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Infinite monkey theorem
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Given an infinite length of time, a chimpanzee punching at random on a typewriter would almost surely type out all of Shakespeare's plays.
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 fact, the monkey would almost surely type every possible finite text an infinite number of times. However, the probability that monkeys filling the observable universe would type a 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).
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
[hide]
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 Real 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[edit]
Direct proof[edit]
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, hence a possible outcome.
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 = ( 1 − 1 50 6 ) n . {\displaystyle 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,[2][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. So the probability of the word banana appearing at some point in an infinite sequence of keystrokes is equal to one.
Infinite strings[edit]
This 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,
∑ k = 1 ∞ P ( E k ) = ∑ k = 1 ∞ p = ∞ , {\displaystyle \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]
Probabilities[edit]
However, for physically meaningful numbers of monkeys typing for physically meaningful lengths of time the results are reversed. 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. 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 as to be inconceivable. 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 every proton in the observable universe were a monkey with a typewriter, typing from the Big Bang until the end of the universe (when protons might no longer exist), they would still need a still far greater amount of time – more than three hundred and sixty thousand orders of magnitude longer – to have even a 1 in 10500 chance of success. To put it another way, for a one in a trillion chance of success, there would need to be 10360,641 universes made of atomic monkeys.[note 6] As Kittel and Kroemer put it in their textbook on thermodynamics, the field whose statistical foundations motivated the first known expositions of typing monkeys,[3] "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."
In fact there is less than a one in a trillion chance of success that such a universe made of monkeys could type any particular document a mere 79 characters long.[note 7]
Almost surely[edit]
Main article: Almost surely
The probability that an infinite randomly generated string of text will contain a particular finite substring is 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)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 (including the separator key), or any other particular sequence of that length, is much lower: (1/90)100. 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. 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)∞ 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.
Correspondence between strings and numbers[edit]
In a simplification of the thought experiment, the monkey could have a typewriter with just two keys: 1 and 0. 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 (010101…), five-sixths (11010101…) and five-eighths (1010000…). 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: those which contain Hamlet and those which do not. 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. 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. The same principles apply regardless of the number of keys from which the monkey can choose; a 90-key keyboard can be seen as a generator of numbers written in base 90.
History[edit]
Statistical mechanics[edit]
In one of the forms in which probabilists now know this theorem, with its "dactylographic" [i.e., typewriting] monkeys (French: singes dactylographes; the French word singe covers both the monkeys and the apes), appeared in Émile Borel's 1913 article "Mécanique Statistique et Irréversibilité" (Statistical mechanics and irreversibility),[1] and in his book "Le Hasard" in 1914. His "monkeys" are not actual monkeys; rather, they are a metaphor for an imaginary way to produce a large, random sequence of letters. Borel said that if a million monkeys typed ten hours a day, it was extremely unlikely that their output would exactly equal all the books of the richest libraries of the world; and yet, in comparison, it was even more unlikely that the laws of statistical mechanics would ever be violated, even briefly.
The physicist Arthur Eddington drew on Borel's image further in The Nature of the Physical World (1928), writing:
If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel.[4][5]
These images invite the reader to consider the incredible improbability of a large but finite number of monkeys working for a large but finite amount of time producing a significant work, and compare this with the even greater improbability of certain physical events. Any physical process that is even less likely than such monkeys' success is effectively impossible, and it may safely be said that such a process will never happen.[3] It is clear from the context that Eddington is not suggesting that the probability of this happening is worthy of serious consideration. On the contrary, it was a rhetorical illustration of the fact that below certain levels of probability, the term improbable is functionally equivalent to impossible.
Origins and "The Total Library"[edit]
In a 1939 essay entitled "The Total Library", Argentine writer Jorge Luis Borges traced the infinite-monkey concept back to Aristotle's Metaphysics. Explaining the views of Leucippus, who held that the world arose through the random combination of atoms, Aristotle notes that the atoms themselves are homogeneous and their possible arrangements only differ in shape, position and ordering. In On Generation and Corruption, the Greek philosopher compares this to the way that a tragedy and a comedy consist of the same "atoms", i.e., alphabetic characters.[6] Three centuries later, Cicero's De natura deorum (On the Nature of the Gods) argued against the atomist worldview:
He who believes this may as well believe that if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the Annals of Ennius. I doubt whether fortune could make a single verse of them.[7]
Borges follows the history of this argument through Blaise Pascal and Jonathan Swift,[8] then observes that in his own time, the vocabulary had changed. By 1939, the idiom was "that a half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum." (To which Borges adds, "Strictly speaking, one immortal monkey would suffice.") Borges then imagines the contents of the Total Library which this enterprise would produce if carried to its fullest extreme:
Everything would be in its blind volumes. Everything: the detailed history of the future, Aeschylus' The Egyptians, the exact number of times that the waters of the Ganges have reflected the flight of a falcon, the secret and true nature of Rome, the encyclopedia Novalis would have constructed, my dreams and half-dreams at dawn on August 14, 1934, the proof of Pierre Fermat's theorem, the unwritten chapters of Edwin Drood, those same chapters translated into the language spoken by the Garamantes, the paradoxes Berkeley invented concerning Time but didn't publish, Urizen's books of iron, the premature epiphanies of Stephen Dedalus, which would be meaningless before a cycle of a thousand years, the Gnostic Gospel of Basilides, the song the sirens sang, the complete catalog of the Library, the proof of the inaccuracy of that catalog. Everything: but for every sensible line or accurate fact there would be millions of meaningless cacophonies, verbal farragoes, and babblings. Everything: but all the generations of mankind could pass before the dizzying shelves—shelves that obliterate the day and on which chaos lies—ever reward them with a tolerable page.[9]
Borges' total library concept was the main theme of his widely read 1941 short story "The Library of Babel", which describes an unimaginably vast library consisting of interlocking hexagonal chambers, together containing every possible volume that could be composed from the letters of the alphabet and some punctuation characters.
