The question of the existence of free subgroups in a given group traces back to a problem posed by von Neumann. When such subgroups exist, their genericity has been the focus of extensive research. In this talk, we formulate and investigate a version of the genericity question for subgroups of the affine group over a field of characteristic zero. Let $G$ denote the affine group over a field $F$, and let $\mu$ be a probability measure on $G \times G$. Consider the left or right random walk $(X_n, Y_n)$ on $G \times G$ driven by $\mu$. I will discuss several results on conditions under which the semigroup generated b $X_n$, $Y_n$ is free, either eventually or infinitely often. The talk is based on a work in progress with Richard Aoun.