Conditional Distributions for Discrete Random Variables

For two discrete random variables $X$ and $Y$ with joint pmf $p_{X,Y}(x,y),$ the conditional probability mass function of $Y$ given $X=x$ is

$$p_{Y\mid X}(y \mid x) = P(Y = y \mid X = x) = \dfrac{p_{X,Y}(x,y)}{p_X(x)},$$

defined whenever $p_X(x) > 0.$

This is ordinary conditional probability: the joint probability of $\{X=x,\,Y=y\}$ divided by the probability of the event $\{X=x\}$ we condition on.

For instance, suppose $X$ and $Y$ have joint pmf

$$\begin{array}{c|cc} & Y=0 & Y=1 \\ \hline X=0 & 0.1 & 0.2 \\ X=1 & 0.3 & 0.4 \end{array}$$

Then $p_X(0) = 0.1 + 0.2 = 0.3,$ so

$$p_{Y\mid X}(1 \mid 0) = \dfrac{p_{X,Y}(0,1)}{p_X(0)} = \dfrac{0.2}{0.3} = \dfrac{2}{3}.$$