Real monkeys[edit]
In 2003, lecturers and students from the University of Plymouth MediaLab Arts course used a £2,000 grant from the Arts Council to study the literary output of real monkeys. They left a computer keyboard in the enclosure of six Celebes crested macaques in Paignton Zoo in Devon in England for a month, with a radio link to broadcast the results on a website.[10]
Not only did the monkeys produce nothing but five total pages[11] largely consisting of the letter S, the lead male began by bashing the keyboard with a stone, and the monkeys continued by urinating and defecating on it. Mike Phillips, director of the university's Institute of Digital Arts and Technology (i-DAT), said that the artist-funded project was primarily performance art, and they had learned "an awful lot" from it. He concluded that monkeys "are not random generators. They're more complex than that. ... They were quite interested in the screen, and they saw that when they typed a letter, something happened. There was a level of intention there."[10][12]
Applications and criticisms[edit]
Evolution[edit]
Thomas Huxley is sometimes misattributed with proposing a variant of the theory in his debates with Samuel Wilberforce.
In his 1931 book The Mysterious Universe, Eddington's rival James Jeans attributed the monkey parable to a "Huxley", presumably meaning Thomas Henry Huxley. This attribution is incorrect.[13] Today, it is sometimes further reported that Huxley applied the example in a now-legendary debate over Charles Darwin's On the Origin of Species with the Anglican Bishop of Oxford, Samuel Wilberforce, held at a meeting of the British Association for the Advancement of Science at Oxford on 30 June 1860. This story suffers not only from a lack of evidence, but the fact that in 1860 the typewriter itself had yet to emerge.[14]
Despite the original mix-up, monkey-and-typewriter arguments are now common in arguments over evolution. For example, Doug Powell argues as a Christian apologist that even if a monkey accidentally types the letters of Hamlet, it has failed to produce Hamlet because it lacked the intention to communicate. His parallel implication is that natural laws could not produce the information content in DNA.[15] A more common argument is represented by Reverend John F. MacArthur, who claims that the genetic mutations necessary to produce a tapeworm from an amoeba are as unlikely as a monkey typing Hamlet's soliloquy, and hence the odds against the evolution of all life are impossible to overcome.[16]
Evolutionary biologist Richard Dawkins employs the typing monkey concept in his book The Blind Watchmaker to demonstrate the ability of natural selection to produce biological complexity out of random mutations. In a simulation experiment Dawkins has his weasel program produce the Hamlet phrase METHINKS IT IS LIKE A WEASEL, starting from a randomly typed parent, by "breeding" subsequent generations and always choosing the closest match from progeny that are copies of the parent, with random mutations. The chance of the target phrase appearing in a single step is extremely small, yet Dawkins showed that it could be produced rapidly (in about 40 generations) using cumulative selection of phrases. The random choices furnish raw material, while cumulative selection imparts information. As Dawkins acknowledges, however, the weasel program is an imperfect analogy for evolution, as "offspring" phrases were selected "according to the criterion of resemblance to a distant ideal target." In contrast, Dawkins affirms, evolution has no long-term plans and does not progress toward some distant goal (such as humans). The weasel program is instead meant to illustrate the difference between non-random cumulative selection, and random single-step selection.[17] In terms of the typing monkey analogy, this means that Romeo and Juliet could be produced relatively quickly if placed under the constraints of a nonrandom, Darwinian-type selection because the fitness function will tend to preserve in place any letters that happen to match the target text, improving each successive generation of typing monkeys.
