Kolmogorov's zero-one law

Kolmogorov's zero-one law states that, for a large set of independent random events (like coin flips), certain long-term outcomes are almost certain to happen or almost impossible—they have probabilities of exactly 0 or 1. These events are called "tail events," meaning they depend on what happens far in the future, beyond any fixed point. For example, whether the overall sequence of coin flips eventually balances out to equal heads and tails. The law tells us that, in infinite independent sequences, these tail events are not just likely—they are almost sure or almost impossible, with no in-between.