac{1}{365}\cdot\frac{1}{365}\cdot365\cdot364$ which is the traditional Bday formula.

But then the situation after asking $3,4 ... n$ people and so on eludes me.

Answer 1

Hint:

For the first time two people share the same birthday to be at the $10$th person, you need:

• the first $9$ people to have different birthdays, and
• the $10$th person to have the same birthday as one of the previous $9$