  We study the probability that an $(n - m)$-dimensional linear subspace in
$\mathbb{P}^n$ or a collection of points spanning such a linear subspace is
contained in an $m$-dimensional variety $Y \subset \mathbb{P}^n$. This involves
a strategy used by Galkin--Shinder to connect properties of a cubic
hypersurface to its Fano variety of lines via cut and paste relations in the
Grothendieck ring of varieties. Generalizing this idea to varieties of higher
codimension and degree, we can measure growth rates of weighted probabilities
of $k$-planes contained in a sequence of varieties with varying initial
parameters over a finite field. In the course of doing this, we move an
identity motivated by rationality problems involving cubic hypersurfaces to a
motivic statistics setting associated with cohomological stability.
