Talk:Infinite monkey theorem

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Swift and Pascal bits are missing

Latest comment: 13 years ago2 comments2 people in discussion

The history section mentions Swift and Pascal in passing, but unlike the Aristotle and Cicero bits, the Swift and Pascal references are never explained or quoted. I found the Swift bit in the popular culture article (which I am working on at the moment), but don't know where to find the Pascal bit. The Swift bit should be added:

• 1782 - Jonathan Swift's Gulliver's Travels (1782) anticipates the central idea of the theorem, depicting a professor of the Grand Academy of Lagado who attempts to create a complete list of all knowledge of science by having his students constantly create random strings of letters by turning cranks on a mechanism (Part three, Chapter five): although his intention was more likely to parody Ramon Llull.

And someone should find and add the Pascal bit. Thanks. Carcharoth 14:27, 12 August 2007 (UTC)Reply

Hi. I have found a book by Swift published 1774 (there are many editions - I found this particular edition on JISC Historic Books). The works of Jonathn. Swift, D.D.: D.S.P.D. with notes historical and critical. pg 176-182 (By J. Hawkesworth, L.L.D. and others. Printed for J. Williams, Dublin Library : Bodleian Library (Oxford) - Accessed via www.jischistoricbooks.ac.uk) In this is the Part III Chapter 5. The machine created by the professor in the academy has words, pronouns, punctuation and all the parts of speech written on individual tablets. The tablets are all put into the machine and the handle is turned. The students of the professor then read off the sentence that is formed and it is written down. "..whereby by his contrivance the most ignorant person at a reasonable charge and with a little bodily labour might write books in philosophy, poetry, politics, law, mathematics, and theology, without the least assistance from genius or study.".

There is a note to this chapter in this edition by Lord Charles Boyle, Earl of Orrery:

"The project for a more easy and expeditious method of writing a treatise in any science, by a wooden engine, is entertainingly satirical; and is aimed at those authors who, instead of receiving materials from their own thoughts and observations, collect from dictionaries and common-place books, an irregular variety, without order, use or design: "ut nec pes nec caput uni, reddatur formae". Orrery." (The latin is a reference to Horace "where the feet and the head have no relation to the other parts").

The addition of the quote from Orrery in this edition of Swift's Works is interesting. It puts Swift and Bentley in the same group. Bentley's Confutation of Atheism (1692) was part of the Boyle Lectures. (See my section below History - references preceding Borel). And according to the page, Gullivers Travels, the book was read by almost everyone when it was published in 1726. Thus almost everyone would have this idea of a machine pumping out sentences from random assortments of words and letters.

In the same edition is Johnathan Swift's essay, A Tritical Essay upon the Faculties of the Mind, about which Swift's footnote reads, "in a farcical, satiric light, designed purely to expose the folly and temerity of those brainless, illiterate scriblers (sic), who are eternally plaguing their contemporaries with a parcel of wild, incoherent, nonsensical trash. Swift." It is in this essay that Swift talks about the jumbling up of letters - but in a similar vein to Bentley without the monkey:

"And if this be so, how can the Epicurean's opinion be true, that the universe was formed by a fortuitous concourse of atoms; which I will no more believe, than that the accidental jumbling of letters of the alphabet could fall by chance into a most ingenious and learned treatise of philosophy. Risum teneatis amici? Horace (latin: "Could you refrain from laughing, my friend?" from the same passage of Horace as Orrery quotes.)

In fact another book by William Wotton Reflections upon ancient and Modern Learning, Wotton complains about what he considers to be Swift's satirical attack on himself and Richard Bentley in Swift's Tale of the Tub. Zorgster (talk) 16:11, 23 January 2012 (UTC)Reply

Later history section?

Latest comment: 18 years ago1 comment1 person in discussion

Is there a possibility for a rigorously sourced selection of examples from the 'in popular culture' article being integrated to this article under the title 'Later history' or 'Recent history'? At the moment, I'm rigorously sourcing the examples and re-ordering them in date order. I'm also turning up papers that mention not the mathematics or the early history and development of the idea, but rather of the current history and usage of the idea in literature and elsewhere. In other words, rather than being a "here are some examples", it would become "here is a history of the later use of the idea". Carcharoth 14:34, 12 August 2007 (UTC)Reply

New paper on this topic - Monkeying Around with Text

Latest comment: 18 years ago1 comment1 person in discussion

Please see Monkeying Around with Text, Terry Butler, University of Alberta, Computing in the Humanities Working Papers, January 2007. I've used this as a reference over at Infinite monkey theorem in popular culture, and I think it will be useful here as well. Carcharoth 18:26, 12 August 2007 (UTC)Reply

Expanded summary for popular culture section

Latest comment: 18 years ago1 comment1 person in discussion

I've now added a summary for the popular culture section, based on my rewrite of the daughter article Infinite monkey theorem in popular culture. See my edit here. Carcharoth 12:11, 16 August 2007 (UTC)Reply

Stephen Ballentyne

Latest comment: 17 years ago6 comments4 people in discussion

Editors of this page may wish to be wary of including material by the philosopher of mathematics Stephen Ballentyne until there is evidence that such an individual exits and has been published. --Mark H Wilkinson ^{(t, c)} 19:12, 16 September 2007 (UTC)Reply

Yes, even though Ballentyne seems to exist, publishing is key. The edits to this article are confused, and if they are based on a published source, I would be very interested in learning which publisher endorsed it and what exactly it said. Melchoir 04:41, 17 September 2007 (UTC)Reply

Apparently this person is now teaching Religious Studies at Uppingham School.[1] Unless he publishes in a reputable journal, and his point of view is discussed by multiple independent published sources, his opinions are just as non-notable and unencyclopedic as those expressed at the local pub.  --Lambiam 07:04, 17 September 2007 (UTC)Reply

Or that could be another individual entirely. The Ballentyne edits have been introduced by two new user accounts, apparently set up for the sole purpose of pushing this issue ([2], [3]).

