Consider two urns, urn A and urn B. In urn A, there are 3 balls numbered 0,1,2. In urn B there are 6 balls numbered 1,2,3,4,5,6. A ball is drawn from urn A, then a ball is drawn from urn B. Define $X_A$ as the number of the ball from urn A and $X_B$ the number of the ball drawn form urn B. What is the joint distribution of $(X_A, X_B)$. Let $Y_1 = X_A X_B$ and $Y_2 = \text{max}\{X_A, X_B\}$. Find the joint distribution of $(Y_1, Y_2)$. Hint: make a table. Find the marginal distribution of $Y_1$ and $Y_2$. Are $Y_1$ and $Y_2$ independent? Justify your answer using part (b).
I tried making the table, not sure if it's correct or not. Also I don't know how to use it to check whether $Y_1$ and $Y_2$ independent or not? $$ \begin{array} {l|l|l|l|l|l|l} & & & & X_B & & & \\ \hline & & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline &0 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 \\ \hline X_A&1 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 \\ \hline &2 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 \end{array} $$ $$ \begin{array} {l|l|l|l|l|l|l} & & & & Y_2 & & & \\ \hline & & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline &0 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 & 1/18 \\ \hline &1 & 1/18 & 0 & 0 & 0 & 0 & 0 \\ \hline &2 & 0 & 1/9 & 0 & 0 & 0 & 0 \\ \hline &3 & 0 & 0 & 1/18 & 0 & 0 & 0 \\ \hline Y_1&4 & 0 & 1/18 & 0 & 1/18 & 0 & 0 \\ \hline &5 & 0 & 0 & 0 & 0 & 1/18 & 0 \\ \hline &6 & 0 & 0 & 1/18 & 0 & 0 & 1/18 \\ \hline &8 & 0 & 0 & 0 & 1/18 & 0 & 0 \\ \hline &10 & 0 & 0 & 0 & 0 & 1/18 & 0 \\ \hline &12 & 0 & 0 & 0 & 0 & 0 & 1/18 \end{array} $$