A different avenue for exploring the analogy between evolution and an unconstrained monkey lies in the problem that the monkey types only one letter at a time, independently of the other letters. Hugh Petrie argues that a more sophisticated setup is required, in his case not for biological evolution but the evolution of ideas:
In order to get the proper analogy, we would have to equip the monkey with a more complex typewriter. It would have to include whole Elizabethan sentences and thoughts. It would have to include Elizabethan beliefs about human action patterns and the causes, Elizabethan morality and science, and linguistic patterns for expressing these. It would probably even have to include an account of the sorts of experiences which shaped Shakespeare's belief structure as a particular example of an Elizabethan. Then, perhaps, we might allow the monkey to play with such a typewriter and produce variants, but the impossibility of obtaining a Shakespearean play is no longer obvious. What is varied really does encapsulate a great deal of already-achieved knowledge.[18]
James W. Valentine, while admitting that the classic monkey's task is impossible, finds that there is a worthwhile analogy between written English and the metazoan genome in this other sense: both have "combinatorial, hierarchical structures" that greatly constrain the immense number of combinations at the alphabet level.[19]
Literary theory[edit]
R. G. Collingwood argued in 1938 that art cannot be produced by accident, and wrote as a sarcastic aside to his critics,
...some ... have denied this proposition, pointing out that if a monkey played with a typewriter ... he would produce ... the complete text of Shakespeare. Any reader who has nothing to do can amuse himself by calculating how long it would take for the probability to be worth betting on. But the interest of the suggestion lies in the revelation of the mental state of a person who can identify the 'works' of Shakespeare with the series of letters printed on the pages of a book...[20]
Nelson Goodman took the contrary position, illustrating his point along with Catherine Elgin by the example of Borges' "Pierre Menard, Author of the Quixote",
What Menard wrote is simply another inscription of the text. Any of us can do the same, as can printing presses and photocopiers. Indeed, we are told, if infinitely many monkeys ... one would eventually produce a replica of the text. That replica, we maintain, would be as much an instance of the work, Don Quixote, as Cervantes' manuscript, Menard's manuscript, and each copy of the book that ever has been or will be printed.[21]
In another writing, Goodman elaborates, "That the monkey may be supposed to have produced his copy randomly makes no difference. It is the same text, and it is open to all the same interpretations...." Gérard Genette dismisses Goodman's argument as begging the question.[22]
For Jorge J. E. Gracia, the question of the identity of texts leads to a different question, that of author. If a monkey is capable of typing Hamlet, despite having no intention of meaning and therefore disqualifying itself as an author, then it appears that texts do not require authors. Possible solutions include saying that whoever finds the text and identifies it as Hamlet is the author; or that Shakespeare is the author, the monkey his agent, and the finder merely a user of the text. These solutions have their own difficulties, in that the text appears to have a meaning separate from the other agents: what if the monkey operates before Shakespeare is born, or if Shakespeare is never born, or if no one ever finds the monkey's typescript?[23]
Random document generation[edit]
The theorem concerns a thought experiment which cannot be fully carried out in practice, since it is predicted to require prohibitive amounts of time and resources. Nonetheless, it has inspired efforts in finite random text generation.
One computer program run by Dan Oliver of Scottsdale, Arizona, according to an article in The New Yorker, came up with a result on August 4, 2004: After the group had worked for 42,162,500,000 billion billion monkey-years, one of the "monkeys" typed, "VALENTINE. Cease toIdor:eFLP0FRjWK78aXzVOwm)-‘;8.t" The first 19 letters of this sequence can be found in "The Two Gentlemen of Verona". Other teams have reproduced 18 characters from "Timon of Athens", 17 from "Troilus and Cressida", and 16 from "Richard II".[24]
A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulated a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:
RUMOUR. Open your ears; 9r"5j5&?OWTY Z0d...
Due to processing power limitations, the program used a probabilistic model (by using a random number generator or RNG) instead of actually generating random text and comparing it to Shakespeare. When the simulator "detected a match" (that is, the RNG generated a certain value or a value within a certain range), the simulator simulated the match by generating matched text.[25]
More sophisticated methods are used in practice for natural language generation. If instead of simply generating random characters one restricts the generator to a meaningful vocabulary and conservatively following grammar rules, like using a context-free grammar, then a random document generated this way can even fool some humans (at least on a cursory reading) as shown in the experiments with SCIgen, snarXiv, and the Postmodernism Generator.
Testing of random-number generators[edit]
Main article: Diehard tests
Questions about the statistics describing how often an ideal monkey is expected to type certain strings translate into practical tests for random-number generators; these range from the simple to the "quite sophisticated". Computer-science professors George Marsaglia and Arif Zaman report that they used to call one such category of tests "overlapping m-tuple tests" in lectures, since they concern overlapping m-tuples of successive elements in a random sequence. But they found that calling them "monkey tests" helped to motivate the idea with students. They published a report on the class of tests and their results for various RNGs in 1993.[26]
In popular culture[edit]
Main article: Infinite monkey theorem in popular culture
The infinite monkey theorem and its associated imagery is considered a popular and proverbial illustration of the mathematics of probability, widely known to the general public because of its transmission through popular culture rather than through formal education.[note 8] This is helped by the innate humor stemming from the image of literal monkeys rattling away on a set of typewriters, and is a popular visual gag.
In The Simpsons episode "Last Exit to Springfield", Mr. Burns shows Homer "a room with a thousand monkeys on a thousand typewriters. Soon they will have written the greatest novel known to man!"
