I have the following problem:
In the country Maxvierk the number $N$ of children of a randomly selected family, and the number $G$ of girls among the children, are given by the following joint probability mass function:
Determine $\mathbb{E}[N|G = g]$ and $\text{Var}[N|G = g]$.
I began by solution as follows:
$$\begin{align} E[N | G = 0] &= \sum_{n = 0}^4 n P(N = n | G = 0) \\ &= (0) \left( \dfrac{1}{5} \right) + (1) \left( \dfrac{1}{10} \right) + (2) \left( \dfrac{1}{20} \right) + (3) \left( \dfrac{1}{40} \right) + (4) \left( \dfrac{1}{80} \right) \\ &= \dfrac{13}{40} \end{align}$$
However, the solution has that $E[N | G = 0] = \dfrac{26}{31}$.
I would greatly appreciate it if people could please take the time to explain my misunderstanding.
You're on the right track. However $E[N|G=0]$ is actually given by a quotient,
$$ \sum_{n=0}^4 n P(N=n \textrm{ and } G=0) \over P(G=0). $$
You've computed the numerator. The denominator is $31/80$, and dividing gives the result you want.