At least we're getting a better class of vandal. --Mark H Wilkinson ^{(t, c)} 09:03, 17 September 2007 (UTC)Reply

It's a shame that the Internet appears to be the sole source of verification for published source materials and that no evidence of Ballentyne's published material exists (as yet) on the Internet. My fellow students and I will strive to correct this omission by writing to the editors of the journals and books he has contributed to. It is sad that material about monkeys urinating on typewriters and repetitive computerised random number experiments are present in this article, instead of the dynamic range of mathematical philosophies that exist on the subject. --Merisalis 04:33, 18 September 2007 (UTC)Reply

Funny thing about the internet is that the work of noted academics tends to turn up in Google searches. For example: Prof JF Toland is exceedingly easy to find. --Mark H Wilkinson ^{(t, c)} 10:41, 18 September 2007 (UTC)Reply

Evolution?

Latest comment: 17 years ago2 comments2 people in discussion

Excuse me, what is the relevance of this to the article?

"Various Christian apologists on the one hand, and Richard Dawkins on the other, have argued about the appropriateness of the monkeys as a metaphor for evolution." —Preceding unsigned comment added by 128.122.20.71 (talk) 14:23, 18 September 2007 (UTC)Reply

Evolution is probably the most common and most important context for the infinite monkey theorem in modern popular culture, and there is a section of the article describing how. The Wikipedia:Lead section provides a one-sentence summary of that section.

I've reverted a bunch of recent edits to the lead that had little basis and removed information. I've also reverted the "However" paragraph from the "Real monkeys" section; the section already states that monkeys are not random number generators, and we don't need arguments that experiments are "unnecessary". Melchoir 01:00, 3 November 2007 (UTC)Reply

I don't get it... :P

Latest comment: 17 years ago5 comments5 people in discussion

Infinity is an unending period of time. Why does the term "almost surely" apply? Shouldnt it just be "certainly"? After all, there are no limits. -- Anonymous Dissident^Talk 09:42, 21 December 2007 (UTC)Reply

Imagine that a fair coin is tossed infinitely often. Denoting the two sides as 0 and 1, this gives an infinite sequence of bits, something like

0111001110001111111111010011110100101001110111100010010111001011011100010011101110101110110010001010...

(space limitations do not allow to show the full sequence). What is the chance of getting exactly this sequence? In fact, all possible sequences are just as likely, such as the sequence

1011010001100111000011101111110110011111011011110011000111011011101111110111001011011110010100011011...,

or, for that matter, the sequence

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000....

So if it is certain that the last sequence will not occur, it is just as certain that any other sequence, such as the two above it, will not occur, including the one that actually is the result of tossing the coin infinitely often. Under the normal meaning of the word certain, that is a contradiction.  --Lambiam 18:56, 21 December 2007 (UTC)Reply

I had the exact identical question back in the section entitled (appropriately enough) "Question". Refer to Dcoetzee's answer there. —Preceding unsigned comment added by 75.163.233.26 (talk) 17:35, 29 March 2008 (UTC)Reply

This looks like a case of approaches infinity vs. infinity. Similarly p(x)->1 not p(x) identical to 1. 68.144.80.168 (talk) 12:30, 19 June 2008 (UTC)Reply

It is definitely not surely. If a monkey presses a button at random, that means it is possible that it can repeatedly press the same button each time. The probability of this continually happening is very small, but it can happen. For this reason it is impossible to 'guarantee' that the works of Shakespeare would ever be produced. It only becomes more likely as more time is given. —Preceding unsigned comment added by 79.68.196.135 (talk) 14:31, 27 June 2008 (UTC)Reply

"Huxley" &c.

Latest comment: 13 years ago6 comments3 people in discussion

Does it not seem a little silly to put a picture of Huxley up when the article does not actually owe much to the man? He simply was attributed wrongly, and had a nice little quip about the bishop. The only other illustration contained in this article illustrates the article beautifully. Huxley, however, has little to do with this. I'm just being picky, but it threw me for a second. It seems like unnecessary emphasis on an only mildly meaningful character (in the context of this article).130.101.14.214 (talk) 19:35, 2 February 2010 (UTC)Reply

The "Evolution" section currently begins:

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.

The footnote for this bold statement includes the following citation: Padmanabhan, Thanu (2005). "The dark side of astronomy". Nature. 435: 20–21. doi:10.1038/435020a.. However, I cannot figure out how that article relates to the Infinite Monkeys, let alone to whether or not "Huxley" invented that formulation. Am I missing something, or has there been an error?

Also:

Borges follows the history of this argument through Blaise Pascal and Jonathan Swift, 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."

Interestingly, when Borges says this, he too attributes it to a "Huxley":

Siglo y medio más tarde, tres hombres justifican a Demócrito y refutan a Cicerón. En tan desaforado espacio de tiempo, el vocabulario y las metáforas de la polémica son distintos. Huxley (que es uno de esos hombres) no dice que los "caracteres de oro" acabarán por componer un verso latino, si los arrojan un número suficiente de veces; dice que media docena de monos, provistos de máquinas de escribir, producirán en unas cuantas eternidades todos los libros que contiene el British Museum.

I would be interested in knowing the history of this attribution. Is there a kernel of truth in it? I guess I'll have to check Respectfully quoted: a dictionary of quotations (the other authority cited in the aforementioned footnote) when I get home, but in the meantime can anyone elucidate this? It seems like we should have some firm sources if we're going to categorically state that the "attribution is incorrect," right? --Iustinus (talk) 23:44, 28 December 2007 (UTC)Reply

Should have specified: since Borges wrote in 39, he's presumably repeating the claim from Jeans. But it's still very interesting that hementions it. --Iustinus (talk) 00:50, 29 December 2007 (UTC)Reply

I think the attribution to Huxley is correct. To Julian Huxley, not Thomas Huxley. Six monkeys, infinite years... (one source: Universal Book of Mathematics). Eddington, Jeans, Huxley were all contemporaries (see God and the Universe) Zorgster (talk) 20:05, 16 January 2012 (UTC)Reply

