Probability, chance and the probability of chance

In our day-to-day discourse on uncertainty, words like belief, chance, plausible, likelihood and probability are commonly encountered. Often, these words are used interchangeably, because they are intended to encapsulate some loosely articulated notions about the unknowns. The purpose of this paper is to propose a framework that is able to show how each of these terms can be made precise, so that each reflects a distinct meaning. To construct our framework, we use a basic scenario upon which caveats are introduced. Each caveat motivates us to bring in one or more of the above notions. The scenario considered here is very basic; it arises in both the biomedical context of survival analysis and the industrial context of engineering reliability. This paper is expository and much of what is said here has been said before. However, the manner in which we introduce the material via a hierarchy of caveats that could arise in practice, namely our proposed framework, is the novel aspect of this paper. To appreciate all this, we require of the reader a knowledge of the calculus of probability. However, in order to make our distinctions transparent, probability has to be interpreted subjectively, not as an objective relative frequency.