Talk:Infinite monkey theorem/Archive 1

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i feel dumb

i always considered myself mildly intelligent. i dont understand a word of this article. im going to go and kill myself now. 208.103.186.197 00:38, 5 October 2006 (UTC)

1/50 ?

"the probability that the first letter is b is 1/50" -- um... what's the 1/50? should it be 1/52, 1/53(to include a whitespace)? or something? Where on earth does 1/50 come from? --Storkk 12:32, 27 August 2006 (UTC)

Okay, I skimmed over "Suppose this typewriter has 50 keys." Still, I think the number should be slightly more intuitively chosen. Like 2 x 26 or (2 x 26 + 10). Just my 2¢. --Storkk 12:34, 27 August 2006 (UTC)

Reductio ad absurdum

The theorem can-not be true. As it predicts a contradiction.

The theorem predicts that there will be typed-out any text of increasing length. Eventually it must type a repeating pattern, say "0101" then a longer repeat; say "01010101", then "01010101010101" ETC.

These repeated texts are predicted to gert longer & longer; for example "010101010101010101010101010101010101" eventually the length of this repeated pattern tends toward infinity. Not only that but infinately many times, for multiple monkeys.

Now only; 1 of 2 things can happen.

1)The pattern hits an upper bound!.

2)There are infinately many repeats of only 1 text!.

Spintronic 16:01, 11 August 2006 (UTC)

Spintronic: you are confused about cardinality. I suggest some basic coursework in mathematics. A countable, infinite sequence of digits has enough "room" for any countable, infinite bounded subsequences. You don't need to single out repeating sequences like "010101". Any sequence of length n will almost surely appear in an infinite random sequence, regardless of how large n is.

To make this clear, consider the sequence 012345678910111213141516171819202122..., in other words, the nonnegative integers concatenated. This includes 0, 01, 010, 0101, and so on for any number of repetitions. But it also includes every other finite sequence of digits, including the works of Shakespeare reduced to binary; with this sequence, it is not even almost surely, it is provably true, and you can actually calculate an upper bound of where the works of Shakespeare occurs. NTK 07:37, 28 October 2006 (UTC)

No I am not confused about cardinality. Sometimes infinity is just lemniscate,

You cannot have an infinite subsequence within an infinite sequence. Heres why!

Have an infinite sequence called R. Within R is an infinite subsequence called p. Within p is another infinite subsequence called q.

R > p > q.

Since, (infinity - 1) = (some finite number). Then;

R-p = q.

Thereby q is a finite number

R-q = p

Thereby p is also a finite number. (Spintronic 23:47, 4 July 2007 (UTC))

Theorem breaks down as complexity grows

The theorem appllies to all complex systems. As such, we can take the "random" nature of atoms "&-or" matter in an "infinate" universe and restate the theorem.

"Given an infinate universe, and an infinately long time, every possible combination of atoms will arise."

Thus; the universe out of "brute force of possible probabilities" has produced the planet we live on with all its complexity.

A man ponders this and realises, that if the theorem is true, there must be a copy of himself in this "infinate" universe. Not only that, but there must be infinately many copies, EXACT copies of himself.

He decides to go and find one of these "exact" copies of himself, (and for simplicity he can travel at any speed, thus arriving practically instantaniously)

He now arrive's at a galaxy that looks "exactly" like the milky way. He approach's a planet that looks "exactly" like the earth. He lands in a country that is "exactly" like he's, and he knock's on the door where he lives. The mans wife answers and informs him that, he had just left a while ago, (exactly the same amount of time ago that he had done),

There is a hair left in a comb on the side, the man analises it in he's dna machine, and it "exactly" matches he's own. (This is a true copy). The stars look the same (they'de have to be) he look's at the history archive and supernova exploded on exactly the same dates as they did back home, back on earth, and in exactly the same places in the sky (theyde have to have done or their realities wouldnt be the same). The man stops' and ponder's this for a moment, and he come's up with an idea, He call's up 5 of he's identical twin brothers, and they all set off in the 6 directions in all axis. He decide's that if they all set off in different directions then they will find 6 copies of their selves no matter which direction they go.

As they set off to the next "identical" world, he crosses one of hes twin brothers along the way he says hello and they pass and carry on. (this must be exactly half way between worlds). When he arrive's at the next "idnetical" world, all 5 of hes twin brothers are arriving from different directions at the same time. "Strange" the man says' to himself, and he ponder's the infinate monkey theorem more closely.

If the infinate monkey theorem is true, then any direction i go in, there will always be a copy of myself. Not only that but every time i set off to meet him(the copy), he also sets off to meet me, so well never meet. Where does he go? He goes to meet the next "identical" copy of myself (according to the theorem there are infinately many of us).

But there is a halfway point, i met my brother there, and not only that, but the stars "MUST" be the same upto this halfway point for every copy, as we all have the same view in the nights sky. So as we "account" for more & more complex systems, the need for greater & greater accuracy in replication also increases. The man come's to 2 conclusions.

'1)There are infinate copies of "only 1" set of stars. And NOT an infinate variety, as the theorem states.'

'2)The theorum breaks down at some point of complexity.'

(If the constraints of the universe present a problem, the idea is still valid, we could imagine an infinately fast computer, simulating every atom in an infinate universe, we still get infinately many copies of only 1 universe.)Spintronic 23:13, 19 January 2006 (UTC)

This is a fun argument, but atoms don't randomly rearrange themselves all the time, the universe has only been around for a few billion years, not nearly long enough to form even mildly interesting arrangements of atoms, and the universe (we believe) has only a finite number of particles, so there's not an infinite number of anything in it. Deco 23:15, 18 January 2006 (UTC)

Thank you!. But, the infinate monkey theorem was originally purported to support darwins theory. As such, the theory implies infinate complexity will arrise in "ANY" given system, where the "components" can rearrange themselves randomly. And i'd bet the dna molecule could pass for a "mildly interesting arrangement". The "constraints" of the universe do not invalidate this argument, as the argument doesnt rely on a physical universe with our exact properties. Hense the idea of the simulated universe at the bottom.Spintronic 23:13, 19 January 2006 (UTC)

Er, the infinite monkey theorem does not support evolution. Evolution is a directed process based on a feedback loop with random elements, but attempting to achieve life by merely randomly rearranging constituent parts would take far longer than the age of the universe. You are attempting to argue against something nobody believes anyway (and this article certainly doesn't say any such thing). Deco 19:27, 19 January 2006 (UTC)

Evolution aside for a minute, the above illustration disproves the IMT, from several viewpoints. Firstly, the illustration, need not resemble our or any physical universe. Namely "the constraints of the universe are irrelivent from a mathematical viewpoint". You could just as easily imagine our friend, being a virtual person contemplating the IMT from within an infinately complex computer simulation"

Eventually as he starts to look for himself (in the simulation) so do all the copies. This is interesting, because a computer simulation would be "'written'" in computer language, and thus 'theoretically' would be typed out by the monkeys of the theorem anyway. As their (the monkeys) typing out the sequence (computer code) where he starts to find his own copy, they would be typing an "infinately long repeated sequence", with "NO" infinate variety, as the theorem states. Anyway;

(as per the article); IMT was "alegidly" inspired by the development of an argument between Thomas Henry Huxley & Samuel Wilberforce, who were debating on the recently published "origin of species". (As the article says). Further; If evolution is the "direct process of a feedback loop with random elements", plz describe the formation of the first d.n.a strand from such a "feedback loop".Spintronic 23:13, 19 January 2006 (UTC)

I'm afraid you're misinterpreting the article. I don't know how DNA evolved, but it did. And I'm not feeding the troll anymore. :-) Deco 00:56, 20 January 2006 (UTC)

Troll? get over yourself mate. Nice to know i was talking with someone who knew hes stuff, wish i had such a blind faith.Spintronic 11:05, 20 January 2006 (UTC)

I don't think you're a troll, but you have a very firm grasp on the wrong end of the stick. The mathematical statements behind the IMT are simply facts. Their consequences may be counterintuitive, especially if you compare them to very-large-but-finite-monkey situations (eg imagining an entire universe of monkeys working for the age of the universe) but if you think the theory is wrong you should work harder to understand it. — ciphergoth 13:19, 21 February 2006 (UTC)

the answer to life, the universe, and everything

"attempting 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."

why don't we try this on wikipedia? also, has anyone ever combined random text generation (though perhaps using a dictionary) with genetic algorithms? something where each revision is human-reviewed, given marks, and then grammar, syntax and phrases from high scoring drafts are kept for future revisions, and where material from unpopular drafts is digarded?

Yes. See [Darwinian Poetry http://www.codeasart.com/poetry/darwin.html]. Derrick Coetzee 18:21, 31 Oct 2004 (UTC)

It has also been tried in music, by John Fulton, a former member of Dillinger Escape Plan. See [Astronaught/Cosmonaught http://remus.rutgers.edu/~jfulton/music.htm]. 207.34.120.71 14:31, 4 October 2005 (UTC)

Various aspects

If it does indeed take an infinite number of monkeys an infinite amount of time to hammer out Hamlet, then wouldn't it be provable that a finite number of monkeys couldn't do it -- even with an infinite amount of time? No. An infinity of infinities is still just the same old infinity. Just one infinity is enough. So one monkey and eternity, or an infinite number of monkeys and not very long at all. However, expecting an infinite number of monkeys to type on a single typewriter may not work. The same theorem of probability says any finite number of monkeys would do the job in some finite amount of time. If you specify a particular finite number of years, the probability that the job would be done before then is less than 1. It approaches 1 as time approaches infinity. The source seems to be Borel's book on probability, published in about 1910. There's also a short story based on this idea that appeared in the New Yorker c. 1940.

