The different interpretations of probability, such as degree of rational belief (subjective), relative frequency (operational) or propensity (physical or dispositional)), often give rise to confusion in deciding which is the most appropriate for a given application. It is the claim of this paper that this confusion is what gives rise to some of the claims of inadequacy of probability theory in designing automatic decision systems as well as making mistakes due to inappropriate metaphors. In this field strong analogies with human decision making provide a guide to system design and although a subjective interpretation of probability in human decision making could be justified and interpreted as “degree of belief” it is difficult to consider this to be correct for a relatively simple autonomous and mechanistic decision module. It is the purpose of this paper to try and dispel the confusion and come to conclusion on the correct interpretation or interpretations for applying probability theory.

Probability theory originated with Edmund Halley and Blaise Pascal. Pascal provided a response to a request for mathematical guidance for games of chance. The only practical help that his results should have provided to the Chevalier de Méré and other gamblers since is to avoid gambling because by working out the odds it can be calculated that the expected win is at best zero and usually negative in any realistic scenario. This advice has not been taken because in the first instance Pascal did not not make it explicit and in the second because most of those who gamble do not or cannot understand the results of probability theory.

Edmund Halley's contribution was on a more practical civic context. As is the case to data local government needed to raise fund. They did so by selling annuities. However without a reliable calculus to estimate the probability if life expectancy many went bankrupt. Halley provided an elementary probability calculus.

There is the additional reason that probability is not enough to reach a decision because decision theory is needed with the restrictions that are put on utility functions for rational decisions.

Despite the negative result in the application for which it was invented, the theory has been developed to provide a rich body of deep and subtle results. Not only that but very useful theorems have been discovered, giving rise to the applied branch of the theory known as statistics. The tools provided are essential to decision making, the insurance business, epidemiology, clinical trials and the evaluation of scientific experiments. Probability theory has shown itself to be essential to basic physics, providing an essential part of quantum mechanics, the theory for our description of the subatomic world, and the theory of how the atomic world gives rise to the macroscopic laws of thermodynamics known as statistical mechanics.

As well as problems of interpretation there is the need to treat the manipulation of the formalism and the statement of the models rigorously. Relatively simple examples show that intuition can be a very poor guide in analysing the consequences of probability models. This rigorous approach has longer term advantages in designing reliable autonomous systems.

Treating probability as the degree of belief held by a system provides an implicit tie to what a system knows. Knowledge is commonly only attributed to intelligent beings but the constructs of modal logic provide definitions and a formalism to describe an idealised form of knowledge which can be attributed to mechanistic systems. This provides a method of addressing the question of what a system knows without needing to define a probability model. In this paper it is argued that the relationship between knowledge, belief, uncertainty and randomness can be clarified by using a unified theory of probability and modality and this lays the foundation for a quite general theory of decision support by an agent with incomplete information about and randomness in its immediate, relevant environment is of great practical importance.

A system with capacity for holding and eventually updating knowledge while acting in face of uncertainty with respect to how its knowledge of the world relates to the true world will be called an agent. In this paper “system” will be used very generally to cover avionics systems, interacting robots, security officers against hackers, telecommunications networks, participants in a battle and even players in a game of “chicken”.