Infinite monkey theorem

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. In fact, the monkey would almost surely type every possible finite text an infinite number of times. The theorem can be generalized to state that any sequence of events which has a non-zero probability of happening will almost certainly eventually occur, given unlimited time. 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. Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. 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. 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. In the early 20th century, Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics. There is a straightforward proof of this theorem. 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. For example, if the chance of rain in Moscow on a particular day in the future is 0.4 and the chance of an earthquake in San Francisco on any particular day is 0.00003, then the chance of both happening on the same day is 0.4 × 0.00003 = 0.000012, assuming that they are indeed independent.