Borel's book also has an account of the Borel-Cantelli theorems. The title of this article may be misread as referring to "infinite monkeys," which is inaccurate. It's not the monkeys that are infinite; each monkey is finite. Rather, it is the number of monkeys that it asserted to be infinite. (Unnecessarily so, if one allows an infinite amount of time.) One should speak of infinitely many monkeys, not of "infinite monkeys."

well, the title of the page is infinite monkey. monkey is hardly plural for monkey, so i think it's a-ok. maybe just the one monkey is infinite, like in age, or like some sort of diety. also, the aspects of an imortal monkey are...amazing to say the least. oh, the applications.

If it does indeed take an infinite number of monkeys an infinite amount of time to hammer out Hamlet, then wouldn't it be provable that a finite number of monkeys couldn't do it -- even with an infinite amount of time? No. An infinity of infinities is still just the same old infinity. Just one infinity is enough. So one monkey and eternity, or an infinite number of monkeys and not very long at all. - don't you have to take into account that any ONE monkey COULD type Hamlet on the FIRST TRY? the possibility is microscopic but it exists..? Zenzizi 04:07, 19 March 2006 (UTC)

---

The title of this article as it now stands Borel's dactylographic monkey theorem does not express the concept as it it commonly known. Indeed typing the above expression into google returns zero results. If the origin of this concept resides with Borel then this should be stated within the body of the article, and the title of the article should be where people would expect to find it as per Wikipedia:Naming conventions (common names). Mintguy

This article is nice! I was just about to link dactylographic in the article to dactylography, as uncommon words should be linked for explanation, but decided to look it up first: dactylography: n. Chiefly US the scientific study of fingerprints for purposes of identification. Why do the monkeys have to have anything to do with fingerprint identification? And if this is true, maybe a short explanation of this could be put into the article. Thanks, snoyes 02:42 Mar 7, 2003 (UTC)

It seems to be a false friend - the French term "dactylographier" means "to type." - Montréalais

Is the theorem really a case of Kolmogorov's 0-1 law? The law tells us that the probability infinitely many copies are typed is either 0 or 1, but doesn't tell us which. Conversely, we know the probability at least one copy is typed is nonzero, but Kolmogorov's law no longer applies since the probability something eventually occurs isn't in the tail sigma-algebra. As I learned it, this theorem was a consequence of the second Borel-Cantelli lemma. Kevinatilusa

Re: "grotesquely incorrect". It's just grammar, chill out. Clearly, I did not mean an infinitely large keystroke. Daniel Quinlan 00:00, Dec 11, 2003 (UTC)

Counting to Infinity Before Speaking Harshly

I reverted an edit that changed a 'graph to

There need not be infinitely many monkeys; a single monkey who executes infinitely many keystrokes suffices. This is inherent in the concept of infinity.

by adding the link and the last sentence. First i must comment that there is a laudable insight in the sentence. Second, i must add that it doesn't belong in the article.

While i'm confident that i know what i'm talking about here, i see some obligation to bear in mind that i am proposing to correct others who have shown similar confidence, and i don't know the content of the Kolmogorov theorem in question, except by what i would called "inferred reputation". It is for that reason that i request someone who has studied it in a formal setting to check my understanding that:

K. addresses limits as a variable approaches infinity rather than transfinite cardinals (or or transfinite ordinals).

I intend in any case to lay out the fact that there is no "concept of infinity" for that equivalence to be inherant in, but three such concepts:

1. A dualist philosophical concept that has only a poetic relationship to mathematics, but is nevertheless durable and popular.
2. A concept of Isaac Newton and Gottfried Leibniz, fundamental to calculus, which deals with the in-fin-ite simply in terms of the absence of end, and uses "infinity" simply as a shorthand for conditions that specify the non-existence of specific limits, and the existence and numerical value of specific limits, without positing any infitite number.
3. A concept of Georg Cantor, of transfinite number, fundamental to discussion of differences in the natures of various infinite collections of abstractions, but almost independent of limit-theory.

I am convinced that this article has suffered seriously by failure to recognize these distinctions, and propose to thoroughly rework the muddled language that has called forth the idea that the seductive concept of transfinite arithmetic helps to understand this subject. But not in the next 12 hours. --Jerzy(t) 09:43, 2004 Mar 2 (UTC)

Jerzy is right. There are multiple concepts of "infinity" (his list is less than complete, but never mind that) and Kolmogorov was writing about a limit as a finite-value variable approaches infinity, as is done in calculus. Transfinite arithmetic should not be brought in to this article. Michael Hardy 20:37, 2 Mar 2004 (UTC)

... but on the other hand, I don't see that the article has suffered from exclusion of these distinctions, except when that comment about what is "inherent in the concept of infinity" was added. Michael Hardy 20:48, 2 Mar 2004 (UTC)

You can state this theorem without reference to limits. If you choose an infinite sequence of characters at random (ie a random map from the natural numbers to the characters), the probability that the map contains Hamlet somewhere in its length is one. In other words, in the space of all such maps, the measure of the set of such maps that do *not* contain Hamlet is zero. This covers the case of a single monkey with infinite time. For infinite monkeys, we consider a map from pairs of natural numbers to characters instead. I've just realised that this answers my question below! — ciphergoth 08:26, 30 September 2005 (UTC)

That's not entirely true, or at least you haven't proven it. The statement I take issue with is that the probability is one for an infinite sequence. There are an infinite number of infinite sequences that don't contain Shakespeare's works, for example an infinite repetition of the character 'a'. In fact, for any infinite sequence of characters that contains Shakespeare's works (except those cases that consist solely of a repetition of Shakespeare with perhaps a finite number of other characters), you can construct an infinite sequence that doesn't simply by deleting those sections. So there are at least as many infinite sequences without Shakespeare's works as with. I would venture to guess (though I have no proof) that the ratio of sequences without Shakespeare's works to those with them is the same as the ratio of possible sequences the same length as Shakespeare's works to one. This is assuming that there's a definative version of Shakespeare's works. Trevor 19:13, 29 August 2006 (UTC)

'Net Shakespeare Simulator

I am not an enthusiast of this activity; in fact, i think those who install it should dedicate their processor cycles to something more socially valuable, like downloading pornography. But we should document it in a way that does more justice to the limitations that it imposes upon itself.

The article invites the inference that the "records" of a dozen or so characters reflect the longest two or three words that both WS and the monkey-engine have put together. My strong impression (from an admittedly brief visit to the site) is that in fact they represent the longest matches between the engine and any string that begins a WS work (or is it a WS play? Do the sonnets count? (Does his "second-best bed" will count?!)) I'm not going to be the one to get the details right, or state them fluently, but i think the present description is inadequate. --Jerzy(t) 01:28, 2004 Mar 31 (UTC)

It's just the plays; a list of exactly which plays heads the FAQ on the site. And you're right about it just being beginnings.
I've had another shot at the paragraph in the article: better now? --Paul A 12:50, 3 Apr 2004 (UTC)

I think that's great! The only phrase that bothers me is

how long it takes the virtual monkeys to produce a complete Shakespearean play

I may be indulging my taste for excessive precision by mentioning it, but i find it a little confusing in that it fails to distinguish between the collective nature of the project (in the sense that any monkey can complete a play) and the individual nature of the completion of any single play by one monkey. I especially fear my excessiveness in this case, in that i have no alternate wording to suggest for making the distinction clearly (tho i'll sleep on it).
One question does occur to me, tho, and its answer might help: isn't one user of the program letting their machine simulate one monkey, and if so, might that focus offer a less tricky wording? --Jerzy(t)

Each user of the program is letting their machine simulate the whole room-full-of-monkeys; one user simulates the room for a while, then the next user simulates it for a while, and so on. (In practice, because it's a random process that doesn't depend on past events, the set-up allows different users to simulate conceptually-subsequent slices of the room's history simultaneously. Ignore that sentence if it's giving you trouble.)

Conceptually, each monkey's pages of typing are added to a single communal pile that is then submitted a page at a time to be checked for matches. (Incidentally, the matching rules require not only that the output match the beginning of a play, but that the play begin at the top of a new page. If a monkey starts typing out a play halfway down the page, it doesn't count.) Consequently, the goal is for the entire play to appear out of the combined output, not out of the output of any one monkey.--Paul A 08:13, 5 Apr 2004 (UTC)

Wow, way f'g cool. The elaborate design described suggests more effort than i thought on the part of the origninators, is moderating my disdain for the project. Tnx for Paul's dogged effort of research and description! --Jerzy(t) 14:39, 2004 Apr 5 (UTC)

I believe that this simulator is just a random number generator, with probabilities updated in real-time based on variables like number of monkeys, monkey years, etc. In other words, the simulator does not simulate typewriters, pages, typed texts, monkeys, and bananas. How it detects a match is like this: when the RNG generates a certain number or numbers in a certain range (the range is calculated based on the calculated probabilities), the simulator then (said that it) "detects a match" and simulates the match (generates random matched text). I also believe that the random keystroke generator is just an accessory unrelated to the simulator and has nothing to do to the RNG (it just adds level of realism). 202.65.112.42 04:10, 31 Oct 2004 (UTC)

I don't know about this simulator. I run it and start getting LOTS of "records", 28 consecutive letters every 5 or 6 seconds, 29 consecutive every few minutes. I could record it if someone recommends a program that can capture your screen to an AVI file or something (for Windows XP). --

--- WARNING: I believe this site, listed as "The Monkey Shakespeare Simulator", is flawed. First, I saw a 'record' of mine, then another, and they were of the same line of the same play. What's more, the line in the article...