In his 1978 radio play, The Hitchhiker's Guide to the Galaxy, Douglas Adams invoked the theorem to illustrate the power of the 'Infinite Improbability Drive' that powered a spaceship. From Episode 2: "Ford, there's an infinite number of monkeys outside who want to talk to us about this script for Hamlet they've worked out."
A quotation attributed [27] to a 1996 speech by Robert Wilensky stated, "We've heard that a million monkeys at a million keyboards could produce the complete works of Shakespeare; now, thanks to the Internet, we know that is not true."
The enduring, widespread popularity of the theorem was noted in the introduction to a 2001 paper, "Monkeys, Typewriters and Networks: The Internet in the Light of the Theory of Accidental Excellence".[28] In 2002, an article in The Washington Post said, "Plenty of people have had fun with the famous notion that an infinite number of monkeys with an infinite number of typewriters and an infinite amount of time could eventually write the works of Shakespeare".[29] In 2003, the previously mentioned Arts Council funded experiment involving real monkeys and a computer keyboard received widespread press coverage.[30] In 2007, the theorem was listed by Wired magazine in a list of eight classic thought experiments.[31]
See also[edit]
Normal number
Hilbert's paradox of the Grand Hotel, another thought experiment involving infinity
Law of truly large numbers
Murphy's Law
The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos, explains the multiverse in which every possible event will occur infinitely many times
The Library of Babel
The Engine
Boltzmann brain
The Infinite Monkey Cage
Notes[edit]
Jump up ^ This shows that the probability of typing "banana" in one of the predefined non-overlapping blocks of six letters tends to 1. In addition the word may appear across two blocks, so the estimate given is conservative.
Jump up ^ The first theorem is proven by a similar if more indirect route in Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 97–100. ISBN 0-387-22833-0.
Jump up ^ Using the Hamlet text from gutenberg, there are 132680 alphabetical letters and 199749 characters overall
Jump up ^ For any required string of 130,000 letters from the set a-z, the average number of letters that needs to be typed until the string appears is (rounded) 3.4 × 10183,946, except in the case that all letters of the required string are equal, in which case the value is about 4% more, 3.6 × 10183,946. In that case failure to have the correct string starting from a particular position reduces with about 4% the probability of a correct string starting from the next position (i.e., for overlapping positions the events of having the correct string are not independent; in this case there is a positive correlation between the two successes, so the chance of success after a failure is smaller than the chance of success in general). The figure 3.4 × 10183,946 is derived from n = 26130000 by taking the logarithm of both sides: log10(n) = 1300000×log10(26) = 183946.5352, therefore n = 100.5352 × 10183946 = 3.429 × 10183946.
Jump up ^ 26 letters ×2 for capitalisation, 12 for punctuation characters = 64, 199749×log10(64) = 4.4 × 10360,783 (this is generous as it assumes capital letters are separate keys, as opposed to a key combination, which makes the problem vastly harder).
Jump up ^ There are ~1080 protons in the observable universe. Assume the monkeys write for 1038 years (1020 years is when all stellar remnants will have either been ejected from their galaxies or fallen into black holes, 1038 years is when all but 0.1% of protons have decayed). Assuming the monkeys type non-stop at a ridiculous 400 words per minute (the world record is 216wpm for a single minute), that's about 2000 characters per minute (Shakespeare's average word length is a bit under 5 letters). There are about half a million minutes in a year, this means each monkey types half a billion characters per year. This gives a total of 1080×1038×109=10127 letters typed – which is still zero in comparison to 10360,783. For a one in a trillion chance, multiply the letters typed by a trillion: 10127×1015=10145. 10360,783/10145=10360,641.
Jump up ^ As explained at "Archived copy". Archived from the original on 2009-10-16. Retrieved 2013-12-04. , the problem can be approximated further. 10145/log10(64)=78.9 characters.
Jump up ^ Examples of the theorem being referred to as proverbial include: Why Creativity Is Not like the Proverbial Typing Monkey. Jonathan W. Schooler, Sonya Dougal, Psychological Inquiry, Vol. 10, No. 4 (1999); and The Case of the Midwife Toad (Arthur Koestler, New York, 1972, page 30): "Neo-Darwinism does indeed carry the nineteenth-century brand of materialism to its extreme limits—to the proverbial monkey at the typewriter, hitting by pure chance on the proper keys to produce a Shakespeare sonnet." The latter is sourced from Parable of the Monkeys, a collection of historical references to the theorem in various formats.
References[edit]
^ Jump up to: a b Émile Borel (1913). "Mécanique Statistique et Irréversibilité". J. Phys. 5e série. 3: 189–196.
Jump up ^ Isaac, Richard E. (1995). The Pleasures of Probability. Springer. pp. 48–50. ISBN 0-387-94415-X. Isaac generalizes this argument immediately to variable text and alphabet size; the common main conclusion is on p.50.
^ Jump up to: a b Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 53. ISBN 0-7167-1088-9.
