its to hard to explain by myself so ill google it and put it here
this is wikipedias link
http://en.wikipedia.org/wiki/Infinite_monkey_theorem
and here are probably the more important parts
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 given 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, but a metaphor for an abstract device that produces a random sequence of letters and symbols ad infinitum. The probability of a monkey exactly typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time even a hundred thousand orders of magnitude longer than the age of the universe is extremely low, but not actually zero.
Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum, through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.
There is a straightforward proof of this theorem. 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 Montreal 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. If we assume that the keys are pressed randomly (i.e., with equal probability) and independently, 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, because events are independent. Therefore, 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 = 1/15 625 000 000 ,
less than one in 15 billion. 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
X_n=\left(1-\frac{1}{50^6}\right)^n.
As n grows, Xn gets smaller. For an n of a million, Xn is roughly 0.9999, but for an n of 10 billion Xn is roughly 0.53 and for an n of 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,[1][note 1] 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.
However, for physically meaningful numbers of monkeys typing for physically meaningful lengths of time the results are reversed. If there are as many monkeys as there are particles in the observable universe (1080), and each types 1,000 keystrokes per second for 100 times the life of the universe (1020 seconds), the probability of the monkeys replicating even a short book is nearly zero. See Probabilities, below.
[edit] Infinite strings
The two statements above can be stated more generally and compactly in terms of strings, which are sequences of characters chosen from some finite alphabet:
Given an infinite string where each character is chosen uniformly at random, any given finite string almost surely occurs as a substring at some position.
Given an infinite sequence of infinite strings, where each character of each string is chosen uniformly at random, any given finite string almost surely occurs as a prefix of one of these strings.
Both follow easily from the second Borel–Cantelli lemma. For the second theorem, let Ek be the event that the kth string begins with the given text. Because this has some fixed nonzero probability p of occurring, the Ek are independent, and the below sum diverges,
\sum_{k=1}^\infty P(E_k) = \sum_{k=1}^\infty p = \infty,
the probability that infinitely many of the Ek occur is 1. The first theorem is shown similarly; one can divide the random string into nonoverlapping blocks matching the size of the desired text, and make Ek the event where the kth block equals the desired string.[note 2]
[edit] Probabilities
Ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of Hamlet. It has a chance of one in 676 (26 × 26) of typing the first two letters. Because the probability shrinks exponentially, at 20 letters it already has only a chance of one in 2620 = 19,928,148,895,209,409,152,340,197,376 (almost 2 × 1028). In the case of the entire text of Hamlet, the probabilities are so vanishingly small they can barely be conceived in human terms. The text of Hamlet contains approximately 130,000 letters.[note 3] Thus there is a probability of one in 3.4 × 10183,946 to get the text right at the first trial. The average number of letters that needs to be typed until the text appears is also 3.4 × 10183,946[note 4], or including punctuation, 4.4 × 10360,783.[note 5]
Even if the observable universe were filled with monkeys the size of atoms typing from now until the heat death of the universe, their total probability to produce a single instance of Hamlet would still be a great many orders of magnitude less than one in 10183,800. As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…", and the statement that the monkeys must eventually succeed "gives a misleading conclusion about very, very large numbers." This is from their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys.[2]
[edit] Almost surely
See also: Almost surely
The probability that an infinite randomly-generated string of text will contain a particular finite substring is 1, but this does not mean the substring's absence is "impossible", despite such an event having a probability of 0. For example, the immortal monkey could randomly type G as its first letter, G as its second, and G as every single letter thereafter, producing an infinite string of Gs; at no point must the monkey be "compelled" to type anything else.
There is a small but nonzero chance that a randomly generated string will consist of the same character repeated throughout; this chance approaches zero as the string's length approaches infinity. There is nothing special about such a monotonous sequence except that it is easy to describe; the same fact applies to any nameable specific sequence, such as "RGRGRG" repeated forever, or "a, b, aa, bb, aaa, bbb…".
If the hypothetical monkey has typewriter with 90 equally-likely keys that include numerals and punctuation, then the first typed keys might be "3.14" (the first three digits of pi) with a probability of 1/90^4, which is 1/65,610,000. Equally probable is any other string of four characters allowed by the typewriter, such as "GGGG", "mATh", or "q%8e". The probability that 100 randomly typed keys will consist of the first 99 digits of pi, or any other particular sequence, is much lower: 1/90^100. If the monkey's allotted length of text is infinite, the chance of typing the entirety of the digits of pi is 0, which is just as possible as typing nothing but Gs (also probability 0).
The same applies to the event of typing a particular version of Hamlet followed by copies of itself ad infinitum; or Hamlet immediately followed by all the digits of pi; these specific strings are equally infinite, they are not prohibited by the terms of the thought problem, and they each have a prior probability of 0. In fact, any particular infinite sequence the immortal monkey types will have had a prior probability of 0, even though the monkey must type something.
This is an extension of the principle that a finite string of random text has a lower and lower probability of being a particular string the longer it is (though all specific strings are equally unlikely). This probability approaches 0 as the string approaches infinity. At the same time, the probability that the sequence contains a particular subsequence (such as the word MONKEY, or the 12th through 999th digits of pi, or a version of the King James Bible) increases as the total string increases. This probability approaches 1 as the total string approaches infinity, and thus the original theorem is correct.
cant really explain it any better but here is the show i was watching
http://topdocumentaryfilms.com/to-infinity-and-beyond/