Flauius. Hence: home you idle CrmS3RSs
jbnKR IIYUS2([;3ei'Qqrm'

...was also the line that set my own records. TWICE. I got past "Creatures". Some sort of seeding repeated, or maybe the generator isn't real, I don't know. Also, I set 41's every few seconds, and occasional 42's. --Falos 00:11, 22 November 2006 (UTC)

What about older precursors, like Galileo or the hermeticists?

The emphasis of this article on modern thinkers is pretty odd, considering that this is infact a recurring conundrum throughout human history. Not just Galileo, but the umpteen hundred beautiful names of god in Islamic tradition. Oh, I just remembered, maybe something on Arthur C. Clarke too... --Cimon

Huxley/Wilberforce debate

I have removed the following passage about the 1860 debate between T. H. Huxley and the Bishop of Oxford (Samuel Wilberforce), because it is so far as I can see unsustainable:

Wilberforce began the debate and, after making several scientific points regarding the plausibility of Darwin's work, concluded with William Paley’s argument that a watch implies the existence of a watchmaker, and similarly design in nature implies the existence of a Designer.

Huxley then arose and put forward his now well-known argument that six eternal monkeys or apes typing on six eternal typewriters with unlimited amounts of paper and ink could, given enough time, produce a Psalm, a Shakespearean sonnet, or even a whole book, purely by chance that is, by random striking of the keys.

In the course of his presentation Huxley pretended to find the 23rd Psalm among the reams of written gibberish produced by his six imaginary apes at their typewriters. He went on to make his point that, in the same way, molecular movement, given enough time and matter, could produce Bishop Wilberforce himself, purely by chance and without the work of any Designer or Creator.

No transcript of the Huxley/Wilberforce debate exists, but typewriters were not in commercial production at the time, and though prototypes did exist and might have been known to members of the British Association, it is unlikely that Huxley would have relied on that. No mention of the infinite monkey theorem can be found in Huxley's own accounts of the debate (for excerpts, see[1]), and the account given here differs markedly from contemporary descriptions. In my view the association of the infinite monkey theorem with the Oxford debate is an urban myth born of the fact that there really was some by-play about apes - but this was the famous exchange in which Wilberforce asked Huxley which side of his family the ape ancestry was on, and Huxley replied that he'd sooner be descended from a creature of mean intelligence than one of high intelligence who misused it in the way the Bishop had.

seglea 09:38, 30 Aug 2004 (UTC)

Gian-Carlo Rota

"Gian-Carlo Rota wrote in a textbook on probability (unfinished when he died):" but later completed by a cadre of determined simians...

Explanation of my last edit

A few months back I created the section titled "pedantic usage note" a few months ago after someone objected to the inclusion of that usage note in an earlier part of the article, and seemed to think it was unduly pedantic. I think it's useful, especially in view of some of the terminological confusion I've seen elsewhere on Wikipedia since then that could have been avoided if those who were confused had seen this fact pointed out. But now someone has said that if the article admits to being pedantic, then there's probably more pedantry "hidden [sic!]" elsewhere in it, and it should therefore not be a featured article! I think that criticism has no merit. I did not remove any pedantry, but I did remove that which may confuse the overly literal-minded: the word pedantic. Here's how that section now appears:

Usage note

To some lay persons, "infinite monkeys" and "infinitely many monkeys" may be synonymous; to the mathematician, the former is incorrect because each monkey individually is finite.

Michael Hardy 21:22, 21 Sep 2004 (UTC)

Arthur Eddington

What is the context for the quote by Arthur Eddington? In particular, what sort of vessel was he talking about, and what manner of extreme probabilities was he illustrating by invoking this concept? --[[User:Eequor|ηυωρ]] 04:43, 23 Sep 2004 (UTC)

Orthography -- explanation of my recent edit

The so-called theorem is not a misnomer; rather, the name give to the theorem is a misnomer. That name is "the infinite monkey theorem", including the definite article. If the first sentence were about the theorem itself rather than about the phrase, then of course I would not highlight the definite article along with the rest of the phrase, since it's not part of the title phrase. When writing about a phrase rather than using the phrase to write about what it refers to, one italicizes it (see Wikipedia:Manual of Style). Usually it's better to right about the thing that the term refers to rather than about the term, e.g. "A dog is an animal that barks" rather than "Dog refers to an animal that barks." When there are divergent meanings or when for some other reason it is better to write about the term rather than about the thing, one italicizes the term. And notice that of course I included the indefinite article when writing about dogs but I did not include it when writing about the word "dog"; it would be silly to say that a dog is a word referring to an animal that barks. A dog is an animal, not a word. Michael Hardy 19:56, 29 Sep 2004 (UTC)

I just changed it from

The "infinite monkey theorem" is a misnomer....

to

"The infinite monkey theorem" is a misnomer....

Here's why. A while back I read an assertion that said

Blogs are short for web logs.

That is absurd. It could have said

"Blogs" is short for "web logs".

and then it would have made sense: it is the word that is (singular!) short for "web logs". Loosely, it could have said

Blogs is short for web logs.

and I wouldn't have thougth the person who wrote it didn't understand what he was writing. But to write

Blogs are short for web logs.

looks like failure to distinguish between writing about the word and writing about the things themselves. Now what if he had written

"The blogs" is short for "the web logs".

That would have been correct. Contrast this with

The "blogs" is short for the "web logs.

That makes no sense. What is the definite article doing there if it's not part of the expression being asserted to be short for "the web logs"?

For the same reason, if this article had said

"Infinite monkey theorem" is a misnomer....

it would make sense, and if says

"The infinite monkey theorem" is a misnomer....

it would make sense. But if the word "the" is not part of the expression being asserted to be a misnomer, then what is it doing there? Michael Hardy 01:37, 31 Oct 2004 (UTC)

Infinite monkeys

Although this is not how it was originally stated, is it not the case that if an infinite number of monkeys begin typing random characters from a finite alphabet, one of them (indeed, infinitely many of them) will immediately produce any given text? Derrick Coetzee 03:16, 31 Oct 2004 (UTC)

Huh, you're right. It is guaranteed that a finite number of monkeys will produce the text given an infinite amount of time. But given an infinite amount of monkeys, then they will produce the text immediately. Might want to mention that in the article somewhere. Fishal 04:19, 31 Oct 2004 (UTC)

No it's not right. Assuming that the monkeys each type at one character per second, it will take them twenty seconds to produce all the twenty character texts and two hundred seconds to produce all the two hundred character texts. The minimum time taken for a particular text to be typed is directly proportional to the length of the text. Thus the works of Shakespeare will take much longer to appear than the works of Dashiell Hammett. I have amended the article accordingly. -- Derek Ross | Talk 18:58, 2004 Oct 31 (UTC)

This is exactly what I meant. It's just that it's hard to say it precisely without being verbose. Derrick Coetzee 19:51, 31 Oct 2004 (UTC)

Dactylography

Just to shed some more light on the term "dactylography". The origin of the word is neither French or English, it is, of course, Greek.

dactylo (pronounced thaktilo) means finger graphy (pronnounced grafi) means writing.

daktilografo is a verb in Greek meaning typing (writing with fingers).

Regards, Konstantinos

Lottery odds

19,928,148,895,209,409,152,340,197,376, roughly equivalent to the probability of buying 4 lottery tickets consecutively and winning the jackpot each time.

This feels at first like a good real-life illustration of something very improbable, but it for me it always invites the question: "which lottery"? The UK national lottery, for instance, has odds of about 14 million to one against winning, and the term "jackpot" is a bit confusing, too, because the prize is often shared among several winners.

I can't think of a better real-world example for now, though: randomly choosing one of the world's six billion people a certain number of times in a row, perhaps? I don't have the maths to know how many times... Mswake 14:43, 31 Oct 2004 (UTC)

Does it really matter if you have a perfect, concrete example or not? The words and their connotations carry the point across well enough, in my opinion. However, since you want suggestions, maybe being struck by lightning X times in a row, or dropping a bag full of coins and having them all land on their sides (vertically, I mean)? Anyway. (Sorry I don't have a username, I'll get one soon enough...) 65.94.228.36 21:10, 25 Mar 2005 (UTC)

About the Monkey

"A single monkey who executes infinitely many keystrokes will eventually type out any given text, and an infinite number of monkeys will immediately produce all possible texts simultaneously."

