A Tiny Mathematical Proof Against Evolution [AKA - Million Monkeys Can't Type Shakespeare]

Nutters.org ^ | 13-Dec-1995 | Brett Watson

[Southack]

There is a recurring claim among a certain group which goes along the lines of "software programs can self-form on their own if you leave enough computers on long enough" or "DNA will self-form given enough time" or even that a million monkeys typing randomly on a million keyboards for a million years will eventually produce the collected works of Shakespeare.

This mathematical proof goes a short distance toward showing in math what Nobel Prize winner Illya Prigogine first said in 1987 (see Order Out of Chaos), that the maximum possible "order" self-forming randomly in any system is the most improbable.

This particular math proof deals with the organized data in only the very first sentence of Hamlet self-forming. After one examines this proof, it should be readily apparent that even more complex forms of order, such as a short story, computer program, or DNA for a fox, are vastly more improbable.

So without further adue, here's the math:

[Southack]

The Mathematics of Monkeys and Shakespeare

or "Monkey Claims Copyright on Hamlet: Film at 11."

A constrained rant by The Famous Brett Watson, 13-Dec-1995.

Introduction

I don't know who it was that first talked about the possibility of monkeys typing randomly on typewriters producing Hamlet entirely by chance, but it is an argument that I have often heard. "Sure it's unlikely," I'm told, "but given enough time and enough monkeys, it would happen."

This argument is actually quite sound -- given enough time and enough monkeys, one could eventually produce "Hamlet" by accident. The fact that it is intuitively sound is the argument's greatest problem, because it means that people generally don't bother checking the exact figures. This is a shame, because it is one of those rare areas of speculation where the exact figures can be calculated.

The other problem is that large numbers are rarely understood. Allow me to tell a little joke to demonstrate.

The Orders of Magnitude Gag

During the US government's "Strategic Defense Initiative" program, better known as "Star Wars", leading scientists on the project were asked to report their progress to the Minister of Defense. So they gathered their data and brought it together in a presentation. In short, they had discovered that the whole problem of shooting down a nuclear missile at such a great distance was a very tricky problem indeed. They intended to explain to the minister what an impossible problem it was -- well beyond the capabilities of current technology. During the course of their presentation, the following exchange took place.

SCIENTIST: ...and so you can see, Mr Minister, that in order to achieve an acceptable hit-rate against the missiles, our instruments need to be accurate to one part in ten to the ninth. So far, the best we have been able to achieve is one part in ten to the fifth.

MINISTER: That's tremendous! We're over half way there!

Our poor politician completely failed to understand the meaning of the scientist's statement. Ten to the fifth is not more than half of ten to the ninth. Ten to the fifth is 100,000 and ten to the ninth is 1,000,000,000. Perhaps our scientist friend would have done better to let the politician see all those zeros -- and then translate it into terms of a budget increase!

This brings me to the question of probability. Just as large numbers are widely misunderstood, there are very few people who appreciate just how unlikely things can be. The monkeys and typewriters scenario sounds possible until you examine the math. What follows is an in-depth discussion of the mathematics of the situation -- it's very long, but interesting, I think.

Monkeys Produce Hamlet: Feasibility Study

Let's imagine a very simple typewriter that has only the 26 upper-case letters, a space bar and five punctuation characters (a total of 32 buttons). It doesn't even have a carriage return -- it does an automatic return when the required number of letters have been typed, and it has an infinite roll of paper being fed through it. We have a monkey that knows how to press the keys and will do so in a totally random manner indefinitely. All in all, we have a little bit of machinery, but no real intelligence in the system. We want our monkey to type the following snippet: "TO BE OR NOT TO BE, THAT IS THE QUESTION."

The probability of this happening is quite simple to calculate, and this will in turn give us some idea of how many monkeys and typewriters we need for a reasonable chance of success. Place your bets now -- our monkeys are fast typists and can type the required number of characters in a single second (there are 41 keystrokes)! On average, how long will it be before one of our monkeys produces a line matching the above sentence?

