  In this paper we study the generic, i.e., typical, behavior of finitely
generated subgroups of hyperbolic groups and also the generic behavior of the
word problem for amenable groups. We show that a random set of elements of a
nonelementary word hyperbolic group is very likely to be a set of free
generators for a nicely embedded free subgroup. We also exhibit some finitely
presented amenable groups for which the restriction of the word problem is
unsolvable on every sufficiently large subset of words.
