18.2 The axioms of probability
In the previous section, we have talked about probability as an extension of mathematical logic. Just like formal logic, probability has its axioms, which we need to understand to work with probability models. Now, we are going to seek the answer to a fundamental question: what is the mathematical model of probability and how do we work with it?
Probabilities are defined in the context of experiments and outcomes. To talk about probabilities, we need to define what we assign probabilities to. Formally speaking, we denote the probability of the event A by P(A). First, we’ll talk about what events are.
18.2.1 Event spaces and σ-algebras
Let’s revisit the six-sided example from the previous section. There are six different mutually exclusive outcomes (that is, events that cannot occur at the same time), and together they form the event space, denoted by Ω:
In general, the event space is the collection of all mutually exclusive...
