Prediction of membrane permeability to small molecules represents an important aspect of drug discovery. First-principles calculations of this quantity require an accurate description of both the thermodynamics and kinetics that underlie translocation of the permeant across the lipid bilayer. In this contribution, the membrane permeability to three drugs, or drug-like molecules, namely, 9-anthroic acid (ANA), 2',3'-dideoxyadenosine (DDA), and hydrocortisone (HYL), are estimated in a pure 1-palmitoyl-2-oleoylphosphatidylcholine (POPC) and in a POPC:cholesterol (2:1) mixture. On the basis of independent 2-5-μs free-energy calculations combined with a time-fractional Smoluchowski determination of the diffusivity, the estimated membrane permeabilities to these chemically diverse permeants fall within an order of magnitude from the experimental values obtained in egg-lecithin bilayers, with the exception of HYL in pure POPC. This exception is particularly interesting because the calculated permeability of the sterol-rich bilayer to HYL, in close agreement with the experimental value, is about 600 times lower than that of the pure POPC bilayer to HYL. In contrast, the permeabilities to ANA and DDA differ by less than a factor of 10 between the pure POPC and POPC:cholesterol bilayers. The unusual behavior of HYL, a large, amphiphilic compound, may be linked with the longer range perturbation of the lipid bilayer it induces, compared to ANA and DDA, suggestive of a possibly different translocation mechanism. We find that the tendency of lower permeabilities of the POPC:cholesterol bilayer relative to those of the pure POPC one is a consequence of increased free-energy barriers. Beyond reporting accurate estimates of the membrane permeability, the present contribution also demonstrates that rigorous free-energy calculations and a fractional-diffusion model are key in revealing the molecular phenomena linking the composition of a membrane to its permeability to drugs.