In this chapter we recapitulate the beginnings of probability theory. The reader to whom this subject is completely new may wish first to consult a more leisurely introduction, such as McColl (1997).
There are different schools on the meaning of probability. For example, it is argued that a statement such as ‘The Scottish National Party has a probability of 1/5 of winning the election’ is meaningless because the experiment ‘have an election’ cannot be repeated to order. The way out has proved to be an axiomatic approach, originated by Kolmogorov (see Figure 9.1) in 1933, in which all participants, though begging to differ on some matters of interpretation, can nevertheless agree on the consequences of the rules (see e.g. Kolmogorov, 1956b). His work included a rigorous definition of conditional expectation, a crucial and fruitful concept in current work in many areas and applications of probability.
Sample spaces and events
Model 9.1 We begin with the idea that, corresponding to an experiment E, there is a set S, the sample space, consisting of all possible outcomes. In the present context an event A is a set of outcomes, that is A ⊆ S. Then it is a matter of definition that, if E is performed with outcome a, the event A occurs if and only if a ∈ A.
Often, but not always, the outcomes are conveniently represented by numbers, as illustrated in examples below.