1.2 Interpretations

The formal probability system is a model whose usefulness can only be established by examining its structure and determining whether patterns of uncertainty and likelihood in any practical situation can be represented adequately. This system is consistent with many probability assignments, just as the notion of mass is consistent with many different mass assignments to sets in the basic space. The defining properties (P1), (P2), P(3) and a number of derived properties provide consistency rules for making probability assignments. One cannot assign negative probabilities or probabilities greater than one. The sure event is assigned probability one. If two or more events are mutually exclusive, the total probability assigned to the union must equal the sum of the probabilities of the separate events. Any assignment of probability consistent with these conditions is allowed. One may not know the probability assignment to every event. A typical applied problem provides the probabilities of members of a class of events (perhaps only a few) from which to determine the probabilities of other events of interest.Early work on probability began with a study of relative frequencies of occurrence of an event under repeated but independent trials. This approach has not been entirely successful mathematically. In the model we adopt, there is a fundamental limit theorem, known as Borel's theorem, which may be interpreted “if a trial is performed a large number of times in an independent manner, the fraction of times that event occurs approaches as a limit the value P(A). Establishing this result (which we do not do) provides a formal validation of the intuitive frequency notion that lay behind early attempts to formulate probabilities. However, there are many applications of probability in which the relative frequency point of view is not feasible, involving unique non repeatable trials.