Well, there are 32 keys, so starting at any moment, the chances of our monkey getting the first keypress right are one in 32. Not good, but we have fast monkeys and lots of time. However, once it has got the first keystroke right, we also need the second keystroke to be right, otherwise we are back to square one. The chances of it getting the first and second keystrokes right are only one in (32*32 = 1024). Only one chance in 1024, but still lots of time to get it right. To get the first three characters right will be a one in (32*32*32 = 32768) chance. Each time it presses a key, there is a one in 32 chance that it will be correct. To get our little snippet of Hamlet, it will need a total of 41 consecutive "correct" keystrokes. This means that the chances are one in 32 to the power of 41. Let's look at a table of values.

   Keys   Chances (one in...)
   ------------------------------------
    1     32
    2     32*32 = 1024
    3     32*32*32 = 32768
    4     32*32*32*32 = 1048576
    5     32^5 = 33554432
    6     32^6 = 1073741824
    7     32^7 = 34359738368
    8     32^8 = 1099511627776
    9     32^9 = 3.518437208883e+013
    10    32^10 = 1.125899906843e+015
    ...
    20    32^20 = 1.267650600228e+030
    ...
    30    32^30 = 1.427247692706e+045
    ...
    41    32^41 = 5.142201741629e+061
    ...
    204   32^204 = 1.123558209289e+307

The last figure is included only because it is the largest value that the MS Windows calculator can handle -- it's doing better than my hand-held Casio (old faithful!) which only goes up to 1e+99. Okay, so these figures are pretty vast, but we have a lot of monkeys and they can type fast. So how long will it take, on average, for one of my monkeys to type a line matching that sentence? Hard question. Let's get an idea of how long we are talking here. How many lines can my monkey type in a year, given that it types at a rate of one line per second?

  1 line per second
  * 60 seconds per minute = 60 lines per minute
  * 60 minutes per hour = 3600 lines per hour
  * 24 hours per day = 86400 lines per day
  * 365.24 days per year = 31556736 lines per year

Okay, now for the moment of truth. We know how many possible different lines can be produced, hence how likely it is for us to get the right one at random (because only one is right). We can calculate the chances of getting the quote in a year most easily by calculating the chances of missing on every attempt: the chances of getting the quote will be 100% minus the chances of missing on every attempt. I need a really amazingly precise calculator to do this because the chances of missing are so close to 100% that most calculators will round it off to 100%. The calculation is as follows.

 probability of missing on one attempt = 1 - 1/(32^41)
 ...of missing for a minute straight = (1 - 1/(32^41)) ^ 60
 ...of missing for an hour straight = ((1 - 1/(32^41)) ^ 60) ^ 60
 ...of missing for a day straight = (((1 - 1/(32^41)) ^ 60) ^ 60) ^ 24
 ...for a year straight = ((((1 - 1/(32^41)) ^ 60) ^ 60) ^ 24) ^ 365

If you have access to Unix, you can calculate this with the dc command, but be warned that it may take quite a while to calculate and annoy other users because the computer is so slow. Use of the nice command is suggested. The syntax, should you care to try, is as follows. Type the dc command, then type the following lines.

  99k
  1 1 32 41 ^ / - 60 ^ 60 ^ 24 ^ 365 ^
  p

The figure that is eventually printed will be the probability (expressed as a value between zero and one) of our monkey not typing our little phrase from Hamlet in the space of one year's worth of continuous attempts. The answer that it prints looks like this:

0.999999999999999999999999999999999999999999999999999999386721844366784484760952487499968756116464000

Notice all the nines? Even to fifty or more significant figures, this reads 100%. Okay, so realistically, there is no way that our monkey can do its job in a year. Maybe we should start talking centuries? Millenia? As I understand it, common scientific wisdom suggests that the universe is about 15 billion years old (although they may have revised their dating since I last heard about it). We can easily extend our current figure of one year to count many years. Our calculator will be much faster if we break the calculation down to powers of two and just use the "square" operation, so let's choose a nice even power of two like 2^34, which is about 17 billion (17,179,869,184 to be precise). The new figure is:

