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Infinite monkey theorem

"The infinite monkey theorem" is a popular (but perhaps misleading) name for an idea from Emile Borel's[?] book on probability, published in 1909. The book introduced the concept of "dactylographic monkeys", which, when seated in front of typewriter keyboards, hit keys at random. Borel exemplified a proposition in the theory of probability called Kolmogorov's zero-one law by saying that the probability is 1 that such a monkey will eventually type every book in France's National Library. Strictly speaking, what Borel was illustrating was only a special case of Kolmogorov's zero-one law, the more general statement of which had not yet been given. Subsequent restatements by other people have replaced the National Library with the British Museum and the Library of Congress; a popular retelling says that the monkeys would eventually type Shakespeare's plays.

(The word "dactylographic" appears in the English translation of Borel's book, and seems to be an Anglicization of a French word for typewriting, but in English, "dactylography" means the study of fingerprints.)

An incorrect usage prevailing among some non-mathematicians refers to "infinite monkeys" where "infinitely many monkeys" would be correct usage. Each of the monkeys by itself is finite; it is only the number of monkeys that that usage takes to be infinite. In another sense, the latter usage is also incorrect since just one monkey suffices; there is no need for infinitely many.

In Inflexible Logic by Russell Maloney, a short story that appeared in the New Yorker in 1940, the protagonist felt that his wealth put him under an obligation to support the sciences, and so he tested that theory. (He had heard the British-Museum version.) His monkeys immediately set to work typing classics of fiction and nonfiction. The rich man was amused to see unexpurgated versions of Samuel Pepys' diaries, of which he owned only copies of bowdlerized editions.

Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died):

"If the monkey could type one keystroke every nanosecond, the expected waiting time until the monkey types out Hamlet is so long that the estimated age of the universe is insignificant by comparison ... this is not a practical method for writing plays. (We cannot resist the temptation to quote from A.N. Whitehead, 'I will not go to infinity'.)"

Popular culture references to this theorem include The Simpsons (in one episode, Montgomery Burns has his own room full of dactylographic monkeys, one of which is chastised for mistyping a word in the opening sentence of A Tale of Two Cities) and The Hitchhiker's Guide to the Galaxy (Ford Prefect and Arthur Dent, under the influence of a device that makes highly improbable events occur, are ambushed by an infinite number of monkeys who want their opinion on the monkeys' script for Hamlet).



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