Counting probability question

Question

I have a pretty basic statistics problem on multinominal distribution. Any feedback would be appreciated.

If I have n marbles that can fall into three buckets (a, b, c) each with probability of $\eta_a$, $\eta_b$, $1-(\eta_a+\eta_b)$ . What is the probability of both buckets (a and b) to have at least one marble?

Thanks,

Further clarification: I would like to estimate the probability that both buckets (a and b) to have at least one marble each.

Answer 1

For convenience denote $\eta_c:=1-\eta_a-\eta_b$.

Let $A$ denote the event that bucket $a$ contains at least one marble and let $B$ denote the event that bucket $b$ contains at least one marble.

Then: $$P(A\cap B)=1-P(A^{\complement}\cup B^{\complement})=1-P(A^{\complement})-P(B^{\complement})+P(A^{\complement}\cap B^{\complement})=$$$$1-(\eta_b+\eta_c)^n-(\eta_a+\eta_c)^n+\eta_c^n$$