As an introduction, recall that if two events are statistically independent, then the probability of both happening equals the product of the probabilities of each one happening independently.

There is a straightforward proof of this theorem. In the early 20th century, Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics. Jorge Luis Borges traced the history of this idea from Aristotle's On Generation and Corruption and Cicero's De Natura Deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, up to modern statements with their iconic simians and typewriters. One of the earliest instances of the use of the 'monkey metaphor' is that of French mathematician Émile Borel in 1913, but the first instance may have been even earlier. Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. In this context, 'almost surely' is a mathematical term meaning the event happens with probability 1, and the 'monkey' is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. The theorem can be generalized to state that any sequence of events that has a non-zero probability of happening will almost certainly occur an infinite number of times, given an infinite amount of time or a Universe that is infinite in size. In fact, the monkey would almost surely type every possible finite text an infinite number of times. The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare.

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