Let $(X,Y)$ be a pair of jointly distributed discrete RVs taking values in $\mathscr{X}\times\mathscr{Y}$ (where both alphabets are finite) with joint pmf $$p_{xy}(x,y):=P(X=a,Y=b), \ a\in\mathscr{X},b\in\mathscr{Y}$$>Then the joint entropy of $X$ and $Y$ (denoted $H(X,Y$)) is given by $$\begin{align*} H(X,Y)&=-\sum_{a\in\mathscr{X}}\sum_{b\in\mathscr{Y}}p_{xy}(x,y)\log_2p_{xy}(x,y)\\ &=E_{p_{xy}}[-\log_2p_{xy}(x,y)] \end{align*}$$

For $(X,Y,Z)\sim p_{XYZ}$ on $\mathscr{X}\times\mathscr{Y}\times\mathscr{Z}$, the conditional joint entropy of $X$ and $Y$ given $Z$ is $$\begin{align*} H(X,Y|Z)&=E_{p_{XYZ}}[-\log_2(p_{XY|Z}(X,Y|Z))]\\ &=-\sum_{z\in\mathscr{X}}\sum_{b\in\mathscr{Y}}\sum_{c\in\mathscr{Z}}p_{XYZ}(a,b,c) \log_2(p_{XY|Z}(a,b|c)) \end{align*}$$