probability axioms

Probabilities are just numbers we attach to events to say how likely they are, but they cannot be any numbers we please. To keep the whole system consistent, the mathematician Kolmogorov laid down three simple ground rules — the axioms — that every honest assignment of probabilities must obey. Everything else in probability, from Bayes' theorem to the central limit theorem, is built on top of these three rules.

The three axioms are: (1) every probability is at least zero — you can never have a negative chance; (2) the probability of the whole sample space is exactly 1 — something on the menu is certain to happen; (3) if two events cannot both happen (they are mutually exclusive), the chance that one or the other happens is just the sum of their separate chances. From these you can derive familiar facts: P(not A) = 1 - P(A), no probability exceeds 1, and for overlapping events P(A or B) = P(A) + P(B) - P(A and B), where the last term removes the double-counted overlap.

For an actuary the axioms are the guardrails that keep a pricing model coherent. If the probabilities of all mutually exclusive outcomes do not add to 1, the model is leaking or inventing risk and the premium will be wrong. They are also why you can never quote a 110% chance of a claim, and why the probabilities of "claim" and "no claim" must always sum to one.