Ray effects, an inherent problem in the formulation of the discrete ordinates approximation to the transport equation, are studied. In particular, the effectiveness of using Monte Carlo procedures to generate a first or second collision source is investigated. Monte Carlo procedures provide a general methodology that can be applied to the discrete ordinates solution of complex problems in either two-dimensional or three-dimensional geometries for which ray effects are likely to occur. The Monte Carlo method, which is intrinsically free of ray effects, performs the transport of the source particle to the first collision sites, at which estimates for the uncollided fluxes are made. The uncollided fluxes are then used to compute the first collided fluxes. The uncollided, collided, or first collided fluxes are calculated as first or second collision scattering sources in a format suitable for input into the DORT two-dimensional and TORT three-dimensional discrete ordinates codes. The computational time and precision requirements of the Monte Carlo calculation are analyzed. The results show that significant improvements are achieved in the solution of test problems when using the estimated first collision source and that ray effects are virtually eliminated when using the estimated second collision source.