In the previous chapter the axiomatic approach provided us with a mathematical model based on the triplet (S, ℱ, P(·)) which we called a probability space, comprising a sample space S, an event space ℱ (σ-field) and a probability set function P(·). The mathematical model was not developed much further than stating certain properties of P(·) and introducing the idea of conditional probability. This is because the model based on (S, ℱ, P(·)) does not provide us with a flexible enough framework. The main purpose of this section is to change this probability space by mapping it into a much more flexible one using the concept of a random variable.

The basic idea underlying the construction of (S, ℱ, P(·)) was to set up a framework for studying probabilities of events as a prelude to analysing problems involving uncertainty. The probability space was proposed as a formalisation of the concept of a random experiment ℰ. One facet of ℰ which can help us suggest a more flexible probability space is the fact that when the experiment is performed the outcome is often considered in relation to some quantifiable attribute; i.e. an attribute which can be represented by numbers. Real world outcomes are more often than not expressed in numbers. It turns out that assigning numbers to qualitative outcomes makes possible a much more flexible formulation of probability theory. This suggests that if we could find a consistent way to assign numbers to outcomes we might be able to change (S, ℱ, P(·)) to something more easily handled.