0.999999999999999999999999999999999999999999989463961512816564762914005246488858434168051444149065728

The chances of failure are still essentially 100%, even after 2^34 years. Hmmm. It doesn't look like were are going to get very far with this, but just for the heck of it, let's see if we are any better off with a lot of monkeys. Let's not hold back here -- I hypothesize 17 billion galaxies, each containing 17 billion habitable planets, each planet with 17 billion monkeys each typing away and producing one line per second for 17 billion years. What are the chances of the phrase "TO BE OR NOT TO BE, THAT IS THE QUESTION." not being included in the output?

0.999999999999946575937950778196079485682838665648264132188104299326596142975867879656916416973433628

I'd bet money on that. It's about 99.999999999995% sure that they would fail to produce the sentence. Are you astounded? It's such a trivial requirement, right? Just one puny sentence. And yet the figures keep coming up "impossible". Where have we made a mistake? We have fallen into the same trap as the politician who was the subject of my joke, way back up there. We have failed to appreciate the sheer magnitude of the problem. Let's look at it one more time.

The number of 41-character strings that are possible with a 32-character alphabet is 32^41. According to dc, this value is as follows.

51422017416287688817342786954917203280710495801049370729644032

In case you don't feel like counting, this value is 62 digits long. In our hypothesising above, we imagined 17 billion galaxies, each with 17 billion planets, each with 17 billion monkeys, each of which was producing a line of text per second for 17 billion years. How many lines of text did we wind up producing in this experiment? The math is as follows:

2^34 * 2^34 * 2^34 * 2^34 * 365 * 24 * 60 * 60

And the answer is as follows:

2747173049143991138247931294711870033017962496000

Once again, in case you don't feel like counting, the answer is 49 digits long. Now, there is no guarantee that our monkeys are going to type something different every time, but even if we managed to rig up the experiment so that they never tried the same thing twice, they have still only produced 1/18,718,157,355,362 of the possible alternatives. The denominator in that fraction is 14 digits long, by the way. It's a figure that's vastly bigger than anything you would come across in the real world. Is it any wonder, in light of that, that it is so damn hard to get the right answer by accident?

Conclusion

In light of this, I find it impossible to believe that "chance" had anything to do with the process that created life. How can I suppose that Shakespeare himself was the result of a random process when it is quite clearly impossible for even a trivial fragment of his work to have arisen by chance? No sir, I see information all around me, and I conclude that it is the product of a far, far greater intelligence.

Information is the product of intelligence, not chance.

Postscript

If you read the thesis above, you will agree at least that if the universe did arise by chance, it must be truly infinite and in the continuous process of trying out new alternatives. That, and the universe as we know it is an incomprhensibly unlikely fluke. I find it much simpler (and much more "natural", I might say) to suppose that there is a God who is greater than the universe who made it, much like I made this document.

Comments on this article are welcome.

See Also

More Monkey Business

A followup to this essay by the same author, dated 2000-07-28. Gives mathematical insights into other monkey math problems and answers questions frequently raised in regards to this essay.

---

Author: The Famous Brett Watson <famous@nutters.org>
History: [1995-12-13] Written as an email to Peter Lowe.
History: [1996-01-09] Converted to HTML 2.0 with minor alterations.
History: [1998-10-16] Corrected contact info.
History: [2000-08-15] Updated to HTML 4.0 with cosmetic changes. Added link to "More Monkey Business".
IPRs: This document is Public Domain (P) 1995.
Acknowledgements: the author wishes to acknowledge Peter Lowe as the person who inspired this essay.

[Texaggie79]

This hurts Mr. T's head!!!

[Hodar]

[AKA - Million Monkeys Can't Type Shakespeare]

I believe the saying that an infinite number of monkeys, with an infinite number of typewriters, can in fact recreate any piece of literature (Bible, Shakespeare, Mark Twain, Stephen King, etc.). To limit this to simply 1 million does not convey the thought of what the term 'infinite' means.

