  The usual way to investigate the statistical properties of finitely generated
subgroups of free groups, and of finite presentations of groups, is based on
the so-called word-based distribution: subgroups are generated (finite
presentations are determined) by randomly chosen k-tuples of reduced words,
whose maximal length is allowed to tend to infinity. In this paper we adopt a
different, though equally natural point of view: we investigate the statistical
properties of the same objects, but with respect to the so-called graph-based
distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups
(and finite presentations) are determined by randomly chosen Stallings graphs
whose number of vertices tends to infinity. Our results show that these two
distributions behave quite differently from each other, shedding a new light on
which properties of finitely generated subgroups can be considered frequent or
rare. For example, we show that malnormal subgroups of a free group are
negligible in the graph-based distribution, while they are exponentially
generic in the word-based distribution. Quite surprisingly, a random finite
presentation generically presents the trivial group in this new distribution,
while in the classical one it is known to generically present an infinite
hyperbolic group.
