A fair six-sided die is rolled 30 independent times, let X be the number of ones and Y the number of twos.
What is the joint PMF of X and Y?
Question: Should I think of this as binomial distribution or trinomial and why? It seems like trinomial since there is number of one, number of two and the rest numbers as three categories. I am not sure.
Let $\Omega = \{1,2,3,4,5,6\}^{N}$ be the sample space. If for a given $\omega\in\Omega$, $X(\omega)=n$ and $Y(\omega)=m$, then $\mathbb P(\{\omega\}) = \left(\frac16\right)^{n+m}\left(\frac23\right)^{N-n-m}$. There are $\binom Nn\binom{N-n}m$ outcomes with $X(\omega)=n$ and $Y(\omega)=m$, and hence $$ \mathbb P(X=n,Y=m) = \binom Nn\binom{N-n}m \left(\frac16\right)^{n+m}\left(\frac23\right)^{N-n-m},\ \ n,m\geqslant 0, n+m\leqslant N. $$