[arielb]

but if you took that monkey who wrote Hamlet-what are the odds he'd continue to write Romeo and Juliet?

[Jay W]

even that a million monkeys typing randomly on a million keyboards for a million years will eventually produce the collected works of Shakespeare.

Not even 20 million AOL subscribers have come anywhere close to producing anything remotely approximating the works of Shakespeare after 10 years of emails, message board postings, and instant messages!

[Hodar]

You are also assuming that each experiment is being performed serially, instead of gazillions of experiments being performed simultaneously.

[isthisnickcool]

Are we talking about African or European monkeys? And what is it's typing speed velocity?

[mykej]

Yawn. This is the same old thermodynamics argument, and that's been shot down time and time again.

If you don't believe that order arises form disorder, you don't believe that plants grow, or that fetuses develop into grown up humans. More orderly states arise from less orderly states ALL THE TIME.

Please try to come up with something new.

[cracker]

Ping!... Don't forget to visit the Crevo List for all the latest!

Decided to take this elsewhere? Are you going to respond to the argument that this has nothing to do with evolution?

[freedomlover]

Of course it only took one monkey to come up with Earth in the Balance.

[freedomlover]

To limit this to simply 1 million does not convey the thought of what the term 'infinite' means.

So true. The same applies to comparison of typing to the creation of life.

[gfactor]

the problem with this analogy is that it only makes one attempt -- ie, there is no re-evaluation -- to use the parlance of genetic algoriths, no fitness function. that is a key part of the iteration: make random guesses, check for fitness, modify highest fitting etc... I'd like to see what the chances are if these monkeys had some appropriate fitness function. something based on an english dictionary.

but the key is this, its not all randomness, but ITERATED randomness.

[Khepera]

To find all articles tagged or indexed using Straight Americans Speaking up (SASU™), click below:
	click here >>>	SASU™	<<< click here
Master Bump List (To view all FR Bump Lists, click here)

Give 'em Truth then, Give 'em Hell!

[monkey]

The Shakespeare-didn't-write-the-plays crowd will now claim it was the orangutan who wrote Hamlet.

[Texas_Jarhead]

ROFL!! Zingg... Good one.

[Southack]

OLD LINK

[cracker]

From the last thread:
It is DATA "self-forming", not chemicals that are in question. Whether the data that we are looking for is the first sentence of Hamlet or the first gene in DNA, Watson's math applies equally.

Except,
1. Nobody suggests that DNA emerged fully formed - it was proceded by an age of RNA-based chemistry.
2. Pre-RNA molecules/structures would still have to be self-replicating, thus introducing a variety of selection pressures that would accelerate the rate of information retention for re-use in a subsequent iteration.
3. How much "data" is contained in a self-replicating compound? Is it more or less than in a sentence of Hamlet? Your Hamlet string has behind it a whole language, with idiom and abstract meaning, embeded in a complex cultural context. The compound only needs to specify how to make a copy of itself.

[RightWhale]

Assuming the generation is entirely random. Like a chemical process, there are rules, which monkeys notoriously ignore.

[knarf]

"I believe the saying that an infinite number of monkeys, with an infinite number of typewriters, can in fact recreate any piece of literature (Bible, Shakespeare, Mark Twain, Stephen King, etc.)."

I don't.

The concept of thought, translated into/by language, requires intelligence that monkeys (and sometimes even men ... /8^) seem to lack.

[Sabertooth]

Not even 20 million AOL subscribers have come anywhere close to producing anything remotely approximating the works of Shakespeare after 10 years of emails, message board postings, and instant messages!

That's because I'm on Earthlink.

[in_troth]

Depends on how long it takes for him (and his progeny) to evolve into shakespeare!

[ThinkLikeWaterAndReeds]

This does prove that monkeys can type as well as white men can jump but doesn't address the process of evolution. You did hit on monkeys though, and that is a great start to argue for evolution. Throw away your calculator and apply reason.

