In various contexts of statistical problems one deals with the measurement of a “risk”—a risk of dying, a risk of becoming ill, and so forth. In applied vital statistics various indices are used to measure such risks, and typically these arenotprobabilities. In formal mathematical statistical analyses, typically theyaremeasured as probabilities. A special problem arises when more than one risk must be measured at the same time. Even if they are independent in a technical statistical sense, the presence of one risk will complicate the measurement of another. This situation has been referred to as “competing risks.” In this paper we apply an analysis of competing risks to some questions that arose in the study of the relation of smoking to lung cancer.