1.1 Probability systems  (Page 4/4)

Some of these properties, such as (P4) , (P5) , and (P6) , are so elementary that it seems they should be included in the defining statement. This would not be incorrect, but wouldbe inefficient. If we have an assignment of numbers to the events, we need only establish (P1) , (P2) , and (P3) to be able to assert that the assignment constitutes a probability measure. And the other properties follow as logical consequences.

Flexibility at a price

In moving beyond the classical model, we have gained great flexibility and adaptability of the model. It may be used for systems in which the number of outcomes is infinite(countably or uncountably). It does not require a uniform distribution of the probability mass among the outcomes. For example, the dice problem may be handled directly byassigning the appropriate probabilities to the various numbers of total spots, 2 through 12. As we see in the treatment of conditional probability , we make new probability assignments (i.e., introduce newprobability measures) when partial information about the outcome is obtained.

But this freedom is obtained at a price. In the classical case, the probability value to be assigned an event is clearly defined (although it may be very difficult to performthe required counting). In the general case, we must resort to experience, structure of the system studied, experiment, or statistical studies to assign probabilities.

The existence of uncertainty due to “chance” or “randomness” does not necessarily imply that the act of performing the trial is haphazard. The trial may be quite carefullyplanned; the contingency may be the result of factors beyond the control or knowledge of the experimenter. The mechanism of chance (i.e., the source of theuncertainty) may depend upon the nature of the actual process or system observed. For example, in taking an hourly temperature profile on a given day at a weather station, the principalvariations are not due to experimental error but rather to unknown factors which converge to provide the specific weather pattern experienced. In the case of an uncorrected digitaltransmission error, the cause of uncertainty lies in the intricacies of the correction mechanisms and the perturbations produced by a very complex environment. A patient at a clinicmay be self selected. Before his or her appearance and the result of a test, the physician may not know which patient with which condition will appear. In each case, from the point of viewof the experimenter, the cause is simply attributed to “chance.” Whether one sees this as an “act of the gods” or simply the result of a configuration of physical or behavioral causestoo complex to analyze, the situation is one of uncertainty, before the trial, about which outcome will present itself.

If there were complete uncertainty, the situation would be chaotic. But this is not usually the case. While there is an extremely large number of possible hourly temperature profiles, a substantialsubset of these has very little likelihood of occurring. For example, profiles in which successive hourly temperatures alternate between very high then very low values throughout the day constitutean unlikely subset (event). One normally expects trends in temperatures over the 24 hour period. Although a traffic engineer does not know exactly how many vehicles will be observed in a giventime period, experience provides some idea what range of values to expect. While there is uncertainty about which patient, with which symptoms, will appear at a clinic, a physiciancertainly knows approximately what fraction of the clinic's patients have the disease in question. In a game of chance, analyzed into “equally likely” outcomes, the assumption of equal likelihoodis based on knowledge of symmetries and structural regularities in the mechanism by which the game is carried out. And the number of outcomes associated with a given event is known, or may bedetermined.

In each case, there is some basis in statistical data on past experience or knowledge of structure, regularity, and symmetry in the system under observation which makes it possible to assignlikelihoods to the occurrence of various events. It is this ability to assign likelihoods to the various events which characterizes applied probability. However determined, probability is a number assigned to events to indicate their likelihood of occurrence . The assignments must be consistent with the defining properties (P1) , (P2) , (P3) along with derived properties (P4) through (P9) (plus others which may also be derived from these). Since the probabilities are not “built in,” as in the classical case, a prime role ofprobability theory is to derive other probabilities from a set of givenprobabilites .