6-ConceptsProbability Page 169 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 169Intelligence and engineering applications, but their use in economics
and finance has so far been limited.
Let’s now examine probability as the key representation of uncer-
tainty, starting with a more formal account of probability theory.OUTCOMES AND EVENTS
The axiomatic theory of probability is based on three fundamental con-
cepts: (1) outcomes, (2) events, and (3) measure. The outcomes are the
set of all possible results of an experiment or an observation. The set of
all possible outcomes is often written as the set Ω. For instance, in the
dice game a possible outcome is a pair of numbers, one for each face,
such as 6 + 6 or 3 + 2. The space Ω is the set of all 36 possible out-
comes.
Events are sets of outcomes. Continuing with the example of the
dice game, a possible event is the set of all outcomes such that the sum
of the numbers is 10. Probabilities are defined on events, not on out-
comes. To render definitions consistent, events must be a class ℑ of sub-
sets of Ω with the following properties:■ Property 1. ℑ is not empty■ Property 2. If A ∈ ℑ then AC ∈ ℑ; AC is the complement of A with
respect to Ω, made up of all those elements of Ω that do not belong to
A
∞■ Property 3. If Ai ∈ ℑ for i = 1,2,... then ∪ Ai ℑ ∈
i = 1Every such class is called a σ-algebra. Any class for which Property 3 is
valid only for a finite number of sets is called an algebra.
Given a set Ω and a σ-algebra G of subsets of Ω, any set A ∈ G is said
to be measurable with respect to G. The pair (Ω,G) is said to be a mea-
surable space (not to be confused with a measure space, defined later in
this chapter). Consider a class G of subsets of Ω and consider the small-
est σ-algebra that contains G, defined as the intersection of all the σ-
algebras that contain G. That σ-algebra is denoted by σ{G} and is said
to be the σ-algebra generated by G.
A particularly important space in probability is the Euclidean space.
Consider first the real axis R (i.e., the Euclidean space R^1 in one dimen-
sion). Consider the collection formed by all intervals open to the left and
closed to the right, for example, (a,b]. The σ-algebra generated by this