I have found an article from Irishman's Diary (The Irish Times, May 18, 1939. Accessed via ProQuest Historical Newspapers: The Irish Times (1859-2007)). There is a short piece titled "Monkeys by the Million", the author reports listening to a radio program in which the host mentioned a friend from Aberdeen told him "if you had a sufficiently large number of monkeys thumping the keys of an equally large number of typewriters for an indefinite period of time, you eventually would produce all Shakespeare's plays.". A Scottish friend of the author disputes this, "..because if you covered the earth with typewriters and monkeys, with a patch three-foot square in the middle for the Editor of the Irish Times to sit on - if it would hold him - and to take observations; and if you set these said monkeys knocking the keys of each typewriter about a line a minute, say 40 characters, the chances that at end of the year one of them had produced 'To be or not to be, that is the question' is about one in a million million million million million million million". This therefore differs from the comment above that "by 1939, the idiom was" - it is highly likely that by 1939 the idiom had taken a multitude of forms. Zorgster (talk) 17:28, 21 January 2012 (UTC)Reply

From The Manchester Guardian Sept 12th 1928, pg 5, The British Association:

The article discusses the talk given by Professor Frederick George Donnan at the Annual Meeting. He is discussing a discovery made by physiologist A V. Hill regarding cell respiration and the constant need for oxygen. It is not clear whether the Guardian reporter is making this comment, or whether Prof. Donnan said it:

"... but according to the statistical theory of probability if we waited long enough anything that was possible, no matter how improbable, would happen. If six monkeys were set before six typewriters it would be a long time before they produced by mere chance all the written books in the British Museum, but it would not be an infinitely long time." (Guardian, Sep 12 1928, pg 5) Zorgster (talk)

From The Guardian - Feb 5th 1927 - A Definition of Extreme Improbability:

This article discusses a talk on Feb 4th 1927 by Arthur Eddington at the 2nd Gifford Lectures titled The Nature of the Physical World (of which the book is referenced in the main article), in which he discusses entropy in the universe. Eddington's metaphor for entropy is of air spreading out in a box and the article adds (or reports) that the chance of all the air ending up in one half is very low... "The chances against this are greater than that an army of monkeys drumming on an array of typewriters should by accident compose all the books in the British Museum.". This attribution is earlier than that cited in the main article, but I cannot confirm it is the words of the reporter or Eddington. Zorgster (talk) 19:11, 21 January 2012 (UTC)Reply

assumptions

Latest comment: 17 years ago1 comment1 person in discussion

It must be made very clear here what we take on faith in our definition of "infinite". Even if "infinity" as a logical abstraction is comfortable and acceptable in the exact branches of mathematics, bringing it into statistics raises innumerable difficulties, not all of which are mathematical. --VKokielov (talk) 05:05, 4 February 2008 (UTC)Reply

One of the worst Wikipedia articles ?

Latest comment: 16 years ago6 comments6 people in discussion

And it also has an error - in the Direct Proof section, please note the while the individual keystrokes are independent (by assumption), the blocks of 6 letters are NOT, since they overlap. and if they not overlap, there is the possibility of |1QpBAN| |ANAhlp|, which is not accounted for. Either way, the proof is incorrect.

zermalo (talk) 20:02, 17 March 2008 (UTC)Reply

Your claim is not constructive. Please point out which particular sentence you believe is incorrect by directly quoting it here. Melchoir (talk) 01:32, 18 March 2008 (UTC)Reply

I think this is covered pretty well by footnote 1. Does it need to be in the body of the article? Algebraist 14:29, 5 May 2008 (UTC)Reply

Whether or not the text is organized into blocks is irrelevant to the proof. Organizing the text into blocks of, say, 6 implies that we will be making 1/6th as many samples which in turn implies that the time needed will be 6 times longer. The samples are still independant. Let us examine the alternative. No blocks: There are two variations to this problem. Case 1: Will "poem" be replicated. In this case, overlapping samples can be considered independant. 'Nuff said. Case 2: searching for the first instance of "poem". This case is harder, (as samples demonstrate dependance), but we can still fall back on the blocks (or even case 1). 68.144.80.168 (talk) 12:48, 19 June 2008 (UTC)Reply

The proof is correct, because if the probability of BANANA occurring on a block boundary is 1, then any more likely event has the same probability. There's no need for a precise analysis here. Dcoetzee 01:45, 28 June 2008 (UTC)Reply

Oh, dear! I hadn't noticed the claim. It is wrong, but the errors also really do not matter. See the "Definitely wrong but morally right" section infra. JoergenB (talk) 20:20, 7 October 2008 (UTC)Reply

The Brainiac Experiment

Latest comment: 17 years ago3 comments3 people in discussion

When I came to this page I immediately thought of an experiment conducted on Brainiac testing this idea. They sat several monkeys and several "Brainiacs" down at computers, and the closest they could get to Shakespeare's works was when one monkey typed "alas" in the whole six hours. I would put this on the page, but I don't think I'd do a very good job of it and I'm relatively new to Wikipedia, so please could someone else who knows more about what they're doing add this? Thanks:) Lowri (talk) 17:27, 22 July 2008 (UTC)Reply

Do you have a reliable source for this story? Surely anyone of the "Brainiacs" could have typed To be or not to be; in fact almost any English-speaking "Joe Shmoe" would know that much Shakespeare. What have the computers to do with it?  --Lambiam 00:04, 27 July 2008 (UTC)Reply

Is the story itself not reliable enough? Well, it doesn't matter anyway as their experiment really didn't prove anything, they probably just did it for the "lulz". 193.44.6.146 (talk) 21:24, 31 July 2008 (UTC)Reply

How tiny is very tiny?

Latest comment: 17 years ago7 comments4 people in discussion

In a recent edit, the sentence

"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."

was changed to:

"The probability of a monkey typing a given string of text as long as, say, Hamlet, is very tiny, but not zero."

The edit summary stated:

"This is redundant and wrong ".