Jump up ^ Arthur Eddington (1928). The Nature of the Physical World: The Gifford Lectures. New York: Macmillan. p. 72. ISBN 0-8414-3885-4.
Jump up ^ Eddington, Arthur. "Chapter IV: The Running-Down of the Universe". The Nature of the Physical World 1926–1927: The Gifford Lectures. Archived from the original on 2009-03-08. Retrieved 2012-01-22.
Jump up ^ Aristotle, Περὶ γενέσεως καὶ φθορᾶς (On Generation and Corruption), 315b14.
Jump up ^ Marcus Tullius Cicero, De natura deorum, 2.37. Translation from Cicero's Tusculan Disputations; Also, Treatises On The Nature Of The Gods, And On The Commonwealth, C. D. Yonge, principal translator, New York, Harper & Brothers Publishers, Franklin Square. (1877). Downloadable text.
Jump up ^ The English translation of "The Total Library" lists the title of Swift's essay as "Trivial Essay on the Faculties of the Soul." The appropriate reference is, instead: Swift, Jonathan, Temple Scott et al. "A Tritical Essay upon the Faculties of the Mind." The Prose Works of Jonathan Swift, Volume 1. London: G. Bell, 1897, pp. 291-296. Google Books
Jump up ^ Borges, Jorge Luis. "La biblioteca total" (The Total Library), Sur No. 59, August 1939. Trans. by Eliot Weinberger. In Selected Non-Fictions (Penguin: 1999), ISBN 0-670-84947-2.
^ Jump up to: a b "No words to describe monkeys' play". BBC News. 2003-05-09. Retrieved 2009-07-25.
Jump up ^ "Notes Towards the Complete Works of Shakespeare" (PDF). vivaria.net. 2002. Archived from the original (PDF) on 2013-01-20. Retrieved 2014-07-11.
Jump up ^ "Monkeys Don't Write Shakespeare". Wired News. Associated Press. 2003-05-09. Archived from the original on 2004-02-01. Retrieved 2007-03-02.
Jump up ^ Padmanabhan, Thanu (2005). "The dark side of astronomy". Nature. 435 (7038): 20–21. Bibcode:2005Natur.435...20P. doi:10.1038/435020a. Platt, Suzy; Library of Congress Congressional Research Service (1993). Respectfully quoted: a dictionary of quotations. Barnes & Noble. pp. 388–389. ISBN 0-88029-768-9. CS1 maint: Multiple names: authors list (link)
Jump up ^ Rescher, Nicholas (2006). Studies in the Philosophy of Science. ontos verlag. p. 103. ISBN 3-938793-20-1.
Jump up ^ Powell, Doug (2006). Holman Quicksource Guide to Christian Apologetics. Broadman & Holman. pp. 60, 63. ISBN 0-8054-9460-X.
Jump up ^ MacArthur, John (2003). Think Biblically!: Recovering a Christian Worldview. Crossway Books. pp. 78–79. ISBN 1-58134-412-0.
Jump up ^ Dawkins, Richard (1996). The Blind Watchmaker. W.W. Norton & Co. pp. 46–50. ISBN 0-393-31570-3.
Jump up ^ As quoted in Blachowicz, James (1998). Of Two Minds: Nature of Inquiry. SUNY Press. p. 109. ISBN 0-7914-3641-1.
Jump up ^ Valentine, James (2004). On the Origin of Phyla. University of Chicago Press. pp. 77–80. ISBN 0-226-84548-6.
Jump up ^ p.126 of The Principles of Art, as summarized and quoted by Sclafani, Richard J. (1975). "The logical primitiveness of the concept of a work of art". British Journal of Aesthetics. 15 (1): 14. doi:10.1093/bjaesthetics/15.1.14.
Jump up ^ John, Eileen and Dominic Lopes, editors (2004). The Philosophy of Literature: Contemporary and Classic Readings: An Anthology. Blackwell. p. 96. ISBN 1-4051-1208-5. CS1 maint: Multiple names: authors list (link) CS1 maint: Extra text: authors list (link)
Jump up ^ Genette, Gérard (1997). The Work of Art: Immanence and Transcendence. Cornell UP. ISBN 0-8014-8272-0.
Jump up ^ Gracia, Jorge (1996). Texts: Ontological Status, Identity, Author, Audience. SUNY Press. pp. 1–2, 122–125. ISBN 0-7914-2901-6.
Jump up ^ Newyorker.com Acocella, Joan "The Typing Life: How writers used to write" The New Yorker April 9, 2007, a review of The Iron Whim: A Fragmented History of Typewriting (Cornell) 2007, by Darren Wershler-Henry
Jump up ^ Inglis-Arkell, Esther (June 9, 2011). "The story of the Monkey Shakespeare Simulator Project". io9. gizmodo. Retrieved February 24, 2016.
