Axioms of Probability
Three simple rules due to Kolmogorov make probability a consistent measure and generate every other identity.
Kolmogorov's axioms
Modern probability rests on three axioms for a probability measure P defined on events of a sample space S:
- Non-negativity: P(A) ≥ 0 for every event A.
- Normalization: P(S) = 1.
- Countable additivity: for pairwise disjoint events A1, A2, …, P(∪ Ai) = Σ P(Ai).
Consequences
Everything else follows. The complement rule P(Aᶜ) = 1 − P(A) comes from A ∪ Aᶜ = S. The empty event has P(∅) = 0. Monotonicity says if A ⊆ B then P(A) ≤ P(B), because probability cannot decrease as an event grows.
The inclusion-exclusion rule
For events that may overlap, additivity must correct for double-counting: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For three events you add the singles, subtract the pairs, and add back the triple. This pattern generalizes to any finite number of events.
Why axioms rather than intuition
Intuition about chance is unreliable, especially with rare events or conditioning. The axioms give a fixed reference against which any calculation can be checked: a result that violates additivity or normalization is simply wrong. This matters in engineering risk analysis, where the temptation to add probabilities of overlapping failure modes leads to inflated or impossible totals.
The axioms say nothing about how to assign the numbers — that is the job of a model, of data, or of a symmetry argument. They only guarantee that once assigned, the numbers behave consistently.