I don't see what is wrong with this, assuming the age of the universe is not significantly more than 10¹⁰⁰⁰⁰⁰ years and that the monkey does not type significantly faster than 10¹⁰⁰⁰⁰⁰ characters per second. Please enlighten me.  --Lambiam 23:41, 12 August 2008 (UTC)Reply

You might want to knock a zero off the second exponent (I'm not sure of the average word length of Hamlet, so I'm being conservative), but otherwise You're right. Algebraist 23:52, 12 August 2008 (UTC)Reply

I see the full edit summary was This is redundant and wrong, especially interesting to be wrong only 2 lines after explaining the perils of reasoning in this exact way...:), referring to 'the perils of reasoning about infinity by imagining a vast but finite number, and vice versa'. Perhaps Diza was imagining an infinite age of the universe? Algebraist 23:55, 12 August 2008 (UTC)Reply

Yes, the shortened version, while true, is a bit of an empty statement - it's a bit obvious. But the original is also vague - we've no idea what "the chance of" is actually referring to, or even what "the experiment" is. The theorem is only meaningful in an idealised, thought-experiment sense. I suspect any re-write eliminating this vagueness would render it too wordy for an introductory paragraph. I can't imagine calculations about the universe or the laws of physics having any place here - there's plenty of scope for them in the main body. I'd say the sentence is both redundant and vague, and the section works better without it.Bob D (talk) 00:14, 13 August 2008 (UTC)Reply

The antecedent of "it" in the original is obviously "a monkey typing a given string of text as long as, say, Hamlet". In the preceding sentences of the lede it has just been explained that the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces a random sequence of letters ad infinitum, which in an infinite amount of time will almost surely produce a given text, such as the complete works of William Shakespeare. At least to me, it appears so obvious that the experiment is to let an abstract device produce a random sequence of letters for an indefinite amount of time that I don't think this needs to be explicated.  --Lambiam 13:05, 17 August 2008 (UTC)Reply

Since 'it' has no reference to a time scale or rate, so there is no meaning to the 'chance of it occurring' within a given time.Bob D (talk) 06:28, 18 August 2008 (UTC)Reply

The "chance of it occurring" in question is, of course, a non-constant function of the typing rate. This does not make it meaningless, nor does it prevent us from observing that it is miniscule in any experiment. Melchoir (talk) 07:01, 18 August 2008 (UTC)Reply

130.000 (or actually more) ?

Latest comment: 16 years ago1 comment1 person in discussion

Take Hamlet from http://www.gutenberg.org/dirs/etext98/2ws2610.txt and pipe it through

perl -pe 's/\[.*?\]//; s/[\s\,\.\-\;]//g'|wc

and you'll see it's just a little less than 130.000 characters.

Not that it matters much, but still :) Rkarlsba (talk) 03:55, 3 September 2008 (UTC)Reply

Definitely wrong but morally right

Latest comment: 16 years ago4 comments4 people in discussion

The section Direct proof contains a fallacious statement (whence of course its proof also is not quite correct). At the same time, the error is of no importance for the thesis of the article; instead of the proposed exactly exponential expression, you get an approximately exponential expression, not with the same but with a rather similar base, and thus the conclusion is not at all involved. Thus, the argument ought to be rephrased. I'm afraid there is no doubt about the results in themselves; they are the most simple cases in studies of growths of dimension sequences related to finitely presented algebras, which were studied in the '70's e.g. by Victor Ufnarovski, Warren Dicks, and myself. The simple, purely combinatoric situations, independently later have been rediscovered by combinatorians (coming to the same conclusion, of course); and I suspect that the simple "one forbidden word" enumeration problem also has been covered by independent discoveries several other times, both before and after the first publication of it that I know about (by Govorov in 1972).

Here is the result, and an outline of the proof. A much more general result, covering any finite number of "forbiodden words", was proved and puublished by V. E. Govorov in the Mat. Zametki 12 (1972), pp. 197-204. I concentrate on "BANANA" and 50 letters, however.

Assume given an alphabet of 50 letters, and among these the letters A, B, and N. For any ("European") natural number n, let a_n be the number of strings of length n in the 50 letters, that do not contain BANANA as a subword; let us call "BANANA" forbidden, and the strings without an occurrence thereof allowed. Then, for ${\displaystyle 0\leq n\leq 5}$ , clearly ${\displaystyle a_{n}=50^{n}}$ . For ${\displaystyle n\geq 6}$ , ${\displaystyle a_{n}}$  fulfils the recursion formula

${\displaystyle a_{n}=50a_{n-1}-a_{n-6}}$ .

The reason is simply this: We may prepend any letter to any one of the ${\displaystyle a_{n-1}}$  allowed strings of length ${\displaystyle n-1}$ , and in that manner we get ${\displaystyle 50a_{n-1}}$  strings. Obviously, these strings include all the allowed strings of length n. However, some of them are forbidden; namely those beginning with "BANANA". For any one of these forbidden strings, the last ${\displaystyle n-6}$  letters will form an allowed string, say S (since otherwise already the string ANANAS would be forbidden, before prepending the initial "B"). Thus, out of the ${\displaystyle 50a_{n-1}}$  candidates for allowed strings, exactly ${\displaystyle a_{n-6}}$  fail.

Now, the most neat way to sum up these recursive properties are by the generating formal power series (as it ought to be called), alias the generating function (as it usually is called). In fact, it is not hard to see that

${\displaystyle \sum _{n=0}^{\infty }a_{n}x^{n}={1 \over 1-50x+x^{6}}\,}$ ,

by means of the usual methods taught in an elementary coursis including combinatoric enumeration by means of formal power series. However, our purpose here is slightly different; we want an estimate of the probability for finding or not finding "BANANA" as a subword of an arbitrary length n string. Assuming that the letters are chosen i.i.d. and with equal probabilities, all the ${\displaystyle 50^{n}}$  strings are equally probable; whence the probability of not having "BANANA" as a subword is exactly ${\displaystyle a_{n}/50^{n}}$ . Now, as also taught in these elementary enumeratoric combinatorics courses, a rational expression for a formal power series, such as the one supra, may be converted to a polynomial expression for the coefficients, in terms of the roots of denominator polynomial of the rational expression; or, more precisely, of the associated "auxiliary equation"

${\displaystyle y^{6}-50y^{5}+1=0\,}$ .

A sixth degree equation is a bit hard to solve by elementary means (as Abel proved); but it is easy to see that this one does not have double roots (by taking the g.c.d of it and its derivative). This yields, that there are constants ${\displaystyle c_{1},\ldots ,c_{6}}$ , such that for any ${\displaystyle n\geq 0}$  we have

${\displaystyle a_{n}=\sum _{j=1}^{6}c_{j}r_{j}^{n}\,}$ ,

where the ${\displaystyle r_{j}}$  are the roots of the auxiliary polynomial. The absolutely largest of these roots, say ${\displaystyle r_{1}}$ , is a positive real number; and thus already for moderately large n the quotient ${\displaystyle a_{n}/50^{n}}$  will be rather close to ${\displaystyle (r_{1}/50)^{-n}}$ . However, as you may see by simple means, the auxiliary equation does not have any rational root, whence a fortiori

${\displaystyle r_{1}\neq 50-50^{-5}\,}$ .