Jump up ^ Marsaglia G.; Zaman A. (1993). "Monkey tests for random number generators". Computers & mathematics with applications. Elsevier, Oxford. 26 (9): 1–10. doi:10.1016/0898-1221(93)90001-C. ISSN 0898-1221PostScript version
Jump up ^ Robert Wilensky, speech at a 1996 conference. "Quote Details: Robert Wilensky: We've heard that a..." The Quotations Page. Retrieved 2012-01-18. [unreliable source?]
Jump up ^ Monkeys, Typewriters and Networks Archived 2008-05-13 at the Wayback Machine., Ute Hoffmann & Jeanette Hofmann, Wissenschaftszentrum Berlin für Sozialforschung gGmbH (WZB), 2001.
Jump up ^ Ken Ringle, "Hello? This is Bob", Washington Post, 28 October 2002, page C01.
Jump up ^ Notes Towards the Complete Works of Shakespeare – some press clippings.
Jump up ^ Greta Lorge, "The Best Thought Experiments: Schrödinger's Cat, Borel's Monkeys", Wired Issue 15.06, May 2007.
External links[edit]
Ask Dr. Math article, August 1998, Adam Bridge
The Parable of the Monkeys, a bibliography with quotations
Planck Monkeys, on populating the cosmos with monkey particles
PixelMonkeys.org – Artist, Matt Kane's application of the Infinite Monkey Theorem on pixels to create images.
RFC 2795 – April Fools' Day RFC on the implementation of the Infinite monkey theorem.
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Cobie Smulders
From Wikipedia, the free encyclopedia
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Cobie Smulders
Smulders at the 2013 San Diego Comic-Con
Born
Jacoba Francisca Maria Smulders
(1982-04-03) April 3, 1982 (age 36)
Vancouver, British Columbia, Canada
Residence
New York City, New York, U.S.
Occupation
Actress, Model
Years active
2002–present
Spouse(s)
Taran Killam (m. 2012)
Children
2
Jacoba Francisca Maria Smulders[1] (born April 3, 1982[2][3]) is a Canadian actress and model. She is best known for her starring role as Robin Scherbatsky on the comedy series How I Met Your Mother (2005–2014). She is also known for playing S.H.I.E.L.D. agent Maria Hill in the Marvel Cinematic Universe, appearing in the films The Avengers (2012), Captain America: The Winter Soldier (2014), Avengers: Age of Ultron (2015), Avengers: Infinity War (2018), and Spider-Man: Far From Home (2019).
Contents
1 Early life
2 Acting career
3 Charitable endeavors
4 Personal life
5 Filmography
5.1 Film
5.2 Television
6 Theater
7 Awards and nominations
8 References
9 External links
Early life[edit]
Smulders was born in Vancouver, British Columbia to a Dutch father and an English mother. She was named after her great-aunt, from whom she gained the nickname "Cobie".[4] Smulders describes herself as "a fluent listener" of French.[5]
Smulders worked in modeling, which she later said she "kind of hated", adding that the experience made her hesitant about pursuing acting as a career: "You know you go into these rooms, and I've had the experience of people judging you physically for so long and I was over that. But then it was like, 'Oh no, I have to actually perform. I have to do well, and I have to have a voice, and I have to have thoughts now.'"[6]
In her youth, Smulders aspired to be a marine biologist.[5] Describing her transition to her eventual professions, she said,
I studied acting throughout high school, then modeling took over because it brought more opportunity. When I quit modeling, coming back to Vancouver, I registered at the University of Victoria. At the same time, that summer, I was taking acting classes and decided to give that a year or two instead of going back to school. I always wanted to go back to school. I was looking into taking some classes part-time, but marine biology involves a lot of lab time.[5]
Acting career[edit]
Smulders at the CBS Comedies Premiere Party in September 2008
Smulders' first acting role was as a guest in the Showtime science fiction series Jeremiah, and she has appeared in several television series since, including a recurring role on The L Word. Her first permanent series role was in the short-lived ABC series Veritas: The Quest, which ran for one season. After the cancellation of Veritas, Smulders was cast as television reporter Robin Scherbatsky on the CBS sitcom How I Met Your Mother in 2005, remaining on the show for its entire nine-season run.
In June 2010, Smulders made her Off-Broadway debut in Love, Loss, and What I Wore at The Westside Theatre.[7]
Smulders played Maria Hill in the 2012 film The Avengers. She received training from a Los Angeles SWAT team trainer to handle guns in order to portray the character.[8] Smulders reprised the role in two episodes of the television series Agents of S.H.I.E.L.D.,[9] and in the films Captain America: The Winter Soldier (2014), Avengers: Age of Ultron (2015), Avengers: Infinity War (2018), and Spider-Man: Far From Home (2019).[10][11] Joss Whedon has suggested he had considered her for the role of Wonder Woman in his draft of the eponymous film, which did not go into production.[12] Smulders did voice a Lego version of Wonder Woman in the 2014 animated film The Lego Movie. It was the first time the Wonder Woman character had a theatrical film appearance.[13]
In 2013, Smulders had a supporting role in the romance film Safe Haven. She also starred in the comedy-drama Delivery Man and They Came Together.[14] In July 2015, it was reported that Smulders had exited the made-for-TV film Confirmation because she had broken her leg; Zoe Lister-Jones was then confirmed to replace her in the role of Harriet Grant.[15]
In 2016, she appeared in the comedy-drama The Intervention, and with Tom Cruise in the action-adventure film Jack Reacher: Never Go Back. In 2017, she played the recurring character "Mother" in the Netflix series A Series of Unfortunate Events.
