Talk:Infinite monkey theorem

Why is the part about a monkey pissing on a keyboard on this page?

It's the last paragraph. What does this have to do with anything?

-G

Some researchers gave some monkeys a keyboard to see what they'd do with it and, well, they peed on it. RAmen, Demosthenes 21:28, 15 February 2007 (UTC)

Zero or one?

The Probabilities section states:

The text of Hamlet, even stripped of all punctuation, contains well over 130,000 letters which would lead to a probability of one in 3.4×10¹⁸³⁹⁴⁶... The mere fact that there is a chance, however unlikely, is the key to the "infinite monkey theorem", because Kolmogorov's zero-one law says that such an infinite series of independent events must have a probability of zero or one. Since we have shown above that the chance is not zero, it must be one.

Couldn't one just as easily argue that, "since we have shown above that the chance is not one, it must be zero"? -- noosphere 22:59, 26 November 2006 (UTC)

Well, you could, if you had shown the probability is not one. But you haven't. Good thing too, since in fact it is one. --Trovatore 23:29, 26 November 2006 (UTC)

Per the above, the probability the quoted argument was referring to was "one in 3.4×10¹⁸³⁹⁴⁶", which is not one. -- noosphere 04:14, 27 November 2006 (UTC)

That's the probability per trial. The zero-or-one is the probability of at least one success, given infinitely many independent trials. --Trovatore 05:01, 27 November 2006 (UTC)

Then where in the Probabilities section is there a demonstration that the probability of at least one success, given infinitely many independent trials is not zero? I only see mention of the probability per trial. -- noosphere 06:19, 27 November 2006 (UTC)

Well, if you perform infinitely many trials, you've certainly performed one trial. So the chance of success in infinitely many trials can't be less than the chance in a single trial, which is calculated as "one in 3.4×10¹⁸³⁹⁴⁶" (copied from your text above), and that is greater than zero. --Trovatore 06:22, 27 November 2006 (UTC)

That's convincing. But it's not part of the argument given in that section. Perhaps it would make the article clearer if something along the lines of your argument should go in to that section instead of having it claim that "we have shown above that the chance is not zero" when no such demonstration is present. -- noosphere 07:05, 27 November 2006 (UTC)

No proof gives every detail; they'd be unreadable. But in any case the proof by direct calculation, which doesn't need the zero-one law as a black box, is given earlier in the article. The separate proof as an application of the zero-one law is a debatable organizational choice (one of the problems of article-by-committee) but I don't think that leaving out that particular step is its biggest problem. --Trovatore 07:16, 27 November 2006 (UTC)

For the record, this discussion refers to this text, which has since changed dramatically. To be precise, I changed it.

From what I gather of David A. Williams' Probability with martingales and of our article Kolmogorov's zero-one law, the probability of "at least one success" is not the kind of "tail event" to which the law applies. It applies only to the statement that a given text occurs infinitely many times, whose probability is not a priori bounded below by a positive number. After Williams admits that the zero-one law solution to the monkey problem does not differentiate between 0 and 1, he advocates using the second Borel-Cantelli lemma to prove that the probability is 1 after all. This is done in our article.

Anyway, I removed both references to Kolmogorov in the article, so there is no longer a problem. Melchoir 08:25, 3 March 2007 (UTC)

Interesting; thanks for tracing that down. The proof sounded reasonable and the conclusion is true, so I hadn't bothered to check the conditions for the application of the zero-one law. This outcome vindicates both Noosphere (that the proof was invalid) and me (that leaving out the step Noosphere objected to was not the biggest problem in the exposition). --Trovatore 08:54, 3 March 2007 (UTC)

And I'm mostly interested that the outcome vindicates me (in that reasonable-sounding things still demand verification). Melchoir 09:08, 3 March 2007 (UTC)

The paragraph at the top is certainly terse, and the application of the zero-one law is redundant, but that doesn't mean that there is no relationship between the zero-one law and this topic. The original author probably had an interpretation like this in mind: Let the random variable X be such that an observation of X requires randomly generating exactly as many letters as the length of Hamlet. The outcome is X = 1 if the text generated is Hamlet, X = 0 otherwise. In short, break up the typewriter output into consecutive blocks and replace each block with 0 or 1 depending on whether it equals Hamlet.

