Editing Infinite monkey theorem (section)

===Almost surely===
{{Main|Almost surely}}
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The probability that an infinite randomly generated string of text will contain a particular finite substring is&nbsp;1. However, this does not mean the substring's absence is "impossible", despite the absence having a prior 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. (To assume otherwise implies the [[gambler's fallacy]].) However long a randomly generated finite string is, there is a small but nonzero chance that it will turn out to 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-...", or "Three, Six, Nine, Twelve…".

If the hypothetical monkey has a 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)<sup>4</sup>, 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 (including the separator key), or any other ''particular'' sequence of that length, is much lower: (1/90)<sup>100</sup>. If the monkey's allotted length of text is infinite, the chance of typing only the digit of pi is 0, which is just as ''possible'' (mathematically probable) as typing nothing but Gs (also probability 0).

The same applies to the event of typing a particular version of ''Hamlet'' followed by endless copies of itself; or ''Hamlet'' immediately followed by all the digits of pi; these specific strings are [[infinite set|equally infinite]] in length, 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. Thus, the probability of the monkey typing an endlessly long string, such as all of the digits of pi in order, on a 90-key keyboard is (1/90)<sup>∞</sup> which equals (1/∞) which is essentially 0. 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.