Let $q$ be an irreducible polynomial over $\mathbb{Z}.$ What is the probability that $q$ has at least one root modulo a prime $p?$
For quadratic $q,$ the probability should be about a half by quadratic reciprocity in combination with completing the square. The probability that a general function from $\mathbb{F}_p$ to itself has a root is about $\frac{e-1}{e}.$ Therefore, one would guess as the degree of the polynomial goes to infinity, that the probability approaches this number.