The calculation in the article showed that the probability that X = 1 is positive, and so the probability that X = 0 is not 1. Now think about infinite sequences of observations of X and put the usual measure on these using the probabilities of X. The set of such sequences that contain infinitely many ones is a tail set (so the zero-one law does apply), although the measure of its complement is easy to compute (it's zero) and so there is no need to apply the zero-one law to find out the measure of this set is 1. Its complement is the set of infinite sequences that contain only finitely many 1s, and this is the countable union of a sequence of measure zero sets, one measure zero set for each possible final location of X = 1 in the sequence. There is also no need to apply the Borel-Cantelli lemma, since this is a direct calculation. CMummert · talk 13:02, 9 March 2007 (UTC)

Yes, in that sense the infinite monkey theorem's result is certainly consistent with what the zero-one law tells us about the problem. The Borel-Cantelli lemma is more relevant to the article than the zero-one law, even though they are both redundant, because the former gets us p=1. Melchoir 19:02, 9 March 2007 (UTC)

Erm...

Would it still count if they typed out Christopher Marlowe instead? Totnesmartin 23:06, 28 November 2006 (UTC)

I think the last 3 paragraphs of Intuitive proof sketch should be deleted. They really aren't relative to the theorem as the theorem isn't concerned with capitalization or the possibility of jamming a typewriter.161.184.194.100 09:09, 12 December 2006 (UTC)

I think the last three paragraphs are essential to the article, not only because it's hilarious, but also because it shows the reality of the expression. People want to know if it has been proven in reality. poopsix 02:32, 5 January 2007 (UTC)

Probability section removal

I removed the following text from the Probability section:

But the problem as stated ignores "boundary" counditions. It is physically impossible to breed and sustain infinitely many monkeys (even if only "theoretical"), and similarly impossible to provide immortality to any of the monkeys, or to keep the monkeys on task for the required time. But of even more consequence, no machine is self repairing, nor has a life cycle sufficient for the problem, thus the probability is 0, not 1.

It is impossible to "re-create" an art-form of reasonable complexity with any random-process; if it were not so, there would be no such thing as plagiarism, nor methods of determining document authenticity.

Even if one could randomly create meaningful artistic works, there still needs to be judgement processes to determine what is of value that should be kept, and what is worthless that should be eliminated.

For a good computer program random text generator on the fastest computers available, the "answer" is that the machine is in all likelyhood going to "crash" before producing the desired results, even if the program takes into acount word probability and grammar logic.

The first paragraph fails to recognize that we are not talking about a specific incident, but instead an abstract idea. The second is just wrong; it negates the entire article. The third is an opinion statement, and the fourth goes back to talking about a specific incident again.BlueSoxSWJ 17:26, 15 December 2006 (UTC)

guaranteed to occur, or only almost guaranteed to occur

2007-01-09, 16.19 Ason: "(→Probabilities - Removed "almost" in: "To consider that an event this unlikely is almost guaranteed to occur given infinite time can give a sense of the vastness of infinity.")"

2007-01-09, 19.30 Trovatore: "(but it *isn't* guaranteed. It just has probability one. That's better than any guarantee you'll ever get in this physical life -- but this is mathematics.)"

An even that is unlikely to happen during a definite time, but not impossible, WILL happen if given infinite time. Actually it will happen an infinite number of times, regardless of how unlikely the event is. Therefore it CAN be said that the event is guaranteed to occur given infinite time. If you don't like the word guaranteed, you could say the event WILL occur given infinite time.

Using the word almost, implies that there is a probability that the event will never occur, and THAT is wrong.