Now, I know that arguments involving probabilities often invoke hot sentiments; and I also know that nobody likes being told that someone else knows much more about these subjects. I'm really not trying to bully anyone; but I did write part of my ph.d. thesis about these things, and have published a few articles about this later, whence I would be lying if I told you that this just is a guess. I've really tried to explain why you get the results, perhaps too lengthily; I'm absolutely prepared to discuss them further in detail, here or on my user talk page; but, as I said, the differences are actually not that important. The base of the exponential expression is a number somewhat smaller than 1, that's all that matters here, actually. I'd like to rewrite the text slightly, weakening the claims a bit (and not including the Govorov et al. precise results with proofs, if you don't mind), but providing a correct and sufficient estimate. However, I won't try this, if there are too many "unresolved issues" about the mathematics left. JoergenB (talk) 21:50, 7 October 2008 (UTC)Reply

What's wrong with the direct proof as it stands? Algebraist 11:54, 8 October 2008 (UTC)Reply

The entire point of the simple proof is just to demonstrate a simple case of the theorem and why it's true in that case. It doesn't need to provide an accurate estimate of the probability, only to bound it from below, and then show that that bound tends to 1. A more accurate estimate may be relevant, but for this initial example it's overkill and defeats the point of having a simple example. Dcoetzee 21:45, 8 October 2008 (UTC)Reply

I believe JoergenB meant we should use the phrase "Sketch of proof" if we don't meant to have the exact details to be correct. K61824 (talk) 05:10, 10 May 2009 (UTC)Reply

Link 24: the monkey Shakespeare simulator

Latest comment: 15 years ago2 comments2 people in discussion

This site bombarded me with Java errors and made Firefox crash. Is this a common issue with the site? Should the link be removed if the site is unusable because of this? - 84.27.9.117 (talk) 22:21, 25 October 2008 (UTC)Reply

might be just you. worked fine on ieFirl21 (talk) 15:13, 3 September 2009 (UTC)Reply

Kolmogorov Complexity and monkeys typing on a computer

Latest comment: 16 years ago3 comments2 people in discussion

One of the most interesting sidenotes to the infinite monkey theorem is the fact that the monkeys would be significantly faster if they used a computer instead of a typewriter. I first heard about this in Cover and Thomas, "Elements of Information Theory".

Given that the Information entropy of the English language is about 1.5 bits per character, Shakespeare's works could probably be compressed by a factor of at least three. Thus, the probability that the monkeys come up with a compressed version of Hamlet (or a computer program which prints Hamlet) is much higher than the probability that they produce the full text.

I just find that interesting... what do others think about it? If there are enough people that motivate me (write to j dot b dot w at gmx dot ch) I'll create a Wikipedia account and write it down properly. —Preceding unsigned comment added by 128.178.83.79 (talk) 11:00, 19 December 2008 (UTC)Reply

Yes, that's interesting - I've added your paragraph (with minor changes) to the Probabilities section as a footnote to the text.Bobathon (talk) 13:48, 11 January 2009 (UTC)Reply

It may be appropriate to create a section for this somewhere within the article - it has a significant bearing on all calculations relevant to this subject (though clearly not on the final "finite but non-zero" conclusion)Bobathon (talk) 13:54, 11 January 2009 (UTC)Reply

Note

Latest comment: 16 years ago1 comment1 person in discussion

There are some broken equations in the Solution section. Could somebody more knowledgeable than I fix that? Thanks! –Juliancolton ^{Tropical Cyclone} 02:53, 25 February 2009 (UTC)Reply

Weasel misconception

Latest comment: 16 years ago1 comment1 person in discussion

I have corrected a misconception in the article which stated that "Dawkins has his Weasel program produce the Hamlet phrase METHINKS IT IS LIKE A WEASEL by typing random phrases but constantly freezing those parts of the output which already match the goal". The bold text is incorrect; in the context of this article, it's not a big deal, but it is part of the toolkit used by those who oppose evolution, and is plainly wrong.

The current Weasel program article does not clearly address the issue, but it is well covered in the discussion. The correct letters appear to lock because the program chooses the best match from mutated progeny (so mutations which make a good letter bad will usually not be the next parent). However, a video shows that the program does not lock correct letters, and the words used by Dawkins in The Blind Watchmaker make it obvious that his simulations apply random mutations to all locations. Johnuniq (talk) 08:55, 2 May 2009 (UTC)Reply

How was the specific probability calculated?

Latest comment: 15 years ago8 comments4 people in discussion

Just wondering, how was the number 3.4 × 10^183,946 obtained? This needs to be explained somewhere, or else the number should be removed for being unverifiable original research. ··gracefool☺ 18:54, 8 May 2009 (UTC)Reply

It's just 26^130000. Algebraist 20:00, 8 May 2009 (UTC)Reply

How do you work that out? ··gracefool☺ 10:38, 9 May 2009 (UTC)Reply

It was clear from the immediately preceding text that 26^130000 was the number that should have appeared there, so I just checked (using Google calculator) that it was. Algebraist 18:37, 9 May 2009 (UTC)Reply

To calculate n = 26^130000 we take log of both sides: log(n) = 1300000*log(26) = 183946.5352

Therefore n = 10^0.5352 * 10^183946 = 3.429 * 10^183946 Johnuniq (talk) 00:29, 10 May 2009 (UTC)Reply

Thanks. So including punctuation the figure is about 10^360783 (26 letters x2 for capitalisation, + 12 for punctuation characters = 64, log(64)*199749 characters). That makes a big difference to the number! ··gracefool☺ 01:18, 10 May 2009 (UTC)Reply

I've added this stuff to the article. ··gracefool☺ 05:36, 25 June 2009 (UTC)Reply

