The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, Out(Fn), and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.