Um... I would have assumed that the point of using infinitely many monkeys over a single monkey typeing for an infinite amount of time is that, eventually, the finite monkey will die. func(talk) 15:05, 31 Oct 2004 (UTC)

It comes down to the Zero-One law again. There is a positive, though trivially small, chance that a particular monkey sitting down at the typewriter types everything he is supposed to type perfectly. If that monkey is immortal, has a typewriter that won't break down, an infinite amount of paper and ink, and doesn't have to worry about the Big Crunch or the Last Judgement, he has infinitely many independent tries at getting it "right". The Zero-One law then kicks in -- the probability that he will ever get it right, given this infinite number of tries, has to be zero or one. Since the probability isn't zero -- indeed the chance wasn't zero if he only got one chance -- the probability must be one.

If infinitely many monkeys sit down at their infinitely many typewriters, each has an independent (miniscule) chance of getting his "assignment" right on the first try. Since that chance is non-zero, and there are infinitely many independant attempts (each of the infinitely many monkey trying once) to do it, the chance that some one of the infinitely many monkeys gets it right on the first try must be one. Worse, as long as there are not infinitely many texts, we can put every other monkey doing Shakespeare, every other monkey of the ones left doing Faulkner, every other monkey not yet assigned doing Hemingway, every other monkey not yet assigned doing the Bible, and so on for as many works as you want to produce, and be guaranteed that in one "try", one monkey in each group will have produced his assigned text.

Diagram to explain that last bit

Monkeys
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 ...
Assigned to Shakespeare
  x   x   x   x   x     x     x     x      x     x     x     x    x     ...
Assigned to Faulkner
    x       x         x           x           x           x           x  ...
Assigned to Hemingway
        x                  x                        x                    ...
Asigned to the Bible
                 x                                              x        ...
Still unassigned
o                                       o                                ...

Pretty mind-boggling, huh? The joys of infinity! (The quoted text above is probably confusing to the reader, though. If my explanation worked for you, I could have a go at fixing it -- or if it really worked, you could fix it yourself.) Mpolo 15:50, Oct 31, 2004 (UTC)

Monkey research

Iremembered a researh of which six monkeys and six typewriters are locked up in a room. The result is monkeys banging on the typewriters and one mokey typeing the same letter over and over. ("bbbbbbbbbbbbbbbbbbbbbbbbbbbb")

unno whether i should put it. SYSS Mouse 16:41, 31 Oct 2004 (UTC)

It's already there... in "Attempts at simulation". According to the article, it was an s. Mpolo 16:54, Oct 31, 2004 (UTC)

Zero-One law

The Zero-One law is really irrelevant here. Kolmogorov's zero-one law is a deep result having to do with (Kolmogorov's own) formulation of probabilities on sequences. Here, however, we have a trivial calculation that can be done with high school mathematics, and was known (in this way or the other) to Pascal. Gadykozma 17:13, 31 Oct 2004 (UTC)

I wouldn't go so far as to say the zero-one law is irrelevent; this is certainly a demonstration of it. I've added a simpler proof from basics, though, that should hopefully be very easy to understand. Derrick Coetzee 18:24, 31 Oct 2004 (UTC)

Proof section

It should be noted that the Proof section actually calculates the probability that the monkey will type the word banana with its first letter appearing at a position divisible by six within the first 6n characters rather than just the probability that the monkey will type the word banana within the first 6n characters. While this does not affect the proof aspect of the calculation, it does make an unlikely event seem even more unlikely than it really is. -- Derek Ross | Talk 20:05, 2004 Oct 31 (UTC)

Wait a moment

If an _infinite_ number of monkeys were typing on typewriters at least one would have to produce the entirety of Hamlet immediately (or at least with a delay of however long it takes this monkey to type it out) according to the logic of the zero-one law. -- Anonymous Reader

That's what the article says, isn't it? Michael Hardy 21:42, 1 Nov 2004 (UTC)

So how long would it take you to find the monkey in question? Suppose you start to read the first few letters, discount that monkey, move to the next one. Will it take a limitless amount of time, or less, on average? N12345n 15:00, 2004 Nov 2 (UTC)

It will, of course, take a finite amount of time. Assuming monkey n is the first one who types it successfully, you'll only have to examine slightly more than n letters to find it on average, since most of the monkeys will screw up on the first letter. This n would probably be very large though. Deco 23:08, 2 Nov 2004 (UTC)

Interesting point. Assuming the article's 50 key typewriter, 2% of the monkeys will get the first letter correct, 2% of those will get the second letter correct, 2% of those will get third letter correct, etc. So you will have to examine somewhere around (n + n / 49) characters. Derek Ross | Talk 04:13, 2004 Nov 3 (UTC)

Not a proof

That isnt a proof. An informal explanation, but not a proof.

It's intended to be palatable to people with little formal background. I changed the name to Proof sketch to avoid suggesting it's a complete, general proof. Deco 23:12, 2 Nov 2004 (UTC)

Best paragraph in Wikipedia

In 2003, scientists at Paignton Zoo and the University of Plymouth, in Devon in England reported that they had left a computer keyboard in the enclosure of six Sulawesi Crested Macaques for a month; not only did the monkeys produce nothing but five pages consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it.

Bravo. Tempshill 22:51, 2 Nov 2004 (UTC)

I second that - that paragraph is so good it's worthy of a place somewhere in uncyclopedia as well. 81.133.29.244 11:09, 11 April 2006 (UTC)

wow i had no idea there was such a thing as uncyclopedia, that's for linking! I'm going to be on this page for hours and yes I agree, out of this entire article, that paragraph gave me the most enjoyable laugh. I support the moving of this article!Ironstove 01:16, 9 September 2006 (UTC)

so good I just blogged it. i've been to paignton zoo and seen those macaques - they now have an agent they seem to have got over their fifteen minutes of fame just fine. The noticeboard at the zoo didn't mention the experiment though.

Anal-retentive 'pedia

NB: the animal pictured here is a chimpanzee, which is an ape, not a monkey.

Wow. Yes, it is an encyclopedia, but come on...

Yeah, I mean, really. It's better to just leave that out so people can point at us and go "how stupid are those editors, and how can they call that an encyclopedia", right?

You should be glad Wikipedia does its utmost to be accurate. :-) JRM 19:26, 2005 Mar 20 (UTC)

I don't know, considering that whether the creature doing the typing is a monkey or not makes absolutely no difference to the theorem, it might be going a little overboard. It might be better to include it much more briefly and subtlely. I'm revising to that effect. Deco 20:03, 20 Mar 2005 (UTC)

Now that is subtle. We won't let anyone make a monkey out of Wikipedia. JRM 20:24, 2005 Mar 20 (UTC)

Infinity Generation

For a long time, I've thought about Infinity Generation - that is, the generation of every possible permutation in a given man-made system. Since such systems are by their nature, finite, they cannot have infinite permutations. However, such systems can appear to humans to be infinite, since humans are unable to perceive or comprehend infinity directly. The level to which any system could appear to humans as infinite would be known as it's granular limit - a term which I coined after thinking about how when we look at a newspaper photo from far enough away, it appears solid, but when we look close enough we see the actual dots (or granules) that form the image. The granular limit would be the limit at which we could be close enough to see and comprehend the image as one entity, but not actually perceive its granules.

Coming back to the point in discussion. I first conceptualized Infinity Generation when thinking about simple images on computers. Take for example a standard Digital Camera image of 640x480 pixels (total = 307,200) pixels. A low-colour format might make use of the GIF standard of 256 colours. To calculate all possible permutations of that Digital Camera image, one takes the number of pixels and raises it to the possible values that each pixel might have (ie: 307,200 ^ 256 = 6.0232391645008958015139896503459e+1404). With today's computing power, we could create a programme which would extrapolate all of those possible permutations of that image within a reasonable timeframe. Since an image of 640x480x256 would have an acceptable granular limit, it could easily be recognisable as any kind of object, person, whatever that exists (or doesn't exist) in our modern world. After generating every image permutation, we would have effectively generated Infinity - we would have within our images, every object, etc. that has, will, does, can't or never will exist.

Taking this one step further, we could apply the Infinity Generation to texts, such as Shakespeare. We would configure the computer programme to a suitable length (ie: The length of a Shakespearean Play), and after a while, we would have every possible permutation of that length of text. Not only would we have the Shakespearean Play, we would have every text that fitted into the configured space. Text would be a better place to start than images, since images are made up of millions (or billions) of colours, and text has only the usual 26 characters (52 if counting uppercase), plus the usual punctuation, white space, etc. If left to run long enough, the Generators would come up with every thing that was, will or is written.

Furthermore, we could expand our Infinity Generation to Films, Music and more.

At the present time, such Infinity Generators would be notoriously hard to implement without some kind of AI, since we would be like Leonardo chipping away the stone, to reveal the statue beneath - we would have to separate the 99.999999999999999 (or whatever) % of chaff, to reveal the wheat - the comprehensible stuff.

What we would do with all this stuff, even I don't know. But look at it this way - amongst all the chaff, the wheat of unimaginable power would lie - a formula for nuclear fusion, a cure for diseases. Plus all the new music, films, texts etc. Truly, from Infinity comes Creativity....

