Let $X_1$, $X_2$, $X_3$ be independent Binomial($3, p$) random variables.
Let $Y_1 = X_1 + X_3$, $Y_2 = X_2 + X_3$
Let $Z_1 = \begin{cases} 1 & \text{if $ Y_1 = 0$} \\ 0 & \text{Otherwise} \end{cases} $
$Z_2 = \begin{cases} 1 & \text{if $ Y_2 = 0$} \\ 0 & \text{Otherwise} \end{cases} $
I'm trying to find the joint pmf of $(Z_1, Z_2)$
$Y_1$ and $Y_2$ are Binimoial($6, p$) as the X's are independent
The probabilities of $Y_1 = 0$, $Y_2 = 0$ are $(1-p)^6$ I believe.
The issue is $Y_1$ and $Y_2$ are not independent as they both contain $X_3$, meaning I don't think I can just multiply the pmf of $Z_1$ with the pmf of $Z_2$. How would I find the joint pmf in this case?