Given two discrete RVs $X$ and $Y$ with joint pmf $p(x,y),x\in\mathscr{X},y\in\mathscr{Y}$, the conditional pmf of $x$ given that $Y=y$ is denoted by $p_{X|Y}(x|y)$ and defined as $$p_{X|Y}(x|y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{p_{xy}(x,y)}{p_y(y)}, \ x\in\mathscr{X}$$