History[ edit ] The double quotation mark is older than the single. It derives from a marginal notation used in fifteenth-century manuscript annotations to indicate a passage of particular importance not necessarily a quotation ; the notation was placed in the outside margin of the page and was repeated alongside each line of the passage. By the middle sixteenth century, printers notably in BaselSwitzerland had developed a typographic form of this notation, resembling the modern double quotation mark pointing to the right.

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.

If there were as many monkeys as there are atoms in the observable universe typing extremely fast for trillions of times the life of the universe, the probability of the monkeys replicating even a single page of Shakespeare is unfathomably small. 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.

In the case of the entire text of Hamlet, the probabilities are so vanishingly small as to be inconceivable.

The text of Hamlet contains approximatelyletters. The average number of letters that needs to be typed until the text appears is also 3. To put it another way, for a one in a trillion chance of success, there would need to beuniverses made of atomic monkeys.

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 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.

The probability that randomly typed keys will consist of the first 99 digits of pi including the separator keyor any other particular sequence of that length, is much lower: If the monkey's allotted length of text is infinite, the chance of typing only the digits 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 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. At the same time, the probability that the sequence contains a particular subsequence such as the word MONKEY, or the 12th through th 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. Correspondence between strings and numbers[ edit ] In a simplification of the thought experiment, the monkey could have a typewriter with just two keys: The infinitely long string thusly produced would correspond to the binary digits of a particular real number between 0 and 1.

A countably infinite set of possible strings end in infinite repetitions, which means the corresponding real number is rational. Examples include the strings corresponding to one-third …five-sixths … and five-eighths ….

Only a subset of such real number strings albeit a countably infinite subset contains the entirety of Hamlet assuming that the text is subjected to a numerical encoding, such as ASCII.

Meanwhile, there is an uncountably infinite set of strings which do not end in such repetition; these correspond to the irrational numbers. These can be sorted into two uncountably infinite subsets: However, the "largest" subset of all the real numbers are those which not only contain Hamlet, but which contain every other possible string of any length, and with equal distribution of such strings.

These irrational numbers are called normal.

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Because almost all numbers are normal, almost all possible strings contain all possible finite substrings. Hence, the probability of the monkey typing a normal number is 1.Providing study notes, tips, and practice questions for students preparing for their O level or upper secondary examinations.

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