Axiomatic (Set-Theoretic) Probability

Axiomatic (Set-Theoretic) Probability

Earlier classes may have introduced probability informally as $\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ for experiments with equally likely outcomes. The axiomatic approach, due to the mathematician A. N. Kolmogorov, puts probability on a rigorous footing that works for any random experiment, not only ones with equally likely outcomes, and it is built entirely on the set-theoretic language of sample spaces and events developed above.

The Three Axioms

Let $S$ be the sample space of a random experiment. A probability function $P$ assigns to every event $E \subseteq S$ a real number $P(E)$, subject to three axioms:

Any assignment of numbers to events that satisfies all three axioms is a valid probability function — the axioms constrain what is allowed, without forcing one particular formula.

Probability of Equally Likely Outcomes

When a sample space $S = \{\omega_1, \omega_2, \dots, \omega_n\}$ has $n$ outcomes that are all equally likely, consistency with Axioms 2 and 3 forces each simple event to have probability $\dfrac{1}{n}$: since the $n$ simple events $\{\omega_1\}, \dots, \{\omega_n\}$ are pairwise mutually exclusive and their union is $S$, Axiom 3 gives $P(\omega_1) + \cdots + P(\omega_n) = P(S) = 1$; equal likelihood then forces every term to equal $\frac{1}{n}$. For any event $E$ built from $m$ of these outcomes, applying Axiom 3 again gives

recovering the familiar classical formula as a consequence of the axioms, not as a separate definition. …