countable additivity

Countable additivity is the third and most powerful of Kolmogorov's axioms. It says: if you have a list of events, possibly infinitely many, that are pairwise mutually exclusive — meaning no two of them can both happen — then the probability that some event in the list occurs is just the sum of all their individual probabilities. For two disjoint events this is the familiar P(A or B) = P(A) + P(B); the axiom simply insists the same adding-up works for an endless list, A1, A2, A3, and so on.

Why demand it for infinite lists and not just finite ones? Because so many natural questions are infinite by nature. 'How many coin tosses until the first head?' has answers 1, 2, 3, ... with no upper bound; the events 'first head on toss n' are disjoint and infinitely numerous, and to get a total probability of 1 you must be allowed to add up infinitely many terms. Finite additivity alone is too weak to handle these limits cleanly; countable additivity is exactly the strength needed to make sums over countable infinities behave.

Two honest boundaries. First, the events must be genuinely disjoint — if they overlap, you cannot simply add, and you need inclusion-exclusion instead. Second, the word is countable, not arbitrary: the axiom covers lists you can index by the whole numbers (finite or countably infinite), but not uncountable collections like 'all the points in an interval'. That restriction is not a flaw; it is precisely what keeps probability consistent, and it is the seed from which the continuity of probability and, eventually, measure theory grow.