Kolmogorov's zero-one law

In probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, treats of probabilities of certain "tail events" defined in terms of infinite sequences of random variables. Suppose

is an infinite sequence of independent random variables (not necessarily identically distributed). A tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subsequence of these random variables. For example, the event that the series

converges, is a tail event. In an infinite sequence of coin-tosses, the probability that a sequence of 100 consecutive heads eventually occurs, is a tail event.

Kolmogorov's zero-one law states that the probability of any tail event is either zero or one.

In a book published in 1909, Émile Borel stated that if a dactylographic monkey hits typewriter keys randomly forever, it will eventually type every book in France's National Library. That is a special case of this zero-one law.