Given two events A and B, the conditional probability of A given B is the probability of observing A given that B occurred. Its value is obtained by dividing the probability that the events occur together by the probability of B.
For a random pair (or a random vector), the conditional probability distribution models the distribution of a subset of the variables given the value of the others.
Given a probability space $(\Omega,\Sigma,P)$, the conditional probability of the event $A\in\Sigma$ given that the event $B\in\Sigma$ occurred is $$ P(A\ |\ B) = \frac{P(A,\ B)}{P(B)} , $$ i.e., it is the probability that the two events occur simultaneously divided by the probability of the event $B$.
Given a discrete random pair $(X,Y)\in\X\times\Y$, the conditional probability distribution of $X$ given that $Y=y$ is defined for all $y\in\Y$ as $$ P_{X|Y=y} : \Sigma_X \rightarrow [0,1] $$ with $$ P_{X|Y=y} ( x ) = P(X=x\ |\ Y=y) = \frac{P(X=x,\ Y=y)}{P(Y=y)} = \frac{P_{X,Y}(x,y)}{P_Y(y)} , $$ where $P_{X,Y}$ is the joint probability distribution of $(X,Y)$ and $P_Y$ the marginal distribution of $Y$.
Given a continuous random pair $(X,Y)\in\X\times\Y$, the conditional probability density function of $X$ given $Y$ is defined for all $y\in\Y$ as $$ p_{X|Y}(\cdot, y) : \X \rightarrow \R^+ $$ with $$ p_{X|Y}( x \ |\ y ) = \frac{p_{X,Y}(x, y)}{p_Y(y)} , $$ where $p_{X,Y}$ is the joint probability density function of $(X,Y)$ and $p_Y$ the marginal density function of $Y$.
Given a mixed random pair $(X,Y)\in\X\times\Y$ with $\X\subseteq\R$, $\Y\subset\Z$, the conditional probability density function of $X$ given $Y$ is defined for all $y\in\Y$ as $$ p_{X|Y=y}(\cdot, Y=y) : \X \rightarrow \R^+ $$ with $$ p_{X|Y=y}( x \ |\ Y=y ) = \frac{p_{X,Y}(x, y)}{P_Y(y)} , $$ where $p_{X,Y}$ is the joint probability density function of $(X,Y)$ and $P_Y$ the marginal distribution of $Y$.