Given a finite group G, what is the "probability" that a randomly presented group has G as a quotient? This question arises in studying fundamental groups of random 3-manifolds and in studying how often G arises as a Galois group over Q with restricted ramification. The talk will review my earlier work on the case of p-groups (some joint with Jordan Ellenberg, some with Nadya Markin, and some with Michael Bush and Farshid Hajir) and recent work on the general case.