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"If you put an infinite number of monkeys at typewriters, eventually one will bash out the script for Hamlet." (One may also hypothesise a small number of monkeys and a very long period of time.) This theorem asserts nothing about the intelligence of the one random monkey that eventually comes up with the script (and note that the mob will also type out all the possible *incorrect* versions of Hamlet). It may be referred to semi-seriously when justifying a brute force method; the implication is that, with enough resources thrown at it, any technical challenge becomes a one-banana problem.
This theorem was first popularised by the astronomer Sir Arthur Eddington. It became part of the idiom through the classic short story "Inflexible Logic" by Russell Maloney, and many younger hackers know it through a reference in Douglas Adams's "Hitchhiker's Guide to the Galaxy".
See also: RFC 2795.