A probability measure is nothing more than a measure with one extra requirement: the whole space has measure one. Write it \mathbb{P}, so that \mathbb{P}(\Omega) = 1. The complete object,

is a probability space: \Omega the sample space of outcomes, \mathcal{F} the σ-algebra of events, and \mathbb{P} the probability assigned to each event. This triple is the stage on which every random phenomenon in this course plays out.

Kolmogorov's axioms

Stripped to essentials, a probability measure obeys three axioms — the rest of probability is built entirely from them:

• Non-negativity: \mathbb{P}(A) \ge 0 for every event A.
• Normalisation: \mathbb{P}(\Omega) = 1.
• Countable additivity: for pairwise-disjoint A_1, A_2, \dots, \mathbb{P}\!\left(\bigcup_{i} A_i\right) = \sum_{i} \mathbb{P}(A_i).

From these alone come the everyday rules: the complement rule \mathbb{P}(A^{c}) = 1 - \mathbb{P}(A), the fact that \mathbb{P}(\emptyset) = 0, and the inclusion–exclusion formula

The fair die, end to end

Take \Omega = \{1,2,3,4,5,6\}, \mathcal{F} = 2^{\Omega} the power set, and \mathbb{P}(A) = |A| / 6 — each outcome equally likely. Pick an event below; its faces light up and its probability shows as a fraction. Every choice obeys 0 \le \mathbb{P}(A) \le 1.