  The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by
which, from any set of isotropic vectors in Euclidean $d$-space, one can select
a subset of $d$ vectors whose determinant is not too small. Subsequently,
Pelczy\'nski and Szarek improved this lower bound by a factor depending on the
dimension and the number of vectors.
  Pivovarov, on the other hand, determined the expectation of the square of the
volume of parallelotopes spanned by $d$ independent random vectors in
$\mathbb{R}^d$, each one chosen according to an isotropic measure. We extend
Pivovarov's result to a class of more general probability measures, which
yields that the volume bound in the Dvoretzky--Rogers lemma is, in fact, equal
to the expectation of the squared volume of random parallelotopes spanned by
isotropic vectors. This allows us to give a probabilistic proof of the
improvement of Pelczy\'nski and Szarek, and provide a lower bound for the
probability that the volume of such a random parallelotope is large.