Charitable endeavors[edit]
Smulders filmed a PSA with Oceana, an international ocean conservation organization, in 2014.[16][17]
Personal life[edit]
Smulders became engaged to Taran Killam in January 2009.[18] They married on September 8, 2012, in Solvang, California.[19] The couple reside in New York City, New York.[20] They have two daughters; their first, Shaelyn, was born May 16, 2009,[21] and their second, Janita, was born in January 2015.[22][23]
In 2015, Smulders revealed she had been diagnosed with ovarian cancer at age 25, while shooting season three of How I Met Your Mother in 2007. She had surgery to remove two tumors from her ovaries, but the cancer had spread to her lymph nodes, resulting in the need for multiple surgeries over the course of two years.[24][25]
Filmography[edit]
Smulders at the 2014 San Diego Comic-Con International
Film[edit]
Year
Title
Role
Notes
2004 Walking Tall Eye candy
Ill Fated Mary [26]
2005 The Long Weekend Ellen
2006 Escape Psychotic brunette Short film
Dr. Miracles Mrs. Peterson Short film[27]
2007 The Storm Awaits Anabella DeLorenzo Short film[28]
2009 The Slammin' Salmon Tara
2012 The Avengers Maria Hill
Grassroots Clair
2013 Safe Haven Carly Jo Wheatley
Delivery Man Emma
2014 The Lego Movie Diana Prince / Wonder Woman Voice role
They Came Together Tiffany Amber Thigpen
Captain America: The Winter Soldier Maria Hill
2015 Unexpected Samantha Abbot [29]
Results Kat
Avengers: Age of Ultron Maria Hill
2016 The Intervention Ruby
Jack Reacher: Never Go Back Major Susan Turner
2017 Literally, Right Before Aaron Allison
Killing Gunther Lisa McCalla
2018 Avengers: Infinity War Maria Hill Uncredited cameo
2019 Spider-Man: Far From Home Filming
Television[edit]
Year
Title
Role
Notes
2002 Special Unit 2 Zoe Episode: "The Wish"
Jeremiah Deborah Episode: "Thieves' Honor"
2003 Tru Calling Sarah Webb Episode: "Brother's Keeper"
2003 Veritas: The Quest Juliet Droil Main role, 13 episodes
2004 Smallville Shannon Bell/Eve Andrews Episode: "Bound"
2005 Andromeda Jillian Rhade 2 episodes
The L Word Leigh Ostin 4 episodes
2005–2014 How I Met Your Mother Robin Scherbatsky Main role, 208 episodes
2010 How to Make It in America Hayley Episode: "Pilot"
2013–2015 Agents of S.H.I.E.L.D. Maria Hill 3 episodes
2013 Comedy Bang! Bang! Herself Episode: "Cobie Smulders Wears a Black & White Strapless Dress"
2015 Best Time Ever with Neil Patrick Harris Herself Episode: "Tyler Perry"
2016 Animals. Anni Voice role; episode: "Flies."
2017 A Series of Unfortunate Events Mother/Mrs. Quagmire Main role, 8 episodes
Nature Cat Nature Dog Voice role; episode: "Ocean Commotion"
2017–present Friends from College Lisa Turner Main role, 8 episodes
Theater[edit]
Year
Title
Role
Notes
2010 Love, Loss, and What I Wore Principal character June 10, 2010 – June 26, 2010
Westside Theatre
2017 Present Laughter Joanna Lyppiatt April 5, 2017 – July 2, 2017
St. James Theatre
Awards and nominations[edit]
Year
Award
Category
Work
Result
2013 EWwy Award Best Supporting Actress – Comedy How I Met Your Mother Won
2014 People's Choice Award Favorite TV Gal Pals (shared with Alyson Hannigan) How I Met Your Mother Nominated
2017 Theatre World Award Present Laughter Honoree
References[edit]
Jump up ^ "Cobie Smulders on life after "How I Met Your Mother"". CBS News. March 30, 2014. Archived from the original on October 20, 2014. Retrieved October 13, 2014.
Jump up ^ "Cobie Smulders". Los Angeles Times. Archived from the original on January 28, 2015. Retrieved January 28, 2015.
Jump up ^ "Thank you all for the Birthday love!!". Cobie Smulders verified Twitter account. April 3, 2013. Retrieved February 14, 2013.
Jump up ^ "Cobie Smulders on The Late Late Show with Craig Ferguson". YouTube. April 18, 2008. Archived from the original on June 15, 2014. Retrieved August 1, 2013.