I once slapped the keyboard at random and got "iloveyouall". Professor M. Fiendish, Esq. 04:51, 13 September 2009 (UTC)Reply

"Almost surely"

Latest comment: 16 years ago1 comment1 person in discussion

The phrase "almost surely" has ABSOLUTELY NOTHING to do with the monkey metaphor! It's a precise mathematical term with a precise meaning that has NOTHING to do with metaphors. Please fix the article so that it makes sense! —Preceding unsigned comment added by 174.20.91.244 (talk) 18:04, 15 August 2009 (UTC)Reply

Broken external link

Latest comment: 15 years ago2 comments2 people in discussion

The first external link (to the Baltimore Examiner) is broken, and just redirects to the homepage of the Washington Examiner. 170.148.198.156 (talk) 09:52, 9 November 2009 (UTC)Reply

You meant the first link in the External Links, correct? I couldn't find it either. Looks like it got lost when baltimoreexaminer merged into washingtonexaminer. Lots of googling no go. Worse news, see here: http://web.archive.org/web/*/http://www.baltimoreexaminer.com/opinion/The_million_monkey_room.html Looks like baltimoreexaminer never let IA archive their stuff, so it'll be real hard to find again.  :-( —Aladdin Sane (talk) 10:43, 9 November 2009 (UTC)Reply

Probability section fails

Latest comment: 14 years ago9 comments4 people in discussion

That whole section is wp: or, not to mention wrong. It assumes that the monkey is only striking keys that produce letters the monkey could strike any of the function keys or numbers etc. It cites no sources except the bottom where it takes a quote. The main trouble is that this section falsely presents a low probabilty by restricting the origional terms of the theory to the life of the known universe rather then infinity which is how it is supposed to be. Any probability repeated with a time frame of infinity will be come 1, the life of the universe isnt even a warm up phase compared to infinity. This section needs to be removed entirely it is not helpful Smitty1337 (talk) 23:04, 21 April 2010 (UTC)Reply

I agree that OR is an issue but I support the section (WP:IAR or whatever) since it is verifiable and useful to readers. The restriction to 26 letters is just a commonly-made simplification – if the monkey had more than 26 choices the probability would be lower than the effectively zero value shown in the article. No one is denying that in some metaphorical sense randomly striking keys would eventually produce a sentence, but it is valuable to learn that in practical terms the outcome is impossible. The concept has sometimes been used to assert that freak things will eventually occur by blind chance (i.e. physical things in a practical universe). Whereas that is true for many very rare phenomena, it is not true in this universe for an event such as randomly typing a particular book. Also, note that there is a very reliable reference that supports the conclusion of the section. Johnuniq (talk) 02:02, 22 April 2010 (UTC)Reply

this section shows the odds at 3.4 × 10183,946 . if we assume 1 letter per second and 130,000 total seconds thats 2166.66 hours or 90 days roughly 1 quarter of a year, per full book attempted (and this is generous because it doesn't assume failure on letter 3 throws the book out and starts over right there). if the 3.4 × 10183,946 attempts required to get 1 right is all that's statistically probable, then the monkey should write one copy every 76.5 x 10183,946 years which divided by infinity will happen infinite times so in 76.5 x 101839460 years we'd have the 10th copy (collectors edition i presume). I'm being absurd to get my point across, this theorem is meant to prove a point, not be taken literally, the point is that anything that has a probability that is not zero no matter how excessively large the denominator is on that fraction, the concept of infinity makes that number minisucle to the point of if it "can" happen it will happen repeatedly in 345235324532452345234523453245234523452345234532 years (not even .00001% of infinity of course). this is all OR of course which is why i'd never push to say such on the article, but neither should some silly probability section give a false notion of unlikelyhood when infinity makes the chance 100% (eventually) Smitty1337 (talk) 10:33, 22 April 2010 (UTC)Reply

From the reliable source: As Kittel and Kroemer put it, "The probability of Hamlet is therefore zero in any operational sense of an event…". The numbers used for input to the calculation are shown in the "Direct proof" section. Don't you find it interesting that the chance of randomly typing Hamlet is essentially zero, even if you have as many monkeys as there are particles in the universe, and each types 1,000 keystrokes per second for 100 times the life of the universe? Johnuniq (talk) 11:20, 22 April 2010 (UTC)Reply

The trouble with making that statement is that it is without context. The probability may be operationally zero, as the sourcer says, and that is infact true, but the problem is that any probability given infinite repetition will become 1, and if not stopped upon successful completiion, then technically the book will be made and remade infinite times. the statement is true but by itself its misleading because its just the probability without moving to the next logical step of infinite repeatition, and everything above it is wp: or Smitty1337 (talk) 00:43, 23 April 2010 (UTC)Reply

From a Darwinian approach to the question "How long would monkeys type the works of Shakespeare"?, the solution is the period intelligent man has evolved from primates. A monkey , trapped on earth a million years ago with nothing but to mutate itself into an intelligent being, discover words, concoct writing, invent the typewriter and finally produce among its descendants a literary genius would take a few million years. This period has been historically tested and is much much less than the predicted statistical outcome with the assumption that the monkey will genetically remain stagnant without hope for increased brain capacity but just random act in his eternal life. —Preceding unsigned comment added by 149.136.17.253 (talk) 20:16, 1 September 2010 (UTC)Reply

You have missed the point entirely, as the article says right in the lead, its a metaphorical moneky, as in one monkey, not a series of generations, there is no evolution involved in this article and its only meant to imply a concept of probability and infinity. Smitty1337 (talk) 18:29, 2 September 2010 (UTC)Reply

Sorry to engage in forum talk, but while of course you are correct, I think 149.136.17.253 probably knows that, and the point they made is interesting because it highlights how human commonsense can fail when extrapolated too far. Our brains provide a model of the world whereby we are happy to talk about an ideal monkey typing for millions of years (which is fine), but in the time frames under discussion lots of things will change in ways that we struggle to appreciate (the Atlantic ocean is expected to disappear in 200 million years or so, due to plate tectonics; can't find it on en.wiki). Johnuniq (talk) 02:35, 3 September 2010 (UTC)Reply

None of this is right at all. A correct neuron configuration is required to type out a William's Shakesphere Play. Given infinite time, the neuron configuration of the brain will never reach precisely the right locations and synapses as that so the monkey will be typing out an entirely play by chance because it is deterministically a zero probability. In order for an action to occur, a neuron must be fired, and certain neurons specifically. —Preceding unsigned comment added by 209.159.183.114 (talk) 13:33, 20 November 2010 (UTC)Reply

"Experiment"

Latest comment: 14 years ago2 comments2 people in discussion

"Popular interest in the typing monkeys is sustained by numerous appearances in literature, television, radio, music, and the Internet. In 2003, an experiment was performed with six Celebes Crested Macaques. Their literary contribution was five pages consisting largely of the letter 'S'."

