Consider a multinomial distribution $\mathbb{P}$ on $S$ states $\{s_1,\dots,s_S\}$ where $S\in \mathbb{N}$ and $S\geq 2$, with probabilities $\mathbb{P}(s_i)=:p_i$. Now consider $N$ i.i.d. draws $X_1,\dots,X_N$ from $\mathbb{P}$ and suppose that $N\geq S$. What is the probability that each state has occured?
So far I have $$ \mathbb{P}\left(\forall s\in S\, (\exists X_i)\, X_i=s\right) = 1- \mathbb{P}(\exists s\in S \, s\not\in \{X_n\}_{n=1}^N)... $$ But I'm completely stuck
This is the coupon collector’s problem with unequal probabilities. You can find lots of questions and answers relating to this problem under the tag coupon-collector.
By inclusion–exclusion, the probability that each state has occurred after $N$ draws is
$$ \sum_{A\subseteq [S]}(-1)^{|A|}\left(1-\sum_{k\in A}p_k\right)^N\;. $$