Let $G$ denote the affine group over a field $F$. For a probability measure  $\mu$ on $G \times G$, we denote by $(X_n, Y_n)$ the left or right random walk on $G \times G$ driven by $\mu$. In this talk, I will discuss a number of results concerning situations in which the semigroup generated by $X_n$, $Y_n$ is free, either eventually or infinitely often. The talk is based on a work in progress with Richard Aoun.