-- Matt 'devolution' Warne

Chaos Monkeys

This argument assumes that the key-mashings of the monkeys is completely random from one letter to the next. It's possible that if this experiment were tried then perhaps there would emerge discernible monkeymashing patterns that might actually preclude the creation of long, coherent, flawless texts beyond a certain threshold. Simple limitlessness is not enough; in this case you also need random variation. Maybe it would turn out that, out of every X keys hit, a monkey will always wind up hitting two keys that are immediately next to each other after a certain level of fatigue sets in. Or maybe the monkey eventually develops a preference for a certain set of keys and ignores the rest. It's possible that an infinite number of monkeys hitting a keyboard is less like pi and more like the repeating decimal .142857... It's at least possible that there are some regular patterns that would emerge that would ruin this randomness-dependent theorem.

If you can get an infinite set of random key generators, the theorem is sound. But I'm not convinced that monkeys are perfect random letter generators. It may monkeybe that there is a slight flaw in this monkeyplan. Mr. Billion 16:26, 12 May 2005 (UTC)

Problems of the Theory

if you had an infinite number of monkeys, and an infinite amount of time, wouldnt you eventualy get a room full of dead monkeys? I think the monkeys wouldnt acualy type anything, monkeys would usualy just throw the typewriters around, anyway even if they did make the entire of shakespeare, they wouldnt know what to do with it, they would probably use it as loo roll. And you also need an infinite amount of paper. I however am firmly in the believe that Shakespeares works were actualy made by Shakespeare, not an infinite number of monkeys.

By the way an infinite number of monkeys is ALOT OF MONKEYS there is a chance that SOME OF THEM will actually make SOMETHING. There would also be a circus of monkeys producing an oompa loompa style dance, throwing keyboards at one another mysteriously floating due to atmospheric anomlies and all of a sudden they spontaneously combust, and there sub atomic particles will create Shakespeares works, and this will be happening an infinite amount of times. When infinity is used anything and everything can and will happen infinitely, infinity is that mind boggelingly incomprehensibly massive. Many people have a finite veiw of infinity because the concept is just so dificult.

Sheesh. As the article says, just one monkey, with an infinite amount of time, is enough. But the mathematical proposition involved admits a precise statement, and it's not about monkeys. Michael Hardy 19:02, 23 Jun 2005 (UTC)

but nevertheless an infinite amount of time is a long amount of time, the entire works of Shakespeare could spontaneously apear even without monkeys.

Well, again, if one merely states the mathematical proposition without the fanciful stories, it's not about monkeys at all, of course. Michael Hardy 20:54, 23 Jun 2005 (UTC)

Why not just eliminate the monkeys and say that somewhere in the universe at some point in time, the collected works of Shakespeare will spontaneously materialize in some form, due to chaotic particle interaction? The problem of the theory is that is presupposes infinities that don't exist in reality, only in theory. That's why people have a finite view of infinity - cos there's no such thing outside of theory. See below. 24.213.90.49 17:03, 1 October 2005 (UTC)John

Why not just eliminate the monkeys and say that somewhere in the universe at some point in time, the collected works of Shakespeare will spontaneously materialize in some form, due to chaotic particle interaction?

Because that's about physics, and this theorem is not about physics. (And besides, the word "spontaneously" in this context would be absurd.) Michael Hardy 21:55, 1 October 2005 (UTC)

Precise statement

Nowhere in the article is there a precise mathematical statement about what this really is. There are lots of references saying "this is a particular case of <precise theorem>", but the "infinite monkey theorem" itself is never given mathematically in the article. Can this be remedied? Revolver 4 July 2005 22:46 (UTC)

Sure, that's a good idea. The article is supposed to be informal to reach a larger audience than some others, but it still doesn't hurt to be clear what we're talking about. I added some statements. Feel free to tweak them. Deco 4 July 2005 23:21 (UTC)

It's a good start. As long as a precise statement is included, you might as well go for broke: I was looking for something using the words "random variable", "almost surely", etc. In other words, it is stated that the Borel-Cantelli and zero-one theorem are generalisations of this, i.e. this is a special case of these. How is this? In other words, explain what the special case is (at the least, this should be done at the pages of those theorems, since they are at a more advanced level). Revolver 4 July 2005 23:58 (UTC)

Sure, that's a good idea. I've included a very short formal proof using the second Borel-Cantelli lemma which should be more satisfying. Deco 5 July 2005 04:12 (UTC)

For what it's worth, showing the theorem using the zero-one law would amount to showing that "the monkey eventually types the given text" is a tail event, which I'm not really sure at this moment how to do. Seems to involve some independence result. Deco 5 July 2005 04:22 (UTC)

I have a question about the first form of the infinite monkey theorem. I don't believe the E_k are truly independent. Suppose the text is "bbb", the alphabet is {a, b, c, ..., z}. Then, certainly, the event E₁ is not independent of E₂. The probability of E₁ given E₂ should intuitively be 1/26, since ("given E₂") you already know that the second and third letters are "b". Whereas, probability of E₁ in general is (1/26)³. If you restrict to blocks (E₁ = first block of 3 letters, E₂ = second block of 3 letters = 4th, 5th, 6th letters, etc.) then they would be independent. Revolver 5 July 2005 12:49 (UTC)

The explanation of how this relates to the zero-one law doesn't make sense to me, at the moment. The zero-one law articles claims that independence is not needed, yet the article here assumes independence. I'm not sure what the events E_k are supposed to be in the zero-one explanation. Revolver 5 July 2005 12:49 (UTC)

I was unclear. The kth block of n letters is not the block of n letters beginning at position k, but the block of letters beginning at position kn. Because none of these blocks overlap, they are independent. Knowing that one of these blocks equals the given text with probability 1 suffices to demonstrate that some substring does. I think you're referring to the Borel-Cantelli lemma, not the zero-one law, and independence is needed; you're confusing the first and second lemmas, which have different hypotheses and consequences (look farther down). Deco 6 July 2005 08:00 (UTC)

This clears up the independence for me, and yes I was confusing the first and second lemmas. Revolver 6 July 2005 20:03 (UTC)

I tried it....it doesn't work

don't believe it. my friend and i got exotic pet licenses and imported 500 monkeys and were able to get 500 keyboards donated.....well after a year there was not one intelligent thing typed. there were a couple words here and there but i dont know im skeptical of this whole thing. still trying to get rid of the monkeys. leave a line on my talk page if you could use a few dozen--Elysianfields 05:55, 12 August 2005 (UTC)

If you're serious, please consult the second paragraph of this article. Also, how much are those monkeys? I happen to be a monkey farmer and specialise in monkeys that have been traumatised by typing experiments. Deco 22:34, 12 August 2005 (UTC)

The benefits to humankind outweigh the trauma the monkeys suffer. Michael Hardy 23:34, 12 August 2005 (UTC)

Your mistake was to use a finite number of monkeys. Try again with infinitely many monkeys and let us know how you get on. (Alternatively you could use one immortal monkey for infinite time but who can be bothered to wait that long?) — ciphergoth 07:00, August 13, 2005 (UTC)

The useful life of the universe (at least as far as the monkeys are concerned) is approximately 1E14 years (see 1_E19_s_and_more), as after that the last stars cool off and the monkeys can neither be kept warm or fed any more bananas (since bananas require sunlight, which comes from stars). 31 million seconds x 1E14 years x 1 keypunch per second = 31E20 keypunches per monkey in the useful life of the universe (i.e. 1E14 years). Probability therefore requires 21E33 monkeys typing through the useful life of the universe just to type “to be or not to be that is the question”. The last 2 lines of this Hamlet soliloquy, “The fair Ophelia! Nymph, in thy orisons Be all my sins remember'd.” would require the work of 18E67 monkeys without punctuation; or 9E70 monkeys to reproduce it with its proper punctuation, but still without capital letters or carriage returns.

An interesting question at this point is, is it truly Shakespeare without the carriage returns? And without the capital letters? What about the stage directions?

Anyway, there is only 1E53kg of mass in the universe (see 1_E51_kg) with which to create these 9E70 monkeys, their typewriters, and (again) all the paper and ribbons and typewriter repairmen required to keep them going, and that’s assuming you can use up all the universe’s hydrogen and helium to make these things, which again begs the question of where you're now going to find the bananas.

Of course, there's likely fewer than 1E8 monkeys in the universe right now, so... well... you figure out for yourself how long it would take to breed 9E70 from a population of 1E8. Thus, the question of where you're now going to find the monkeys.

Also, you need to keep the monkeys in proximity to each other, because they're taking up all the mass of the universe and otherwise there's no gravity. No gravity would mean that each keypunch will result in the typewriter permanently flying away from the monkey (see Classical_mechanics), and given the low mass-density of the universe (under 1 atom per cubic meter), you've got pretty much no hope of ever getting another letter typed on that particular typewriter by anyone after it flies away, much less 1 keypunch per second. Yes, of course, you could tether each monkey to its typewriter, but first you have to find a tether material to last for 1E14 years. Here's a hint: don't bother trying.

The monkeys also have to be kept as close together as possible to minimize the problem of ripe banana delivery. Unfortunately, keeping 9E70 monkeys in proximity to each other for 1E14 years will mean the probability approaches one that one heck of a poop-throwing fight will wreck the piece of paper that this purported Shakespeare was typed on anyway.

And at that point, you should also consider this theory from the existentialist point of view. Read up on how hard it is on a monkey to be kept in a laboratory. Now imagine that these monkeys instead are being kept in a universe where the only thing they are able to do - anywhere in the universe - is type on a typewriter, eat a banana, bite a repairman, or throw poop at the universe. I'm sure a large number of these monkeys would commit suicide rather than be fated to such a dismal existence - certainly any who evolve a superior intelligence will! But what do you think these monkeys will do over 1E14 years? Evolve. Boy, this is looking even more doomed.

