Multistage models have been used to describe various features of the incidence of cancer including the shape of the age-incidence curve; the influence of age at, duration of, and time since exposure; and the synergistic effect of exposure to multiple carcinogens. However, the models require from five to seven distinct transformations that must occur in a particular sequence. The lack of experimental support for so many events suggests a simpler model involving only two mutational events with a proliferative advantage for intermediate-stage cells. Neither model easily explains the paradoxical phenomenon that protraction of low linear energy transfer (LET) radiation leads to lower risks per unit of total exposure, whereas the reverse occurs for high-LET radiation. In this paper, a three-stage model is considered that consists of two mutations at homologous sites, either or both of which might be induced by radiation, followed by activation of the transformed oncogene, which is not induced by radiation. Single-stranded lesions are potentially repairable, whereas double-stranded lesions may increase the proliferation rate. For low-LET radiation, these two mutations are more likely to occur as the result of independent transversals of a cell by separate quanta of radiation, whereas for high-LET radiation, they are more likely to occur simultaneously as the result of a single particle. The predictions of the model are illustrated for various patterns of exposure and choices of model parameters. Various tests of the proposed model are discussed.