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 a particular chosen text, such as the complete works of William Shakespeare.
In this context, "almost surely" is a
mathematical term with a precise meaning, and the "monkey" is not an actual
monkey; rather, it is a metaphor for an abstract device that produces a random sequence of
letters ad infinitum. The theorem
illustrates the perils of reasoning about infinity by imagining a vast but
finite number, and vice versa. The probability of a monkey
typing a given string of text as long as, say, Hamlet, is so tiny that, were the
experiment conducted, the chance of it actually occurring during a span of time
of the order of the age of the universe
is minuscule but not zero.

There is a straightforward proof of this theorem. If two events are statistically
independent, (i.e. neither affects the outcome of the other), then the probability of both
happening equals the product of the probabilities of each one happening
independently. E.g. if the chance of rain in Sydney on a particular day is 0.3
and the chance of an earthquake in San Francisco on that day is 0.008, then the
chance of both happening on that same day is 0.3 × 0.008 = 0.0024.
Suppose
the typewriter has 50 keys, and
the word to be typed is "banana". Typing at random, 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, because events are independent. So
the chance of the first six letters matching banana is
(1/50) × (1/50) ×
(1/50) × (1/50) × (1/50) × (1/50) = (1/50)6.
For the same reason, the chance
that the next 6 letters match banana is also (1/50)6, and so on.
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
As n grows,
Xn gets smaller. For an n of a million, Xn is 99.99%, but for an n of 10 billion
Xn is 53% and for an n of 100 billion it is 0.17%. As n approaches infinity, the
probability Xn approaches zero;
that is, by making n large enough, Xn can be made as small as one likes.[1][2]
The
same argument shows why at least one of infinitely many monkeys will (almost
surely) 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 to infinity the value of Xn — the probability of the monkeys failing
to reproduce the given text — decreases to zero. This is equivalent to stating
that the probability that one or more of an infinite number of monkeys will
produce a given text on the first try is 100%, or that it is almost certain they
will do so.

A monkey is any member of either the New World monkeys or Old World monkeys, two
of the three groupings of simian primates, the third group being
the apes. There are 264
known extant species of monkey.
The New World monkeys are classified within
the parvorder Platyrrhini, whereas the Old
World monkeys (superfamily Cercopithecoidea) form
part of the parvorder Catarrhini, which also
includes the apes. Thus, scientifically speaking, monkeys are paraphyletic (not a single
coherent group), and Old World monkeys are actually more closely related to the
apes than they are to the New World monkeys.
Because of their similarity to
monkeys, apes such as chimpanzees and gibbons are often
called "monkeys" in informal usage, though they are not monkeys. Conversely, due
to its size (up to 1 m) the Mandrill is often thought to be
an ape, but it is actually an Old World monkey. Also, a few monkey species have
the word "ape" in their common name.