Example 4: A fair cube die, with faces numbered $1, 2, 3, 4, 5, 6$, is thrown twice. If the numbers on the top faces are $m$ and $n$ respectively, then the probability that the graph of the quadratic function $y=x^{2}+m x+n$ intersects the $x$-axis at two distinct points is ( ).
(A) $\frac{5}{12}$
(B) $\frac{4}{9}$
(C) $\frac{17}{36}$
(D) $\frac{1}{2}$
Explanation: The total number of basic events is $6 \times 6=36$, meaning we can get 36 quadratic functions. Using the discriminant of a quadratic equation $\Delta=b^{2}-4 a c=m^{2}-4 n>0$, we know $m^{2}$ $>4 n$. By enumeration, we find that there are 17 pairs of $m, n$ that satisfy the condition, so $P=\frac{17}{36}$.