Within the context of one-group diffusion theory, we discuss the effect of randomness (stochasticity) on the criticality of a bare nuclear reactor. Previous authors have concluded that randomness decreases the critical size for a given amount of fuel, and that such randomness, when in-troduced into a homogeneous critical reactor, leads most probably to a supercritical state. By considering a sufficiently simple stochastic problem so that exact results can be obtained, we judge these prior conclusions to be only partially correct. We show that the effect of randomness on a criticality problem depends on both the nature of the randomness and the ensemble-averaging procedure and interpretation used to describe the reactor in the stochastic setting.