1. This wrongly suggests that it was a scientific experiment, rather than a student art project.
2. I don't see that this is significant enough to mention in the lead. Feezo (Talk) 03:05, 8 February 2011 (UTC)Reply

   I agree that the sentence should be removed from the lead. I suppose the mention in the "Real monkeys" section is warranted, although it adds little of value to the article other than to show that the topic is of general interest. Johnuniq (talk) 05:46, 8 February 2011 (UTC)Reply

Weird sentence in the lede

Latest comment: 14 years ago2 comments1 person in discussion

"The theorem illustrates the perils of reasoning about infinity by imagining a vast but finite number, and vice versa. " - what's "perilous" about it? What, I'm gonna get shot if I think about infinity by imaging a vast but finite number? What the hell is this sentence even supposed to mean? Nonsense.Volunteer Marek (talk) 21:24, 11 April 2011 (UTC)Reply

And what for monkey's sake does the "vice versa" refer to? Reasoning about "finity" by imagining a minuscule but infinite number? Seriously.Volunteer Marek (talk) 21:26, 11 April 2011 (UTC)Reply

almost surely

Latest comment: 14 years ago2 comments2 people in discussion

"almost surely" implies the change the money wont type the play is considerable, but the chance is actually infinitesimal or zero. 173.183.79.81 (talk) 22:35, 14 April 2011 (UTC)Reply

It's a technical term; it's not up to us to change it. It's explained in the text, which is really all we can do. --Trovatore (talk) 22:45, 14 April 2011 (UTC)Reply

typing for infinity

Latest comment: 13 years ago5 comments4 people in discussion

If the apes could type for infinity, and there was a small amount of probability that they could type the works of some author, is it also possible there is a small chance they write something that hasn't been written yet (assuming nobody saw the writings and somehow copied them)? Could the typing somehow predict a future? If they could, they could also predict another piece of writing that will never happen, or never has happened, creating something new, couldn't they? I may be asking something that has been gone over already, but I don't think so. — Preceding unsigned comment added by 66.165.17.192 (talk) 03:47, 3 July 2011 (UTC)Reply

anything about the article ? Arjun^codename024 05:21, 3 July 2011 (UTC)Reply

For the record, there is no "predictive" value to monkeys typing for billions of years, because any process which found some coherent text in the randomness (say, a collection of poetry about probability) would in effect be "writing" that text by searching for it. Overall, readable random texts are way way way less frequent than unreadable ones, unless you have an evolution-like process that keeps the readable stuff and ditches the unreadable, in which case it's a somewhat different puzzle.

However, if we're talking about an infinite length of time, then yes, even one monkey will indeed produce every possible finite sequence of text with probability 1, including readable books that no human had ever written — an infinite number of such books, in fact. There's nothing special about Shakespeare here, it's just used to illustrate this surprising idea. ± Lenoxus (" *** ") 00:32, 13 October 2011 (UTC)Reply

Nothing surprising about it when you consider how long infinite time is. ··gracefool☺ 03:22, 3 November 2011 (UTC)Reply

Lenoxus has it right, but in other words: You can't predict the future with randomness, because to recognize it as something apart from randomness, you have to already know the information in question. If there was no Shakespeare, you couldn't recognize Hamlet as being something special, any more than any of the other millions of readable books likely to be typed before that. If there were a specific prediction, eg. "The World Trade Center will be destroyed on September 11, 2011", how would you tell its accuracy over a million other predictions of the same thing happening on a different date? ··gracefool☺ 03:22, 3 November 2011 (UTC)Reply

The picture in the article is wrong also

Latest comment: 13 years ago2 comments2 people in discussion

That's a chimpanzee, not a monkey. And that's a camera, not a typewriter (I think). — Preceding unsigned comment added by 71.98.215.115 (talk) 05:45, 10 September 2011 (UTC)Reply

It's a very old typewriter: zoom in on it to see the detail (the base, the keys, the paper holding structure at the top). As for the ape historically 'monkey' was used for both apes and monkeys. Certainly when this phrase arose it would be common usage. Even now it is still used informally like that.--JohnBlackburne^words_deeds 12:15, 26 September 2011 (UTC)Reply

Title of first (non-lead) section

Latest comment: 13 years ago2 comments2 people in discussion

Why is the first section under the TOC headed Solution? Solution of what? Problems have solutions, but the theorem is not stated as a problem. I've looked through the history and it appears to have been that way for quite a while so I don't want to change it rashly, but surely we can do better than that. Suggestions? Maybe the inelegance of this word indicates a more structural difficulty with the article, and suggests moving the proof sketch further down? --Trovatore (talk) 20:13, 12 October 2011 (UTC)Reply

Yes, it is odd, although I don't think moving it would be particularly helpful. Perhaps "Analysis"? Also, in "proof of this theorem", the "this" should be replaced with "the infinite monkey". Johnuniq (talk) 00:31, 13 October 2011 (UTC)Reply

Similar concepts

Latest comment: 13 years ago2 comments2 people in discussion

Why aren't there links to similar concepts? Someone could include a link to Bogosort or something, which I believe is a great example.98.119.209.61 (talk) 09:09, 15 November 2011 (UTC)Reply

Good question! I'll add some later, if I can remember to. In the meantime, feel free to be bold. Evanh2008, Super Genius Who am I? You can talk to me... 09:43, 15 November 2011 (UTC)Reply

Why the link at the top about "not to be confused with..."?