^ Jump up to: a b c Johnson, Brian D. (November 16, 2013). "'It's always in my back pocket': Cobie Smulders on Canadian identity". Maclean's. Archived from the original on March 7, 2014. Retrieved December 27, 2013.
Jump up ^ Zuo, Mila (September 1, 2009). "The Elusive Charm of Cobie Smulders". Venus Zine. Archived from the original on July 22, 2011. Retrieved May 28, 2011.
Jump up ^ Wieselman, Jarett (June 10, 2010). "'Love, Loss' and two lovely ladies". New York Post. Archived from the original on October 18, 2010. Retrieved August 13, 2012.
Jump up ^ Sacks, Ethan (April 29, 2012). "'The Avengers': Scarlett Johansson & Cobie Smulders are superwomen of the screen". Daily News. New York City. Archived from the original on October 7, 2014. Retrieved January 30, 2015.
Jump up ^ Goldberg, Lesley (July 19, 2013). "Cobie Smulders' Comic-Con Reveal: Secret 'Agents of SHIELD' Role". The Hollywood Reporter. Archived from the original on October 12, 2014. Retrieved July 19, 2013.
Jump up ^ Graser, Marc (October 29, 2012). "Frank Grillo to play Crossbones in 'Captain America' sequel". Variety. Archived from the original on November 1, 2012. Retrieved October 29, 2012.
Jump up ^ Thompson, Bob (November 7, 2013). "Vancouver's Cobie Smulders is on a roll (with video)". Calgary Herald. Calgary, Alberta. Archived from the original on November 8, 2013. Retrieved November 7, 2013.
Jump up ^ "Satin Tights No Longer". Whedonesque.com. February 3, 2007. Retrieved May 7, 2010.
Jump up ^ "The LEGO Movie review". Retrieved March 24, 2018.
Jump up ^ Cobie Smulders Joins the ‘Starbuck’ Family. NextMovie (2012-08-27). Retrieved on 2013-01-07.
Jump up ^ Petski, Denise (July 9, 2015). "Zoe Lister-Jones Joins HBO Movie 'Confirmation' In Recasting". Deadline Hollywood.
Jump up ^ Wiseman, Jessica (August 6, 2014). "Photos: Rashida Jones, Cobie Smulders, and Angela Kinsey Help Protect Belizean Reefs". Retrieved October 13, 2014.
Jump up ^ Scott, Walter (November 23, 2013). "Cobie Smulders' Detachable Baby Bump". Parade. Retrieved October 13, 2014.
Jump up ^ Alexander, Reagan (January 28, 2009). "HIMYM's Cobie Smulders Flashes Her New Engagement Ring". People. Retrieved November 25, 2009.
Jump up ^ ""SNL's" Taran Killam Marries "How I Met Your Mother's" Cobie Smulders — Exclusive". RumorFix. September 8, 2012. Retrieved September 9, 2012.
Jump up ^ "Cobie Smulders". How I Met Your Mother (official CBS site). Archived from the original on January 22, 2014. Retrieved April 2, 2014.
Jump up ^ Leon, Anya (July 3, 2009). "It's a Girl for Cobie Smulders". People. Retrieved November 25, 2009.
Jump up ^ "Taran Killam and Cobie Smulders Expecting Second Child". People. October 21, 2014. Retrieved October 21, 2014.
Jump up ^ Klassen, Anna (January 26, 2015). "'Unexpected' Star Cobie Smulders & Director Kris Swanberg On Finally Making a Pregnancy Film From the Female Perspective". Retrieved January 30, 2015. I'm in Sundance and I have a three week old baby, so I can certainly relate.
Jump up ^ "'I had tumors on both ovaries': Cobie Smulders reveals cancer scare". Today. April 23, 2015. Retrieved April 26, 2015.
Jump up ^ "Mom of 2 Cobie Smulders Reveals Her Battle with Ovarian Cancer". People. April 23, 2015. Retrieved April 23, 2015.
Jump up ^ Bobbin, Jay (August 13, 2006). "Cobie Smulders - Chicago Tribune". Chicago Tribune.
Jump up ^ Dr. Miracles Episode 3. YouTube. October 2006.
Jump up ^ "The Storm Awaits (episode three)". YouTube. Very Advanced Productions. February 15, 2007. 3:18 minutes in.
Jump up ^ "NBA Star Chris Webber To Exec Produce Cobie Smulders Drama 'Unexpected'". Deadline Hollywood. January 7, 2015. Retrieved January 7, 2015.
External links[edit]
Cobie Smulders on IMDb
Cobie Smulders How I Met Your Mother biography at CBS
Cobie Smulders on Twitter
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WorldCat Identities
BNF: cb15722741g (data)
GND: 140568107
ISNI: 0000 0001 1454 5202
LCCN: no2010092739
MusicBrainz: f16e156c-25b4-4793-a61f-8b76e7c42123
SUDOC: 15582290X
VIAF: 107250250
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cobie_Smulders&oldid=853967448"
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