Abstract: From the outside, black holes can be viewed as rapidly thermalizing quantum many-body systems. The interior of a black hole is more mysterious. We discuss concrete exterior observables which are precisely related to the interior geometry and allow us to probe the growth of the interior in an ensemble of black hole states. The observables of interest diagnose degrees of randomness in the ensemble and have implications for the computational complexity of preparing the ensemble. We construct an ensemble of black hole states, the "Einstein-Rosen caterpillars", which are indexed by an evolution time T. We then show that typical states in this ensemble have (1) an interior geometry which grows linearly with time T and (2) a complexity that grows at least linearly with time T. This establishes a sharp version of the geometry/complexity conjectures arising from holographic models of quantum gravity. Throughout we comment on connections between this story and many related questions in quantum information and many-body physics.