Sum and product of 3 integer random variables

Question

Let $X_A,X_B,X_C\in\{0,1,2,\ldots N\}$ be three integer random variables, such that $X_A+X_B+X_C=N$, where the integer $N$ is constant.

We impose the condition $P(X_AX_B>0)=P(X_A+X_B=0)>0$.

Is there any way to show that this condition imply $P(X_AX_BX_C>0)=P(X_A+X_B+X_C=0)=0$?

Many thanks for your help.

Answer 1

No. Consider the following distributions on the random variables (assuming $N>2$). With probability $\frac12$, $X_A = X_B=0$ and $X_C = N$; with probability $\frac12$, $X_A=X_B=1$ and $X_C=N-2$. Then $P(X_AX_B>0)=P(X_A+X_B=0)=\frac12$, but $P(X_AX_BX_C>0) = \frac12$ while $P(X_A+X_B+X_C=0)=0$.