The probability that infinitely many of the E k occur is 1. Because each block is typed independently, the chance X n of not typing banana in any of the first n blocks of 6 letters is Less than one in 15 billion, but not zero.įrom the above, the chance of not typing banana in a given block of 6 letters is 1 − (1/50) 6. Therefore, the probability of the first six letters spelling banana is The chance that the first letter typed is 'b' is 1/50, and the chance that the second letter typed is 'a' is also 1/50, and so on. Suppose that the keys are pressed randomly and independently, meaning that each key has an equal chance of being pressed regardless of what keys had been pressed previously. 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.Ĭonsider the probability of typing the word banana on a typewriter with 50 keys.
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. 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.
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 which has a non-zero probability of happening will almost certainly eventually occur, given enough time. However, the probability that monkeys filling the entire observable universe would type a single complete work, such as Shakespeare's Hamlet, is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero).
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, such as the complete works of William Shakespeare.
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