And what would such a project would entail for the entropy of the universe? Since we're attempting to convert the mass of the universe into monkeys, typewriters, ribbons, paper, typewriter repairmen and bananas, and then transporting them into one location, wouldn't that mean a terrific increase in entropy for the universe as a whole? And let's just hope that collecting all those monkeys (and typewriters!) (and bananas!) together in one place doesn't cause the whole mass of 'em to implode into one big Supermassive_black_hole, or you'll never know what they wrote.

So, the infinite monkey theorem proves to be mathematically a theoretical certainty, but at the same time a real-world complete impossibility. Shows you how useless mathematics is, eh? I'm pretty sure that all this proves that this Emile Borel fellow must have been a bit bananas.

24.213.90.49 01:44, 30 September 2005 (UTC) John Merrall Who's A Bit Too Much Of A Realist But Is Writing A Provisional Master's Thesis On Just This Sort Of Thing If You Believe What He Says So Please Don't Nick These Ideas

I have a small point... Probability therefore requires 21E33 monkeys typing through the useful life of the universe just to type “to be or not to be that is the question”. I don't think this comes out quite right. You don't need that many monkeys to have them type that, although you might need that many monkeys to make it at all probable... exactly what probability are you aiming for? Just a non-zero effective probability? I'm not sure if I can see the math on this part... I mean, all you NEED for a monkey to type that at the rate of one keypress per second is said monkey at a keyboard and 39 seconds (or only 30 if you're not counting even spaces) .... and, of course, to be really lucky. -JC 07:14, 21 February 2006 (UTC)

Probability requires that before the chances rise to something noticeable. As you say, there's a non-zero chance of it happening immediately with a single monkey. — ciphergoth 13:20, 21 February 2006 (UTC)

What is "something noticeable" though? If you had that many monkeys, what IS the chance you'd have the result of "to be or not to be that is the question"? It just says "Probability requires" but never really goes into the mathematical details of such a requirement - it seems very arbitrary that you just "need 21E33" monkeys, when you really only need one monkey for it to be possible... Either need is the wrong word, or we need to talk a little bit more about exactly what the chances become with that many monkeys and why that number of monkeys was chosen in particular. -JC 23:12, 21 February 2006 (UTC)

I can't work out whether you're pointing up a problem with the article, or just asking a question about what I've said. If the former, which part of the article is problematic? If the latter, you'll probably find it most enlightening if you do the sums yourself - they're not hard. — ciphergoth 11:42, 22 February 2006 (UTC)

It's not the article. You only said "to something noticable" - but I want to know what point that is. It's already a non-zero chance with a single monkey, right? So why does 24.213.90.49 (John Merrall) say that we "need 21E33" monkeys? That is my question. My questions here are in response to John Merrall's post (I hope that much is pretty obvious...) ... -JC 04:44, 23 February 2006 (UTC)

If the alphabet has z letters and you have m monkeys each of which types k keystrokes, then the expected number of times that a passage of length l will be typed is around mkz^{-l} (closer to m(k-l+1)z^{-l} but we usually assume that k >> l). Where that's very small it's also a good approximation of the probability of it being typed even once; where it's bigger you can use the Poisson distribution as an approximation, so the probability of it being typed even once is around 1 - e^{-mkz^{-l}}. From there you should be able to work out how many monkeys are needed to cross whatever probability threshold you want to set. I expect that the estimate of how many monkeys you "need" refers to the point at which mkz^{-l} == 1, at which point the probability of the passage being typed is around 70%, but you can now do the sums yourself and answer any such question. — ciphergoth 10:18, 23 February 2006 (UTC)

Typical mathematicians! You argue over trivia. The point I was trying to make on 30 Sep 95 (my IP address is probably different but it's still me) was, if you were able to convert ALL the available mass of the universe into REAL monkeys who typed through the useful life of the REAL universe, it would take too long to generate that many monkeys using real-world techniques, the monkeys would all go insane or starve or evolve away from monkeydom, all the typewriters would get broken, you'd run into relativistic problems, and so on. I think it's meaningless to say "there's a small non-zero chance even with one monkey", just like it's meaningless to say "an infinite number of monkeys". You don't have infinite time or mass, monkeys are non-infinite creatures in every sense of the word, and I think this might even illustrate that this idealized mathematical problem (using "let's-pretend" monkeys" and a "let's pretend universe") has no meaning because, in translation to the real world, it falls apart from ignoring the second law of thermodynamics. You might as well be talking about the mathematical chance that your smashed teacup will reassemble itself - if you say the chance is nonzero, I reply that it's so trivially nonzero that it may as well be zero. If this theory proves anything, it's that numbers close to zero behave like zero and can be assumed to be zero in the real world.71.19.38.228 12:51, 21 July 2006 (UTC)

British Museum

The monkeys can't type out the contents of the British Museum; most of the exhibits are not text. Is British Library meant here?

For now, I'll remove "others replace the National Library with the British Museum or the Library of Congress" altogether, since I don't think it does any useful work anyway. — ciphergoth 09:29, August 14, 2005 (UTC)

"all the books in the British Museum" is what the story by Russell Maloney said (cited in the article, in the "popular culture" section). Michael Hardy 01:46, 15 August 2005 (UTC)

Dactylographic

I've removed the last instance of the word "dactylographic" from the article. It used to be in here because it appeared in a translation of Borel's work, with a footnote explaining that actually, it meant something different in English to do with fingerprints. That one translator of a book in French chose the wrong word is not of sufficient note for an encyclopaedia article; we should just avoid the translator's mistake and not use the word. — ciphergoth 07:02, August 22, 2005 (UTC)

Sounds good to me. To be fair, the modern sense couldn't have existed at that time, so it wasn't so much incorrect as novel without necessity. It was a curious piece of trivia, but if this translation was neither authoritative nor particular widespread, then I suppose it's not notable. Deco 09:02, 22 August 2005 (UTC)

"if this translation was neither authoritative nor particular widespread"??? I think this particular translation of this particular book is what introduced the "infinite monkey theorem" to the English-speaking world. Michael Hardy 00:54, 24 August 2005 (UTC)

I did say "if". I don't know anything about it. Deco 01:09, 24 August 2005 (UTC)

I think that even though the translation was authoritative and widespread, the phrase "dactylographic monkeys" did not gain much currency, and so doesn't need a mention. — ciphergoth 06:51, August 25, 2005 (UTC)

Section edits

Michael Hardy, I'm afraid I disagree with all your recent edits on this article! First, I don't think "experiments" should go after "literary references" - in fact, I think "literary references" is a good note to end on. Second, though I see your point about the generalization and the misnomer thing not being part of the informal proof sketch, I don't think these deserve sections of their own. Neither can really be expanded to be a section's worth of material - in fact, they can barely get any longer than they already are. I think we should put them each in whatever existing section is the best fit for them. You're right to say that "Intuitive proof sketch" probably isn't it, but a case could be made for both going in "Formal statements".

I'll try an edit along these lines, but please note that this is a very friendly revert and I have the greatest respect for you as a Wikipedian. I'll certainly leave it be if you revert my revert. — ciphergoth 06:51, August 25, 2005 (UTC)

Informal before formal

The normal rule for a mathematics article is that the introductory paragraph should state it informally, and the first section should formally and precisely lay out the topic at hand. However, I don't think that's appropriate here. The infinite monkey theorem is worthy of an article because it is a famous and vivid popularization of a mathematical topic, not because it's of great importance within mathematics. It's not a theorem that any mathematician would directly appeal to - they would appeal to the second Borel-Cantelli lemma. The article should reflect that, and act first and foremost as a resource for those who don't want to immerse themselves in the mathematics of probability.

But, you know, I'm being bold; if you think I'm wrong revert me! — ciphergoth 06:59, August 25, 2005 (UTC)

Weird vandalism by User:80.176.134.178

This edit by User:80.176.134.178 seems to consist of chopping out bits of the article at random. This is a strange form of vandalism - why go to all the effort, when it's just as easy to detect and fix?

No, it is not always easy to detect such vandalism. Subsequent editors might often edit the article without viewing its edit history. This way, they'll inadvertently legitimise the vandal's edit. -- Sundar ^{\talk \contribs} 13:32, August 30, 2005 (UTC)

Even then, it's not too difficult to view all changes since the vandalism, revert to the version preceding it, and merge the new changes back in by hand. I think these people try too hard. Deco 21:45, 30 August 2005 (UTC)

an infinite number of times

In fact, in both cases, the text will almost surely be produced an infinite number of times.

Can someone state this formally? I can see that for any n, the probability that n copies will be produced approaches 1 as t approaches infinity, but I'm not at all sure that I'm happy with the above as an informal way of stating it. — ciphergoth 08:16, 30 September 2005 (UTC)

This is a formal statement, using the formal definition of almost surely. Click the link. Deco 02:38, 16 December 2005 (UTC)

I was waiting for that. Wouldn't an infinite number of monkeys, typing on an infinite number of typewriters, type an infinite number of bibles, or shakespeare's works? In fact, in an infinite environment, if an event is possible once, it is automatically possible an infinite number of times.

