There is compelling biological data to suggest that cancer arises from a series of mutations in single target cells, resulting in defects in cell renewal and differentiation processes which lead to malignancy. Because much mutagenic damage is expressed following cell division, more-rapidly renewing tissues could be at higher risk because of the larger number of cell replications. Cairns suggested that renewing tissues may reduce cancer risk by partitioning the dividing cell populations into lineages comprising infrequently-dividing long-lived stem cells and frequently-dividing short-lived daughter transit cells. We develop generalizations of three recent cancer-induction models that account for the joint maintenance and renewal of stem and transit cells, also competing processes of partially transformed cell proliferation and differentiation/apoptosis. We are particularly interested in using these models to separately assess the probabilities of mutation and development of cancer associated with spontaneous processes and with those linked to a specific environmental mutagen, specifically ionizing radiation or cigarette smoking. All three models demonstrate substantial variation in cancer risks, by at least 20 orders of magnitude, depending on the assumed number of critical mutations required for cancer, and the stem-cell and transition-cell mutation rates. However, in most cases the conditional probabilities of cancer being mutagen-induced range between 7-96%. The relative risks associated with mutagen exposure compared to background rates are also stable, ranging from 1.0-16.0. Very few cancers, generally <0.5%, arise from mutations occurring solely in stem cells rather than in a combination of stem and transit cells. However, for cancers with 2 or 3 critical mutations, a substantial proportion of cancers, in some cases 100%, have at least one mutation derived from a mutated stem cell. Little difference is made to relative risks if competing processes of proliferation and differentiation in the partially transformed stem and transit cell population are allowed for, nor is any difference made if one assumes that transit cells require an extra mutation to confer malignancy from the number required by stem cells. The probability of a cancer being mutagen-induced correlates across cancer sites with the estimated cumulative number of stem cell divisions in the associated tissue (p<0.05), although in some cases there is sensitivity of findings to removal of high-leverage outliers and in some cases only modest variation in probability, but these issues do not affect the validity of the findings. There are no significant correlations (p>0.3) between lifetime cancer-site specific radiation risk and the probability of that cancer being mutagen-induced. These results do not depend on the assumed critical number of mutations leading to cancer, or on the assumed mutagen-associated mutation rate, within the generally-accepted ranges tested. However, there are borderline significant negative correlations (p = 0.08) between the smoking-associated mortality rate difference (current vs former smokers) and the probability of cancer being mutagen-induced. This is only the case where values of the critical number of mutations leading to cancer, k, is 3 or 4 and not for smaller values (1 or 2), but does not strongly depend on the assumed mutagen-associated mutation rate.