Kolmogorov's axioms of probability
/ kol-muh-GOR-off /
Kolmogorov's axioms are the three short rules, set down by Andrey Kolmogorov in 1933, that pin down exactly what a probability is. Rather than arguing about what chance 'really means', they say: a probability is any way of assigning a number P(A) to each event A that obeys these three rules — and then all the familiar facts follow as theorems. This move turned probability from a collection of clever tricks into a clean branch of mathematics.
The three rules are these. Non-negativity: every event gets a probability that is at least zero, P(A) >= 0, since a chance cannot be negative. Normalization: the whole sample space gets probability one, P(Omega) = 1, encoding the certainty that some outcome occurs. Countable additivity: if you have a list of events that are pairwise mutually exclusive (no two overlap), the probability that one of them happens is the sum of their individual probabilities, even when the list is infinitely long. Everything else in the subject — that probabilities lie between 0 and 1, the complement rule, inclusion-exclusion, continuity — is derived from just these three.
What is striking is how little the axioms assume and how much they deliver. They do not tell you what the actual numbers are for a particular die or coin — that comes from a model you supply, whether by symmetry, by data, or by judgment. The axioms only guarantee that whatever numbers you assign hang together consistently. They also stay silent on interpretation: a frequentist and a Bayesian both obey the same axioms while disagreeing about what the number P(A) means. The axioms are the shared rulebook; the interpretations are different ways of reading it.
A fair die: assign P({k}) = 1/6 to each face. Check the axioms — each is >= 0; they sum to 6 × (1/6) = 1 = P(Omega); and for disjoint events like {1} and {2}, P({1} or {2}) = 1/6 + 1/6 = 1/3. The model is valid because it obeys all three rules.
The axioms do not pick the numbers; they only check that whatever numbers you chose are mutually consistent.
The axioms define probability's grammar but not its meaning; what P(A) signifies in the real world is a separate question of interpretation, not settled by the axioms.