I have just typed this an infinite number of times.

However, what about imagined events? Can anything imagined happen an infinite number of times in reality? Francois Bergeron

---

...an infinite number of monkeys will almost surely begin producing all possible texts immediately, with no wait. In fact, in both cases, all possible texts will almost surely be produced an infinite number of times.

This last part is obviously incorrect, or at least misleading. If you give an infinite number of monkeys zero time, they complete zero texts. If you give an infinite number of monkeys a (tiny) finite amount of time, such that they can each produce one letter, they will each produce at most one letter. Of course we could then step in and combine all these letters to produce every possible work (although that's trivial, giving that you have an infinite number of every character), but that's not what's being said. — Asbestos | Talk (RFC) 21:30, 6 January 2006 (UTC)

To be precise, after each monkey types k characters, they will have typed all works of length k an infinite number of times. I guess I'll try to be clearer in the article. Deco 22:13, 6 January 2006 (UTC)

Deco - thanks for that, but I know what "almost surely" means. What I'm looking for is a way of formally stating "will be produced an infinite number of times."

Supposing there are n monkeys each producing n keystrokes. As n increases, the probability that a copy of Hamlet will be produced approaches 1. And the probability that two copies of Hamlet will be produced approaches 1. However, for any n the probability of an infinite number of copies is zero, because there are always only finitely many keystrokes. So how will we formally state this assertion? — ciphergoth 13:26, 21 February 2006 (UTC)

No-one managed to answer this, but here it is. Let l_H be the length of Hamlet, and H_0 ... H_{l-1} be the letters of Hamlet. For a map f from pairs of integers to letters, we say that it contains Hamlet infinitely many times if there exists an infinite set of pairs of integers S such that for all (a, b) in S and all 0 <= i < l_H, f(a, b + i) = H_i. Then we assert that the probability that a random map f contains Hamlet infinitely many times is one.

All this needs is a useful definition of measure over sets of such maps, and a proof... — ciphergoth 09:28, 14 April 2006 (UTC)

twist

Millions of monkeys evolving over millions of years produced humans, which produced all the works of William Shakespeare. And every once in a while, a distant monkey descendent even retypes some of those works.

Question

What does this sentence mean?

The theorem graphically illustrates the perils of reasoning about infinity by imagining a vast but finite number. If every atom in the Universe were a monkey producing a billion keystrokes a second for the entire history of the Universe, it is still very unlikely that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy.

I thought the whole point of the theorem is that eventually, they would write Shakespeare. Don't the quoted sentences (particularly the "very unlikely") directly contradict the "almost surely" of "A single immortal monkey who executes infinitely many keystrokes will almost surely eventually type out any given text"? zafiroblue05 | Talk 01:04, 1 April 2006 (UTC)

No, it doesn't contradict it. An infinite number of keystrokes will invariably eventually type any given text. However, a very, very large, but finite number, isn't necessarily enough (to have a sufficient chance of it, anyway). This sentence is indicating that even a huge, huge, huge number of monkeys (that is, as many as there are atoms in the entire Universe) producing huge amounts of keystrokes in a huge amount of time... still have a very small likelihood of having even one type as far as "slings and arrows" in the soliloquy (that is a bit far in the soliloquy, but this is compared to the entire works of Shakespeare! The required number of monkeys or keystrones for any kind of suitable chance of typing that soliloquy is so vast...
Although I will admit that the theorem doesn't illustrate such perils. In fact, it encourages them, because people think of an infinite number of monkeys as simply a lot of them, all typing along merrily... So I think this sentence should be rephrased to say "It is important to realize, however, the perils of reasoning..." and so on. -JC 06:56, 1 April 2006 (UTC)

Hmmm, I see. I think I just read "the entire history of the Universe" as basically being infinite time right there - maybe it should read "since the Big Bang," or something like that... zafiroblue05 | Talk 17:36, 1 April 2006 (UTC)

Have edited as you suggest - good suggestion, thanks! — ciphergoth 19:03, 1 April 2006 (UTC)

This is poetry

From the article:

"not only did the monkeys produce nothing but five pages (PDF) consisting largely of the letter S, they started by attacking the keyboard with a stone, and continued by urinating and defecating on it."

I don't care if I live until the end of time, that has to be the funniest sentence I'll ever read. J Shultz 08:49, 12 April 2006 (UTC)

Proved it wrong

I proved it wrong. I managed to prove it all wrong. The statement goes that an infinite number of monkeys, typing away for infinity will reproduce the works of Shakespare. My proof that this is untrue is the following... Be prepared to be shocked.... and awed.... at my genius.... it's just a 3 letter word....

MSN (think about it...)

Reuben —Preceding unsigned comment added by 217.22.182.184 (talk • contribs)

See this edit. But you don't have to look that far - look right here... — ciphergoth 22:35, 12 April 2006 (UTC)

The above "disproof" of the hypothesis is false because, although we'll all agree that most MSN users including myself are akin to monkeys, we are not infinite in number. So, sorry, this doesn't invalidate the Infinite Monkey Theorem at all, user 217.22.182.184, and in fact, merely proves that you are firstly a one-time MSN user for how else would you know the nature of MSN users, and secondly, you are a monkey and have managed to edit wikipedia, which is an interesting proof of the IMT. Rolinator 08:19, 14 April 2006 (UTC)

Indeed it suggests a misunderstanding of the term infinite. You cannot be almost infinite. Nil Einne 12:04, 12 August 2006 (UTC)

... sigh ... It suggests many misunderstandings: There is no need (as the article says!) for more than one monkey to be involved. Moreover, if they were infinite in number, then there would be infinitely many monkeys, but not infinite monkeys. Michael Hardy 22:24, 12 August 2006 (UTC)

Question regarding math in opening paragraph

I take issue to this particular statement:

The theorem graphically illustrates the perils of reasoning about infinity by imagining a vast but finite number. If every atom in the Universe were a monkey producing a billion keystrokes a second from the Big Bang until today, it is still very unlikely that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy.

A typical star weighs about 2x10^33 Grams, which is about 1x10^57 atoms of hydrogen per star... That is a 1 followed by 57 zeros. A typical galaxy has about 400 billion stars so that means each galaxy has 1x10^57 X 400,000,000,000 = 5x10^68 hydrogen atoms in a galaxy. There are possibly 80 billion galaxies in the Universe (some estimates say closer to a trillion as an upper estimate but I'll stay conservative), so that means that there are about: 5x10^68 X 80,000,000,000 = 4x10^79 hydrogen atoms in the Universe. But this is definately a lower limit calculation, and ignores many possible atom sources like dark matter.

Using that figure of 4x10^79 and using 10^9 keystrokes a second we arrive at 4x10^88 keystrokes a second or 1.26x10^96 keystrokes a year. Assuming the Universe is ~13.7 billion years (current estimate) and multiplying by our keystrokes/yr. calculation we arrive at 1.728x10^106 total keystrokes. A truly...astronomical...number (haha) as our lower limit. Ok, seriously though, if we look at the reference to shakespeare from above:

"To be or not to be, that is the question: Whether 'tis nobler in the mind to suffer The slings and arrows"

We arrive at 81 letters without punctuation. I'd like to see a breakdown of how ~1.728x10^106 keystrokes is insufficient to perform that operation with some certainty (let's say odds of 50% or greater) and what the specific probability would be. As far as I can tell from Ciphergoth's explanation above, it should be more than sufficient, even with puncuation. J Shultz 05:23, 13 April 2006 (UTC)

• • • edit: my other question was the type of keyboard used...how many keys are there etc.?

Are we also considering a 101 Standard keyboard, or one with navigation and shortcut and F1-F12 buttons as used on a computer? Maybe the advent of computers would substantially reduce the probability of a monkey completing the Soliloquy by several orders of magnitude? :P Rolinator 08:15, 14 April 2006 (UTC)

Perhaps it means starting from the beginning of the play? -JC 08:26, 13 April 2006 (UTC)

No, I meant from the beginning of the soliliquy, and letters only. Your estimate of total keystrokes roughly agrees with mine. However, I get that there are 26^81 ≈ 4x10^114 possible 81-letter sequences, which is much higher, so the probability of that sequence being produced is very low - around 4x10^-9, because when the overall probability of success is very low you can approximate it pretty well simply by multiplying the probability of success at each instance by the number of instances. — ciphergoth 09:31, 13 April 2006 (UTC)

Ok, thank you. What a silly but fun exercise. J Shultz 20:20, 13 April 2006 (UTC)

• sorry one other note, would it be a possible improvement in the intro to give a concrete number as shown in the modified line below;

"...from the Big Bang until today, it is still very unlikely (roughly one chance in 500 million) that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy."

It isn't important, I just thought it was such a ludicrous statement and having a real number of the chances there further demonstrates the impossible number of monkeys required to perform these operations. J Shultz 20:33, 13 April 2006 (UTC)

"Due to the nature of infinity"?

Why at the beginning it is specified "due to the nature of infinity"? It seems useless, non mathematical and difficult to understand.--Pokipsy76 15:14, 11 June 2006 (UTC)

Greatly cut down introduction

I started making a small edit and ended up making a big one - sorry! Call it being bold :-)

As we now tell it, Borel didn't posit the Theorem, and it seems we don't know who did; it seems to be just a part of popular culture.

