5. A die is rolled twice in succession, yielding the numbers $m$ and $n$. The probability that a right-angled triangle can be formed with the points $(0,0)$, $(1,-1)$, and $(m, n)$ as vertices is
5. $\frac{5}{18}$.

Since $1 \leqslant m, n \leqslant 6$, there are a total of 36 possible values for $(m, n)$. The points $(m, n)$ that can form a right-angled triangle with the points $(0,0)$, $(1,-1)$ have the following 10 possible values:
$$
\begin{array}{l}
(1,1),(2,2),(3,3),(4,4),(5,5), \\
(6,6),(3,1),(4,2),(5,3),(6,4) .
\end{array}
$$

Therefore, the required probability is $\frac{10}{36}=\frac{5}{18}$.