Probability: Foundations of

Abstract

Abstract Probability and probability models are mathematical representations of uncertainties. They can be seen either as analogies with urn models or as a logical system which conforms to a set of probability laws or axioms. It is argued that if uncertainties conform to an urn model or to the probability axioms and the probability space is dense enough then there is a correct probability associated with every uncertainty. Because probability laws place constraints on uncertainties there are disputes over where probabilities apply. One view is that probabilities are relative frequencies. This means that probabilities do not apply to unique events as it might be necessary to break down any relative frequency further before it can be applied to a particular event. Nevertheless some conformity between people's judgments and the frequency view of probability are reported. An alternative subjective approach takes probabilities to be measures of belief but it is unclear then why they should conform to probability laws. A partial answer is that if a person X bets in accordance with subjective probabilities which violate these laws then X can be made to lose regardless of the state of the world. Problems with taking probabilities to be measures of belief are discussed and it is concluded that neither the frequency nor the subjective approaches provide absolute standards against which people's uncertainty judgments should be compared. Theorists have attempted to identify objective probabilities but it is unclear how their existence could ever be established. Nevertheless the search for objective probabilities is defended.