Kolmogorov's zeroone law
In
probability theory,
Kolmogorov's zeroone 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 cointosses, the probability that a sequence of 100 consecutive heads
eventually occurs, is a tail event.
Kolmogorov's zeroone 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 zeroone law.