A new numerical method for computing first-flight collision, escape, and transmission probabilities in three dimensions is described. The method consists of subdividing the domain into parts called elements and assuming, as an approximation, that the interaction between a source element and a sink element takes place only along the path that joins their centers of mass. The calculation is repeated with the number of elements increased successively and Richardson extrapolation to an infinite number of elements applied to the sequence of results until convergence to the desired degree of accuracy is attained. Solutions to some test problems indicate that, in general, four steps of repeated Richardson extrapolation are sufficient to yield results with an accuracy comparable to that of existing codes.