The axiomatic approach aims at building a consistent theory of probability and is based on the following logical steps:
- Defining the object of investigation, i.e., events
- Defining an algebra of events, i.e., ways to combine events to describe nontrivial occurrences that we might be interested in
Fig. 5.1 E is a subset of F. - Defining the rules of the game that we need in order to assign a probability measure to each event in a coherent way
5.2.1 Sample space and events
To get going, we should first formalize a few concepts about running a random experiment and observing outcomes. The set of possible outcomes is called the sample space, denoted by Ω. For instance, in dice throwing the set of possible outcomes is
This is a finite set, but we might consider alternative random experiments where the set of possible outcome is an infinite set, such as the whole set of integer numbers. Combining random experiments, e.g., by throwing more dice, or the same one repeatedly, we may define rather complex sample spaces. An important feature of outcomes is that they are mutually exclusive, i.e., they cannot occur together.
As we pointed out before, we need not be only interested in simple events consisting of singletons, i.e., elementary outcomes. We have already met compound events such as
Typically events correspond to statements such as “the outcome is larger than two” or “the outcome is between 3 and 5.” Whatever the case, we see that the elements of these sets are also elements of the sample space Ω. We know from set theory that a set E is a subset of set F, denoted by E ⊆ F when all of the elements of E belong to F (see Fig. 5.1).
DEFINITION 5.1 An event E is a subset of the sample space, i.e., E ⊆ Ω.
Fig. 5.2 An event and its complement.
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