Transport of metabolites between cells and between subcellular compartments is facilitated by special protein channels that form aqueous pores traversing biological membranes. Accumulating evidence demonstrates that solute-specific channels display pronounced binding to the corresponding solutes. In this paper we rationalize this observation by showing that a wide and deep potential well inside the channel is able to greatly increase the transit probability of the particle through the channel. Using a one-dimensional diffusion model with radiation boundary conditions, we give exact analytical expressions for the particle translocation probabilities. We also run Brownian dynamics simulations to verify the model and the quantitative predictions of our theory.