  Consider a uniformly distributed random linear subspace $L$ and a
stochastically independent random affine subspace $E$ in $\mathbb{R}^n$, both
of fixed dimension. For a natural class of distributions for $E$ we show that
the intersection $L\cap E$ admits a density with respect to the invariant
measure. This density depends only on the distance $d(o,E \cap L)$ of $L\cap E$
to the origin and is derived explicitly. It can be written as the product of a
power of $d(o,E \cap L)$ and a part involving an incomplete beta integral.
Choosing $E$ uniformly among all affine subspaces of fixed dimension hitting
the unit ball, we derive an explicit density for the random variable $d(o,E
\cap L)$ and study the behavior of the probability that $E \cap L$ hits the
unit ball in high dimensions. Lastly, we show that our result can be extended
to the setting where $E$ is tangent to the unit sphere, in which case we again
derive the density for $d(o,E \cap L)$. Our probabilistic results are derived
by means of a new integral-geometric transformation formula of
Blaschke--Petkantschin type.
