Summary: A fundamental assumption of the Hodgkin-Huxley model and other conductance-based neuron models is that the average flux of ions through a stochastically gated ion channel is the product of (a) the flux of ions through a channel that is always open and (b) the proportion of time that the gated channel is open. In this paper, we propose and analyze a model of electrodiffusion through a stochastically gated ion channel to investigate the validity of this classical assumption. We find that this assumption is valid for typical physiological parameter regimes, and we also show that it breaks down for parameters outside of typical physiological ranges. Indeed, we show that the flux through a gated channel can be orders of magnitude larger than this classical assumption if either the gating is fast or the potential difference across the membrane is large. Mathematically, our model consists of one-dimensional advection-diffusion equations with a stochastically switching boundary condition. Employing an iterated random function approach, we prove that the solution converges in distribution at large time and find (i) the support of the solution, (ii) analytical formulas for the mean solution and mean flux, and (iii) analytical formulas for the full probability distribution of the solution in various parameter regimes. All of our analysis is accompanied by numerical simulations of the stochastic PDE.