I think I solved this problem but I would like to know if I am right or wrong, I am not quite sure.
We assume that the year has 365 days and the birthdays are uniformly distributed. We want to find out what's the probability that at least 2 of n people share the same birthday. So my sample space would be: $$\Omega = \left \{ (w_{1},... ,w_{n}) | n \geq 2, w_{i}\in{1,... ,365} \right \}$$ $$|\Omega| = 365^{n}$$
Let $A$ be the event that at least 2 people share the same birthday. Let $A_{0}$ be the event that no people share the same birthday/n people were born on different days. $\Rightarrow |A_{0}| = \binom{365}{n}$ This implies: $$P(A) = P(A_{0}^{c}) = 1- P(A_{0}) = 1-\frac{\binom{365}{n}}{365^{n}}$$ Is this true? Or where did I go wrong? The thing that I am really not sure about is the cardinality of $\Omega$.
Thanks for your help