  We show that the probability for a finitely generated subgroup of the modular
group, of size $n$, to be almost malnormal or non-parabolic, tends to 0 as $n$
tends to infinity -- where the notion of the size of a subgroup is based on a
natural graph-theoretic representation of the subgroup.
  The proofs of these results rely on the combinatorial and asymptotic study of
a natural map, which associates with any finitely generated subgroup of
$\textsf{PSL}(2,\mathbb{Z})$ a graph which we call its silhouette, which can be
interpreted as a conjugacy class of free finite index subgroups of
$\textsf{PSL}(2,\mathbb{Z})$.