Latest comment: 13 years ago4 comments3 people in discussion

Why exactly is there a link at the top of the article staying this should not be confused with the hundreth monkey effect? They are totally unrelated things and in my opinion, are not easily confused. If this statement stays, then I think we should also add the following statement: "Not to be confused 12 Monkeys." I mean, there is a number at the front of the statement and it says monkeys... a reader might be confused. Krohn211 (talk) 19:09, 1 December 2011 (UTC)Reply

I agree: if this title were ambiguous it should link to a disambiguation page but it clearly isn't. Further I can't see anyone confusing this with the hundredth monkey effect which seems not to be widely known: from the article it's a discredited crank theory, not part of mainstream science. I've moved it to 'See also' but I have no objection to it being removed altogether.--JohnBlackburne^words_deeds 19:35, 1 December 2011 (UTC)Reply

I am not a heavy wikipedia user so I don't know the goal of the "See Also" section. From my understanding, this section is meant for related topics and I don't believe this is a related topic. If it were up to me, I'd remove all references to it from this article. Looking at the history I can't tell who put this there in the first place. Since I didn't put it there, I don't want to be the one to remove it. But if I had a vote, I'd say remove it altogether. Krohn211 (talk) 03:53, 2 December 2011 (UTC)Reply

The above issue was fixed in this edit by JohnBlackburne. Johnuniq (talk) 06:45, 2 December 2011 (UTC)Reply

Unnecessary paragraph

Latest comment: 13 years ago3 comments2 people in discussion

"Primate behaviorists Cheney and Seyfarth remark that real monkeys would indeed have to rely on chance to have any hope of producing Romeo and Juliet. Monkeys lack a theory of mind and are unable to differentiate between their own and others' knowledge, emotions, and beliefs. Even if a monkey could learn to write a play and describe the characters' behavior, it could not reveal the characters' minds and so build an ironic tragedy"

Summary: monkeys are too dumb to write Shakespeare.

This is taking stating the obvious to a new level; it reads like a tabloid article or something from a waiting room magazine. Perhaps this somehow meets a guideline I don't know about, but surely a fact that no reader is ever likely to not already be aware of expressed in 73 words with "sciency" language and name-dropping of behaviourists does not belong on Wikipedia. Kombucha (talk) 00:05, 23 December 2011 (UTC)Reply

We should lose the name-drop at the least. Kombucha (talk) 00:11, 23 December 2011 (UTC)Reply

Feel free to remove the whole "Real monkeys" section because it is just unrelated commentary (i.e. it is nothing to do with the actual "theorem", and is essentially nonsense, no doubt expressed in impressive language in the original). Johnuniq (talk) 01:49, 23 December 2011 (UTC)Reply

History - references preceding Borel

Latest comment: 13 years ago3 comments2 people in discussion

I am fairly new to editing Wikipedia articles, not new to Wikipedia. I have been researching this topic to find an earlier reference to "the likeliness of monkeys writing great works". I found a reference by Richard Bentley in a sermon (originally 1692/3) regarding the probability that a monkey scribbling away could ever write Hobbes' Leviathan as a comparison to the probability of creation. I dispute that Borel is the first person to use the idea of monkeys typing works by chance in the context of probability. (As an aside, in the French, I have also found 'singe' with the meaning of 'Mime Artist' or 'people who mimic' ("les singes de Balsac (sic)" appeared in a dictionary to denote the plethora of authors all trying to mimic the style of Balzac.)) Bentley's prose is in the context of probability (albeit in terms of creation and not mathematics). True there were no typewriters in 1692, but this merely means that Borel had re-framed the concept of 'monkey scribbles' to 'les singes a frapper' on a typewriter (if it had not already been re-framed previously to him). Also in the re-framing one monkey becomes one million monkeys.

My edit was made in haste... and was removed as 'misplaced and original research'... I would like to discuss this. (Source: Richard Bentley's The folly and unreasonableness of atheism) Zorgster (talk) 04:41, 15 January 2012 (UTC)Reply

It was this edit that was reverted. I haven't wanted to take the time to investigate this issue (I saw your edit and decided to leave it). However, my guess is that the editor who reverted your edit thinks that we would need a secondary source that makes an association between the information you found and the topic of this article. If writing an article at some other place, it would appear under the writer's name and any views in the article would clearly be the views of the author. However, there is no author here, and all statements need to be verifiably related to the topic, rather than likely associations that an editor has located. I have not formed any firm views on the issue, and mention this for background. Johnuniq (talk) 06:06, 15 January 2012 (UTC)Reply

Hi.. thank you for taking the time to explain that. I see what you are saying. I could only speculate myself. ...that the sermon, which was part of the Boyle Lectures and so re-published in 1737 (A Defence of Revealed and Natural Religion), in 1809 (Eight Sermons, Oxford), in 1838 (The Works of Richard Bentley, Vol 3), was possibly used in arguments against atheism - or to strengthen religion. It would have been well circulated amongst scientist and clergy alike. Bentley's 'Confutation of Atheism' also discusses the ideas of Isaac Newton. I would suggest that any scientist, including Huxley, would be familiar with this text. And the argument about monkeys scribbling Hobbes would have been read by many an academic. Huxley may have used it in his crossings with Owen and Wilberforce (in Oxford, 1860) - in debates of Darwinism and religion. The paragraphs around Bentley's mention of the monkey in the 1838 edition, talk a great deal about probability and chance... and as is said in the philosophy of history... we tend, when reading historical prose, to frame the usage of words in the past using our understanding of the present... The human body was seen then (1692) as the creation of the divine... A body was often compared to a book... and the comparison here is that you can deny the 'hand of god' in the creation of man, as much as you could conceive of a monkey ever scribbling Leviathan out of pure chance (Kristine Haugen's Richard Bentley: Poetry and Enlightenment). Again, one can only speculate... and one needs to spend time looking into it... I've used my quota of spare time, too :-) Zorgster (talk) 06:31, 16 January 2012 (UTC)Reply

Possible FAR

Latest comment: 13 years ago1 comment1 person in discussion

Referencing on this article is still sub-par and has been tagged as such for close to 5 months (Criteria 1c); the prose in some parts, such as "in popular culture", is rough (Criteria 1a). This should be fixed, if possible. — Crisco 1492 (talk) 23:58, 31 May 2012 (UTC)Reply