Since I think editing wars are counterproductive, I will not edit the article untill Trovatore or someone else has had reasonable, but not infinite, time to comment.

Ason 11:48, 18 January 2007 (UTC)

I agree. Given infinite time, the probability that the event will occur is 1. That means it is certain. --Aprogressivist 16:19, 18 January 2007 (UTC)

The issue is semantically confusing. I need to grok the semantic difference between mathematical "almost sure" and such ambiguous terms as "guaranteed", "almost guaranteed", "certain", etc. --Aprogressivist 16:33, 18 January 2007 (UTC)

Having read up on the Probability article, it seems consistent to say that the event is certain to happen. The following three phrases are synonymous: P(X) = 1; X is almost sure to happen; X is certain to happen. --Aprogressivist 16:40, 18 January 2007 (UTC)

An event that is certain to happen has probability one. The converse, however, does not hold.

Suppose I pick a real number at random between 3 and 4, with the uniform distribution. Given any particular real number between 3 and 4, what's the probability that I pick it? That probability can only be zero. So the probability that I won't pick it, is 1.

But if you say that it's certain that I won't pick that number, and generalize that argument to all the reals between 3 and 4, you must now conclude that I am certain not to pick any number at all. But that's a contradiction; we assumed that I would pick such a number. --Trovatore 19:31, 18 January 2007 (UTC)

I don't think your generalisation step is valid. I'll work on that in a moment. Besides, as I noted, certain and almost surely are used synonymously in the scope of the Probability article and the Almost Surely article (both mean P(X) = 1). It is therefore consistent to use them interchangeably; if it is inaccurate, it would seem the problem extends beyond the scope of this article. --Aprogressivist 21:18, 18 January 2007 (UTC)

Look at the probability article more carefully, with particular attention to probability#Representation and interpretation of probability values:

The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".

The only thing I can see in the article that might have led you to the conclusions you state is the stuff about Laplace. Laplace was great, but things have moved on a bit in the intervening 230 years. --Trovatore 21:28, 18 January 2007 (UTC)

I concede that certain is not synonymous with almost surely. That being said, to the layperson, I suspect the phrase almost surely is much weaker than it is to the mathematician; it certainly seems so to me. I think the original sentence ("To consider that an event this unlikely is (SOME_QUALITATIVE) to occur given infinite time can give a sense of the vastness of infinity") losses some of its impact to the layperson if it includes the word "almost". I attempted to compromise between impact and accuracy with 'certain (mathematically "almost surely")'; how does that sound? --Aprogressivist 22:10, 18 January 2007 (UTC)

No, I don't agree with compromising on accuracy. It isn't certain, so we can't say it is. --Trovatore 01:26, 19 January 2007 (UTC)

Bear in mind that Wikipedia is not a resource for mathematicians alone, but a resource for the layperson, who will not immediately grasp the semantic distinction between certain and almost surely. The phrase with almost surely has not much strength to the lay person if it is accurate to the mathematician; and if it is accurate to the mathematician it is merely repeating what has already been said in the article already. If you don't wish to compromise, it would seem best to me that it be deleted entirely, because it is somewhat redundant to repeat a mention of almost surely. --Aprogressivist 09:27, 19 January 2007 (UTC)

P.S. It seems we have reached the same conclusion. --Aprogressivist 09:30, 19 January 2007 (UTC)

It remains to be shown, however, that the infinite monkeys problem is not certain. I do not believe this is correct; I think Ason's original contention stands. More succinctly put, given a particular event with a result of nonzero probability, that result is certain to occur in an infinite number of events. --Gnassar 15:51, 25 February 2007 (UTC)

No, that's simply not correct. All you can show probabilistically is that the result has probability 1 of occurring; probability theory has no hope of showing that the outcome is certain. And in fact it is not certain, if the trials are truly independent (as opposed to just probabilistically independent), because if they are truly independent, then any combination of possible outcomes of the individual trials is a possible outcome of the whole (infinite) experiment. --Trovatore 23:15, 25 February 2007 (UTC)

I don't believe that's the case. There is no "any combination of possible outcomes" of an infinite number of trials. All finite combinations of possible outcomes would occur in an infinite number of trials. But defining the infinite set of "combinations" of an infinite number of trials is an overexpansion of choice. —The preceding unsigned comment was added by Gnassar (talk • contribs) 17:22, 6 March 2007 (UTC).

