Probability Spaces: Sample Space, Events & Axioms

A probability model (Definition 10.1) formalises a random experiment with three ingredients. The sample space $\Omega$ is the set of all possible outcomes; its elements are called sample points (or outcomes) and are usually written $\omega$. For a single roll of a die, $\Omega=\{1,2,3,4,5,6\}$ and a typical sample point is $\omega=4$. The events form a collection $\mathcal{F}$ of subsets of $\Omega$: an event $A\in\mathcal{F}$ is a set of outcomes, and we say $A$ occurs when the result of the experiment lies in $A$. 'The roll is even' is the event $A=\{2,4,6\}\subseteq\Omega$. Finally, the probability measure $\mathbb{P}:\mathcal{F}\to\mathbb{R}$ assigns each event a number measuring how likely it is. The triple $(\Omega,\mathcal{F},\mathbb{P})$ is the probability space — the complete description of the experiment, and the object over which every later theorem is stated.

A function $\mathbb{P}$ earns the name probability measure only if it obeys three axioms (Definition 10.1). (1) Range: $0\le\mathbb{P}(A)\le 1$ for every event $A$ — probabilities are never negative and never exceed $1$. (2) Normalisation: $\mathbb{P}(\Omega)=1$ (something in $\Omega$ is certain to happen) and $\mathbb{P}(\varnothing)=0$ (the empty event, 'nothing happens', is impossible). (3) Countable additivity: if $A_1,A_2,A_3,\dots$ are pairwise disjoint (meaning $A_i\cap A_j=\varnothing$ whenever $i\neq j$), then $\mathbb{P}\!\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1}^{\infty}\mathbb{P}(A_i).$ Additivity is the workhorse: the probability of a union of non-overlapping events is the sum of their probabilities. Taking all but finitely many $A_i$ empty gives the finite version (Remark 10.1), $\mathbb{P}(A_1\cup\cdots\cup A_n)=\mathbb{P}(A_1)+\cdots+\mathbb{P}(A_n)$ — the rule that lets us add the probabilities of individual outcomes to get the probability of an event.

Because events are sets, we combine them with set operations, and each has a plain-language meaning (Remark 10.1). For events $A,B$: the union $A\cup B$ is the event that $A$ occurs, or $B$ occurs, or both; the intersection $A\cap B$ (also written $AB$) is the event that both occur; the complement $A^c=\Omega\setminus A$ is the event that $A$ does not occur, collecting every outcome of $\Omega$ outside $A$; and $A\cap B^c$ is the event that $A$ occurs but $B$ does not. Two events are mutually exclusive (disjoint) when $A\cap B=\varnothing$ — they cannot occur together — which is exactly the hypothesis the additivity axiom needs. Every event and its complement partition the sample space: $A\cup A^c=\Omega$ and $A\cap A^c=\varnothing$, so exactly one of $A$, $A^c$ occurs. For three or more events the distributive laws $A\cap(B\cup C)=(A\cap B)\cup(A\cap C)$ and De Morgan's laws $(A\cup B)^c=A^c\cap B^c$, $(A\cap B)^c=A^c\cup B^c$ rewrite compound events — the algebra behind nearly every probability rule to come.

When a sample space is finite and all its outcomes are equally likely, probability reduces to counting. If $|\Omega|=n$, the $n$ single-outcome events are disjoint and their union is $\Omega$, so additivity together with $\mathbb{P}(\Omega)=1$ forces each one to have probability $\tfrac1n$. Summing over the outcomes in an event $A$ then gives Proposition 10.1: $\mathbb{P}(A)=\frac{|A|}{|\Omega|}=\frac{\text{number of outcomes favourable to }A}{\text{total number of outcomes}}.$ This is the classical 'favourable over total' rule, and it turns probability questions into combinatorics — count $|A|$ and $|\Omega|$. The crucial caveat is the hypothesis: the outcomes must be equally likely. A loaded die (Remark 10.2), where a six is twice as likely as any other face, is a perfectly valid probability measure, yet $\mathbb{P}(\{6\})=\tfrac27\neq\tfrac16$, so the counting formula does not apply. Always check the symmetry (fair coins, balanced dice, well-shuffled cards, randomly drawn balls) before counting.