Photon migration in a randomly inhomogeneous, highly scattering and absorbing semi-infinite medium with a plane boundary is considered by a Monte Carlo (MC) technique. The employed MC technique combines the statistical weight scheme and real photon paths simulation, allowing the exclusion of the energy conservation problem. The internal reflection of the scattered radiation on the medium interface is taken into account by allowing the trajectories of photon packets to be split into reflected and transmitted parts. The spatial photon sensitivity profile (SPSP), spatially resolved diffuse reflectance and angular and spatial photon detector weight distributions are considered in terms of Fresnel's reflection/refraction on the boundary of the medium. The effect of the refractive index match is predicted correctly by the MC method and by the diffusion approximation. The results demonstrate that matching of the refractive index of the medium significantly improves the contrast and spatial resolution of the spatial photon sensitivity profile (SPSP). The results of simulation of the spatially resolved diffuse reflectance agree well with the results predicted by the diffusion approximation and the experimental results reported earlier.