Bertini irreducibility theorems via statistics

Let $X \subset \mathbb{P}^n$ be a geometrically irreducible subvariety with $\dim X \ge 2$, over any field.
Let $\check{\mathbb{P}}^n$ be the moduli space parametrizing hyperplanes $H \subset \mathbb{P}^n$.
Let $L \subset \check{\mathbb{P}}^n$ be the locus parametrizing $H$ for which $H \cap X$ is geometrically irreducible. The classical Bertini irreducibility theorem states that $L$ contains a dense open subset of ${\check{\mathbb{P}}}^n$, so the bad locus $L' := \mathbb{P}^n - L$ satisfies $\dim L' \le n-1$.  Benoist improved this to $\dim L' \le \operatorname{codim} X + 1$. We describe a new way to prove and generalize such theorems, by reducing to the case of a finite field and studying the mean and variance of the number of points of a random hyperplane section. This is joint work with Kaloyan Slavov.