Well, it's standard in modern probability theory, whether you like it or not. --Trovatore 17:53, 6 March 2007 (UTC)

Merging "Infinite monkey theorem in popular culture" article

Should the article Infinite monkey theorem in popular culture be merged with this article? (Discuss). --Aprogressivist 16:15, 18 January 2007 (UTC)

This is a social meme that has grown into a Wikipedia Math FA aritlce. If it cannot be moved to the "Media" group in the FA listings, then I would rather this article be merged into the other.--70.231.149.0 03:09, 4 February 2007 (UTC)

Ignore

Ignore. Qpw 13:47, 25 January 2007 (UTC)

Appeals To Intuition

Although I applaud the efforts of whoever it was in writing the intuitive proof of the theorem, I have to say I find the version given here rather unappealing. Essentially it demonstrates that although every finite sequence within an infinite sequence is possible (and indeed highly likely), no finite sequence is almost sure. In that sense it actually disproves the theorem.

An equally appealing disproof of the theorem which exploits the same probabalistic argument would run along the lines of: As the monkey types, the probability that the next key to be hit is the first letter of the target sequence doesn't change, and is always strictly less than one. Therefore it's always possible the monkey will hit a key other than the first letter of the target sequence and therefore there's no certainty that the monkey will start typing the target sequence, and therefore no certainty that it will produce the complete target sequence. Therefore although it's possible the monkey will produce the target sequence, it's not certain that it will. Even allowed infinite time, the probability that the next key to be hit is the start letter of the sequence doesn't change and remains less than one, and therefore the monkey could type forever and never hit the starting letter key.

To be honest I'm not sure how to reword the appeal to inuition to make it more robust, but I'm not sure, as it stands, if it persuades a casual reader of the proof of the theorem. Fizzackerly 13:21, 29 January 2007 (UTC)

Big Bang math checks out

This discussion which for some reason has been removed from this page thrashes out the math for this in some detail; it checks out. I'll remove the fact tag unless someone wants to dispute the linked math. — ciphergoth 19:30, 14 February 2007 (UTC)

A picture

In case you guys want it, here's a screen cap that I upoloaded for Last Exit to Springfield. It shows Mr. Burns' room that is filled with a thousand monkeys working at a thousand typewriters. I figured I'd let someone more familiar with the page put it on if they want to. -- Scorpion 20:38, 15 February 2007 (UTC)

Cicero

Searching "considers this possible will also be able to believe" on Google Web, Books, and Scholar returns only Wikipedia and its mirrors. I for one would like to know where the quotation comes from. Melchoir 23:03, 1 March 2007 (UTC)

…Thanks! Melchoir 00:13, 2 March 2007 (UTC)

1970??

A couple of days ago, I deleted a truly bizarre assertion that the version of this proposition that involves infinitely many monkeys orginated in about 1970. /BEGIN ATTITUDE/Aside from the fact that only a lunatic would think that,/END ATTITUDE/ check out Nevil Shute's novel On the Beach, published in 1957, via Google Books. Michael Hardy 05:01, 5 March 2007 (UTC)

Thanks for the info! The article history indicates that "1970" was not part of the article when it was featured in October 2004. It was added by an anonymous user in October 2005. It lived in the lead section before Ciphergoth moved it into the body in June 2006, where it then remained for another 8 months or so. This all just goes to show what kind of edits accumulate upon featured articles without careful supervision. When I overhauled the article a few days ago, it was in one of the many paragraphs that had been tagged as needing citation, and I had hoped to save as many as possible. But no sources turned up, and now we know why: it was wrong! Melchoir 07:14, 5 March 2007 (UTC)

So what is the answer?

