We propose an approach for estimating the probability that a given small target, among many, will be the first to be reached in a molecular dynamics simulation. Reaching small targets out of a vast number of possible configurations constitutes an entropic barrier. Experimental evidence suggests that entropic barriers are ubiquitous in biomolecular systems, and often characterize the rate-limiting step of biomolecular processes. Presumably for the same reasons, they often characterize the rate-limiting step in simulations. To the extent that first-passage probabilities can be computed without requiring direct simulation, the process of traversing entropic barriers can replaced by a single choice from the computed ("first-passage") distribution. We will show that in the presence of certain entropic barriers, first-passage probabilities are approximately invariant to the initial configuration, provided that it is modestly far away from each of the targets. We will further show that as a consequence of this invariance, the first-passage distribution can be well-approximated in terms of "capacities" of local sets around the targets. Using these theoretical results and a Monte Carlo mechanism for approximating capacities, we provide a method for estimating the hitting probabilities of small targets in the presence of entropic barriers. In numerical experiments with an idealized ("golf-course") potential, the estimates are as accurate as the results of direct simulations, but far faster to compute.