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.