  This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in
a proper group action with contracting elements, with applications towards
relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a
class of statistically convex-cocompact action, we show that an exponential
generic set of $k$ elements for any fixed $k\ge 2$ generates a
quasi-isometrically embedded free subgroup of rank $k$. For $k=2$, we study the
sprawl property of group actions and establish that the class of statistically
convex-cocompact actions is statistically hyperbolic in a sense of M. Duchin,
S. Leli\`evre, and C. Mooney.
  For any proper action with a contracting element, if it satisfies a condition
introduced by Dal'bo-Otal-Peign\'e and has purely exponential growth, we obtain
the same results on generic free subgroups and statistical hyperbolicity.
