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Infinite monkey theorem
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Chimpanzee seated at a typewriter
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. In fact, the monkey would almost surely type every possible finite text an infinite number of times. 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). The theorem can be generalized to state that any sequence of events which has a non-zero probability of happening, at least as long as it hasn't occurred, will almost certainly eventually occur.

In this context, "almost surely" is a mathematical term with a precise meaning, 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. One of the earliest instances of the use of the "monkey metaphor" is that of French mathematician Émile Borel in 1913,[1] 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. 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.


Contents
1 Solution
1.1 Direct proof
1.1.1 Infinite strings
1.2 Probabilities
1.3 Almost surely
1.4 Correspondence between strings and numbers
2 History
2.1 Statistical mechanics
2.2 Origins and "The Total Library"
3 Actual monkeys
4 Applications and criticisms
4.1 Evolution
4.2 Literary theory
4.3 Random document generation
4.4 Testing of random-number generators
5 In popular culture
6 See also
7 Notes
8 References
9 External links
Solution
Direct proof
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.

Suppose the typewriter has 50 keys, and the word to be typed is banana. If the keys are pressed randomly and independently, it means that each key has an equal chance of being pressed. Then, 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. Therefore, the chance of the first six letters spelling banana is

(1/50) × (1/50) × (1/50) × (1/50) × (1/50) × (1/50) = (1/50)6 = 1/15,625,000,000.
Less than one in 15 billion, but not zero.

From the above, the chance of not typing banana in a given block of 6 letters is 1 − (1/50)6. Because each block is typed independently, the chance Xn of not typing banana in any of the first n blocks of 6 letters is

{\displaystyle X_{n}=\left(1-{\frac {1}{50^{6}}}\right)^{n}.}X_n=\left(1-\frac{1}{50^6}\right)^n.
As n grows, Xn gets smaller. For n = 1 million, Xn is roughly 0.9999, but for n = 10 billion Xn is roughly 0.53 and for n = 100 billion it is roughly 0.0017. As n approaches infinity, the probability Xn approaches zero; that is, by making n large enough, Xn can be made as small as is desired,[2][a] and the chance of typing banana approaches 100%.

The same argument shows why at least one of infinitely many monkeys will produce a text as quickly as it would be produced by a perfectly accurate human typist copying it from the original. In this case Xn = (1 − (1/50)6)n where Xn represents the probability that none of the first n monkeys types banana correctly on their first try. When we consider 100 billion monkeys, the probability falls to 0.17%, and as the number of monkeys n increases, the value of Xn – the probability of the monkeys failing to reproduce the given text – approaches zero arbitrarily closely. The limit, for n going to infinity, is zero. So the probability of the word banana appearing at some point in an infinite sequence of keystrokes is equal to one.

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