While the article contains a short discussion of probabilities, it fails to provide an actual answer to the question, "given n typing monkeys, how long will it take to produce the complete works of Shakespeare?". — Loadmaster 23:57, 5 March 2007 (UTC)

More than 10^100000 and less than infinity, regardless of all details. Melchoir 00:26, 6 March 2007 (UTC)

More than 10¹⁰⁰⁰⁰⁰ what? Assuming n monkeys typing at rate r, there is an exact answer. What is it? And why is the answer not in the article? — Loadmaster 18:58, 7 March 2007 (UTC)

"All details" includes the unit of time; more than 10^100000 seconds, years, universe ages, whatever. There are, of course, exact answers, but none of the sources I've read see a need to calculate them to any greater precision. Math books are content with "less than infinity", and physics books are content with "greater than any reasonable period of time". Melchoir 21:41, 7 March 2007 (UTC)

That's not much of an answer is it? "Less than infinity" is any real number - utterly useless as a meaningful answer. I'll say it again: given n monkeys typing at rate r, the article does not provide an answer to the question, "how long will it take them to produce the complete works of Shakespeare?" Surely this is relevant to the article. The closest answer provided is in "Probabilities", which only mentions the text of Hamlet. — Loadmaster 18:51, 8 March 2007 (UTC)

If you're so sure that this is a worthwhile calculation, then you must be confident that it's in the literature. Why don't you go look for it? Surely I'm not the only one with access to Google books around here. Melchoir 19:31, 8 March 2007 (UTC)

Non sequitur. Wikipedia could very well be the first place to provide a definitive answer to the question. — Loadmaster 23:31, 11 April 2007 (UTC)

It could, but it's not supposed to. That's one of the basic ideas of Wikipedia -- it is a tertiary source, and is not supposed to provide new primary sources for anything. --Trovatore 23:49, 11 April 2007 (UTC)

This is...

most definatly a joke. Even with all the external sourcing... I can't help but LOL at it. Quatreryukami 15:46, 7 March 2007 (UTC)

Uh yeah, it kinda sounds not impossible, but definetly improbable. Wikizilla ^{Signme!Complaints Dept.} 01:26, 2 April 2007 (UTC)

External links

I gave my reasons for deleting some of those links and moving and incorporating the others. Are there reasons to keep them now? Melchoir 21:51, 7 March 2007 (UTC)

• No. See WP:EL and WP:NOT. Most of them don't rise to the level of WP:RS, and the sources given cover the necessary info. I deleted most of them again. SandyGeorgia (Talk) 02:30, 8 March 2007 (UTC)

Redirection from De Natura Deorum

Why does the link De Natura Deorum on the works of Cicero page get redirected to this page ?

Is this page really relevant ?

Not relevant enough. That redirect should be deleted. Melchoir 16:50, 19 April 2007 (UTC)

Thought not. I've eventually managed to find out how to add to the RFD page Simonadams 17:29, 20 April 2007 (UTC)

Real Monkeys section

I believe some parts of the Real Monkeys section are fundamentally flawed because the monkeys where not sufficiently repressed from the start by years of education. Many humans, if given the chance, without being fired or punished, would react the same way towards a keyboard. Let us not defame monkeys and act like we're civilized when we're really repressed.(This is all sarcasm in case some hominids don't get it).Septagram 02:24, 5 May 2007 (UTC)

Is this Article a joke?

This is completely ridiculous. Especially the part about real monkeys. Repressed Monkeys? For Christ's sake, they're monkeys! Monkeys can't be repressed. This is a great example of why people don't take Wikipedia seriously - not only that, but some admin must be having fun with this page too, because this is definitely not one of Wikipedia's best articles. LordArros 15:04, 11 May 2007 (UTC)