  Given an irreducible variety $X$ over a finite field, the density of
hypersurfaces of varying degree $d$ intersecting $X$ in an irreducible
subvariety is $1$, by a result of Charles and Poonen. In this note, we analyse
the situation fixing $d=1$ and extend the base field instead of the degree $d$.
We compute the probability that a random linear subspace of the right dimension
intersects $X$ in a given number of points.