Given that, there's no reason to give such prominence to Borel. So I've moved that into a section of its own, credit the original image of many monkeys, many typewriters, and a great deal of time to him, but point out that he and Eddington were actually making the opposite point - that even given resources that seem vast on a human scale the probability of typing Shakespeare is miniscule, and the probability of a significant violation of the Second Law is even smaller than that.

We then implicitly credit the extension of the idea to infinitely many monkeys and infinite time, and the resulting reversal of the conclusion that given this Shakespeare is almost surely produced, to popular culture, since we don't actually know who posited this. Overwhelmingly, the popular culture version of the Theorem which is what the article is about references Shakespeare, so I follow that example. If someone can show that the earliest examples of this idea which reference infinite resources don't reference Shakespeare but some other work, I'd be very interested. I wish we had a citation for "After about 1970".

I think this change improves the flow of the article and de-clutters the introduction, but let me know what you think. — ciphergoth 22:24, 11 June 2006 (UTC)

Did you remove the text concerning the explanation that the monkeys are just a metaphor for an abstract mechanism for generating random strings? This may seem completely obvious, but I think there has been enough confusion on this point that it's worth spelling out. Deco 21:48, 21 July 2006 (UTC)

Worst first sentence ever.

>The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite time will eventually almost surely type the collected works of William Shakespeare, if not every single piece of literature that has ever, and will ever be written.

I understand that "almost surely" has a particular meaning in probability theory (or whatever) but this sentence, in which the adverbial "almost surely" is given with the adverb "eventually", well, surely blows completely.

I tried substituting "one day" but my edit was removed.

How about considering a rewrite?

The only difficulty remaining is in locating the successful monkey.

This is actually not the only difficulty. There is also a chance that you won't find any, even if you've got the perfect shakespeare locator. That's what you gotta love with probability, "at infinity", every single event will have a measure of zero, but that doesn't mean that it doesn't exist.

A chance you won't find a successful monkey IS the difficulty of finding the successful monkey in the first place. I think (and I have put) this sentence back in the article as it is another way to visualize the IMT. 09:18, 29 November 2006 (UTC)

42!

When Arthur and Ford are picked up by the Golden Heart in H2G2 they remain stuck in an improbability field (at a probability of ∞-1) where all manner of bizarre things happen, such as Ford turning into a penguin and Arthur's body warping and stretching. Also they are visited by an infinite number of monkeys with ink stained hands who wave a copy of Hamlet at them. I personally believe this qualifies to be included in the pop culture references in the article, no? 04:16, 18 August 2006 (UTC)

The radio version goes

F/X LOUD GIBBERING OF MONKEYS

ARTHUR: Ford, there's an infinite num,ber of monkeys outside who want to talk to us about this script for hamlet they've worked out.

the improbablity factor given is "Two to the power of fifty thousand to one against and falling", rather than infinity, but then it was only Hamlet.

Almost surely

I've restored the "almost"s to the "almost surely"s that appear throughout the article. I cannot imagine why they were removed; without them the article is inaccurate. Also the Infinite Monkey Theorem is very different from the Random Walk theorem. — ciphergoth 15:49, 16 September 2006 (UTC)

mistake (no mistake, done)

The article states: If every atom in the Universe were a monkey producing a billion keystrokes a second from the Big Bang until today, it is still very unlikely that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy.

In my opinion this is just wrong. That part of the text would be less than 100 characters; In addition, there are less than 40 relevant keys for text on a typewriter keyboard. This means the probability of the monkey typing the correct beginning of the soliloqui is plain and easy ${\frac {1}{{40}^{100}}}$ or better ${\frac {1}{{4}^{101}}}$. The number of atoms in the unverse is said to be maximum $6*10^{79}$. The age of the universe is said to be maximum 13,9 billion years, that's $4.39^{17}$ seconds. A billion keys is $10^{9}$. If you miltiply that, you get a total of $25*10^{105}$ or $2,5^{106}$ keystrokes for our many monkey-friends. This means, they type in 100 characters $2,5^{104}$ times.

This means: In one of $4^{101}$ trys, our moneys type in the right 100 characters. I just calculated, that our monkey has a total of $2,5^{104}$ trys. I am not totally sure, if i calculated correctly (please tell me if i am wrong), but as far as i can see this is far away from "very unlikely". Nerdi 17:42, 18 September 2006 (UTC)

40¹⁰⁰ and 4¹⁰¹ are two very different numbers. If you really want base 4, 40¹⁰⁰ is not far from 4²⁶⁶ —David Eppstein 18:07, 18 September 2006 (UTC)

40¹⁰⁰ is about 10¹⁶⁰. The age of the universe is about 4.19 10¹⁷ -> 2.5 10¹⁰⁶ keystrokes. When your machine write an exponenent, it is the exponent of 10 multiplying the mantissa. pom 18:27, 18 September 2006 (UTC)

At the very least, though, the term "universe" needs to be qualified. To my knowledge there is no consensus upper bound on the number of atoms in the universe, nor even on the question of whether that number is finite. If the passage specified that it's talking about the visible universe, it might be defensible (though it's kind of OR-ish). --Trovatore 20:04, 18 September 2006 (UTC)

That's an estimate for the number of atoms in the entire Universe, as predicted by current theory: see [2]. I think the idea of an infinitely massive Universe is pretty much dead in current cosmology, since it contradicts the Big Bang; what makes you think there's still doubt on that score? — ciphergoth 21:37, 20 September 2006 (UTC)

The math is worked out in "Question regarding math in opening paragraph" above. Interesting that this keeps coming up - I hadn't expected that paragraph to be so controversial! — ciphergoth 21:46, 20 September 2006 (UTC)

The citation does not really state that 6e79 is a hard upper bound. Also it's a bit of an informal, chatty kind of site; I can't take it as establishing a scientific consensus. As I understand it, if the average density of matter in the universe is below the critical density, then the universe has infinite volume. Therefore, if it has positive average density, it must have infinite mass. I don't think this contradicts the big bang; if you run the equations backwards from such a universe, you still wind up with infinite initial density a finite amount of time in the past. --Trovatore 22:03, 20 September 2006 (UTC)

I'm a bit rushed right now, but please not that the use of an asterisk for ordinary multiplication in TeX is like putting catchup on steak. TeX is sophisticated; you can write

$8.34\times 10^{-38},\,$

etc. Michael Hardy 21:15, 18 September 2006 (UTC)

---

Thank you! I correct myself: In one of $4^{266}$ trys, our moneys type in the right 100 characters. It was calculated, that our monkey has a total of $2,5^{104}$ trys. I am not completely sure, but i can see this is very close to "very unlikely". Nerdi 19 September 2006 (UTC)

ciphergoth's cite; finiteness or infiniteness of universe

Ciphergoth offered a website which he claimed put an upper bound on the number of atoms in the universe. If you look at the part of the site most directly addressing the question, which is here, you'll see that it says the following:

The radius of the visible Universe is estimated at 1.7e26 m (18 thousand million lightyears) plus or minus 20 percent or so.

and the calculations go on from there. Note the use of the term "visible Universe". It's true that in the rest of the passage, the author simply calls it "the Universe", but this is apparently convention among astrophysicists (at least, so our observable universe article would have you believe. So I think this citation provides no support at all for the claim that there's a consensus upper bound on the number of atoms in the whole universe. --Trovatore 23:10, 20 September 2006 (UTC)

Regarding Internet Simulator

Although the site only keeps a record up to 24 letters, I felt like checking it out real quick, and my record was 41 letters. This wikipedia article states that it generates AT MAXIMUM 24 letters; should this be changed?

After 2.55116e+79 pages in this session, a monkey typed:

       GLOUCESTER. Now is the winter of our disc?l7QU"G 2&)]4c9JWg2cW'B1ZmV5vqB...

the first 41 letters of which match "KING RICHARD III": _________________________________________________________________________________ GLOUCESTER. Now is the winter of our discontent Made glorious summer by this sun of York; And all the clouds that lour'd upon our house 202.45.98.254 13:27, 25 September 2006 (UTC)

Changing the emphasis

We can simplify the introduction by considering the other infinity instead. If a single monkey is placed in front of a typewriter for an infinite time, the monkey will probably die before it types a single word of Shakespeare. Now, if you wait long enough then another monkey-like creature might eventually rise from the ashes and start to type Shakespeare, but it's a bit counter-intuitive; that's why our monkeys are abstract devices rather than real monkeys.

However, if you place an infinite number of real monkeys in front of an infinite number of real typewriters, then even though most will just urinate on the keyboard before wandering off, infinitely many of them will immediately type the works of Shakespeare in chronological order.

Thus if we consider infinite monkeys and finite time rather than vice versa, we can remove the caveat about the monkeys being abstract machines.

We'd have to add a paragraph to the proof sketch to handle this of course.

— ciphergoth 22:55, 26 October 2006 (UTC)

Problems with this FA

There are alarmingly few inline citations for this Featured Article, and I have not been able to find sources for the claim that "The theorem as it is now stated, with infinite resources, arose in popular culture after around 1970". Lastly, the Infinite Monkey Project in the last paragraph is also unsourced. I think this article is far from being a FARC candidate (at least I hope so), but the possibility of it losing its featured status is real. Kavadi carrier 14:41, 9 November 2006 (UTC)