1.2 Interpretations  (Page 2/5)

There is a variety of points of view as to how probability should be interpreted. These impact the manner in which probabilities are assigned(or assumed). One important dichotomy among practitioners.

• One group believes probability is objective in the sense that it is something inherent in the nature of things. It is to be discovered, if possible, by analysisand experiment. Whether we can determine it or not, “it is there.”
• Another group insists that probability is a condition of the mind of the person making the probability assessment. From this point of view, the laws of probability simply impose rational consistency upon the way one assigns probabilities to events. Various attempts have been made to find objectiveways to measure the strength of one's belief or degree of certainty that an event will occur. The probability $P\left(A\right)$ expresses the degree of certainty one feels that event A will occur. One approach to characterizing an individual's degree of certainty is toequate his assessment of $P\left(A\right)$ with the amount a he is willing to pay to play a game which returns one unit of money if A occurs, for a gain of $(1-a)$ , and returns zero if A does not occur, for a gain of $-a$ . Behind this formulation is the notion of a fair game , in which the “expected” or “average” gain is zero.

The early work on probability began with a study of relative frequencies of occurrence of an event under repeated but independent trials. This idea is so imbedded in much intuitive thought about probability that some probabilistshave insisted that it must be built into the definition of probability. This approach has not been entirely successful mathematically and has notattracted much of a following among either theoretical or applied probabilists. 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 anindependent manner, the fraction of times that event A occurs approaches as a limit the value $P\left(A\right)$ . Establishing this result (which we do not do in this treatment) provides a formal validation of the intuitive notion that lay behindthe early attempts to formulate probabilities. Inveterate gamblers had noted long-run statistical regularities, and sought explanations from their mathematically giftedfriends. From this point of view, probability is meaningful only in repeatable situations. Those who hold this view usually assume an objective view of probability. It is a numberdetermined by the nature of reality, to be discovered by repeated experiment.

There are many applications of probability in which the relative frequency point of view is not feasible. Examples include predictions of the weather, the outcome of agame or a horse race, the performance of an individual on a particular job, the success of a newly designed computer. These are unique, nonrepeatable trials. As the popularexpression has it, “You only go around once.” Sometimes, probabilities in these situations may be quite subjective. As a matter of fact, those who take asubjective view tend to think in terms of such problems, whereas those who take an objective view usually emphasize the frequency interpretation.