It has been known for over 40 years that there are two fundamentally different kinds of detection tasks in the theory of signal detectability. The Type 1 task is to distinguish between events defined independently of the observer; the Type 2 task is to distinguish between one's own correct and incorrect decisions about those Type 1 events. For the Type 1 task, the behavior of the detector can be summarized by the traditional receiver operating characteristic (ROC) curve. This curve can be compared with a theoretical ROC curve, which can be generated from overlapping probability functions conditional on the Type 1 events on an appropriate decision axis. We show how to derive the probability functions underlying Type 2 decisions from those for the Type 1 task. ROC curves and the usual measures of performance are readily obtained from those Type 2 functions, and some relationships among various Type 1 and Type 2 performance measures are presented. We discuss the relationship between Type 1 and Type 2 confidence ratings and caution against the practice of presenting transformed Type 2 ratings as empirical Type 1 ratings.