  Let $X$ be an absolutely irreducible hypersurface of degree $d$ in
$\mathbb{A}^n$, defined over a finite field $\mathbb{F}_q$. The Lang-Weil bound
gives an interval that contains $#X(\mathbb{F}_q)$. We exhibit explicit
intervals, which do not contain $#X(\mathbb{F}_q)$, and which overlap with the
Lang-Weil interval. In particular, we sharpen the best known lower and upper
bounds for $#X(\mathbb{F}_q)$. The proof uses a combinatorial probabilistic
technique.
