Random groups are proposed by Gromov as a model to study the typical behavior of finitely presented groups. They share many properties of the free group, and Knight conjectured that random groups satisfy the zero-one law and have the same first-order theory as the free group. In joint work with Franklin and Knight, we study this zero-one law in other classes of structures. In particular, we consider random presentations in algebraic varieties in the sense of universal algebra. We will discuss some examples where the zero-one law holds and some other examples where the zero-one law fails. We will also discuss some general results.