Suppose $p(0,0)=\frac{1}{18}$, $p(0,1)=\frac{3}{18}$, $p(1,0)=\frac{4}{18}$, $p(1,1)=\frac{3}{18}$, $p(2,0)=\frac{6}{18}$, $p(2,1)=\frac{1}{18}$.
Does $E[X|Y]= E[X]$ and $E[Y]= E[Y|X]$?
Suppose $p(0,0)=\frac{1}{18}$, $p(0,1)=\frac{3}{18}$, $p(1,0)=\frac{4}{18}$, $p(1,1)=\frac{3}{18}$, $p(2,0)=\frac{6}{18}$, $p(2,1)=\frac{1}{18}$.
Does $E[X|Y]= E[X]$ and $E[Y]= E[Y|X]$?