[in_troth]

It isn't that monkeys would recreate stuff because they are smart enough to come up with it or that they would even understand it, it's that one day some monkey would RANDOMLY type out those words (of whatever text) in the exact order.

[Southack]

"Assuming the generation is entirely random. Like a chemical process, there are rules, which monkeys notoriously ignore."

Are you claiming that the Monkey is not a chemical process?

[Cu Roi]

The nature of the 32-key keyboard (convenient function of 2) the assume in their experiment is also important. If the simplified keyboard is based on the QWERTY model, the monkeys have better odds, since the most letters most used in English are in near middle and the monkeys are most likely to hit them.

[Mr. Lucky]

More orderly states arise from less orderly states ALL THE TIME

Just not by accident.

[FreePaul]

I believe the saying that an infinite number of monkeys,

That's the term I've always heard used. I guess that wouldn't fit with the "proof" in this article. Oh well! We'll just have to wait a bit longer.

[NetValue]

"Information is the product of intelligence, not chance. "

If DNA is considered "information" it too is the product of intelligence, i.e., an intellect higher than man's. Once you cross this threshhold, you can appreciate the near perfection of a movable thumb, stereo vision, language, and writing and contemplate the intellect of the designer.

[FreePaul]

...will now claim it was the orangutan who wrote Hamlet.

It was an infinite number of monkeys. You orangutangs keep trying to take credit for it.

[freedomlover]

Did you type that? ;)

[general_re]

You've hit on the right answer, but nobody's paying attention. There's no fitness check. If we posit an environment with selective pressures, where a sentence that is more like the final sentence is favored over a sentence that is less like the final product, then we will very, very quickly arrive at the final sentence - within 40 or 50 generations.

The error in the original post is the assumption that every attempt is a fresh attempt, where we throw everything out from the last run and make a random stab in the dark. If that were the case, the article would be correct. But it isn't - evolution is an additive process, where the last generation represents the starting point for the next generation.

[f.Christian]

Some of the early evolutionists remind me of the guy with homemade wings jumping off of the rocks--cliffs...

I saw a documentary once about this scientist came up through experimenting with worms and insects..."photo-lightism" because they were attracted to light---yeah plants too!

Evolution is social--intellectual public masturbation...with straight faces and---red you know what!

You rub--throw them bone--fossils and they start talking....hearing voices---seeing make-up believe things.

And the USSC certifies it--barbarism as our national science-religion and don't they all act like it too---apes-goons!

[Doomonyou]

Assuming the generation is entirely random. Like a chemical process, there are rules, which monkeys notoriously ignore.

Well use humans instead of monkeys. They know how to type (Rules). If the humans never heard of Shakespeare, Would they be able to produce Hamlet any quicker?

[freedomlover]

No, no, the orangutan wrote It takes a Village.

[Grit]

"It was the best of times, it was the blurst of times." -- C. Montgomery Burns reading the output of his 1000 monkeys at 1000 typewriters.

[Sir Gawain]

the problem with this analogy is that it only makes one attempt -- ie, there is no re-evaluation -- to use the parlance of genetic algoriths, no fitness function. that is a key part of the iteration: make random guesses, check for fitness, modify highest fitting etc... I'd like to see what the chances are if these monkeys had some appropriate fitness function. something based on an english dictionary.

So you're saying this accident called evolution somehow learned from its mistakes?

[sharktrager]

Actuallt, the QWERTY keyboard is an attempt to spread commonly used letters, and was done to keep typewriters from jamming. This can be seen by noting the positioning of the vowels.

Of course, it's also possible the keyboard was arranged by a monkey.

[ladyrustic]

Fetuses developing, and plants growing are not the same thing as order arising from disorder. A fetus is not disordered in any way; nor are seeds or seedlings. Order does not arise by chance from disorder and persist in this universe. Quite the opposite- things that exhibit some order tend to grow more disordered.

[RightWhale]

The monkey is that substance, which exists in itself, that is, exists independently of any other conception.

A million monkeys will ignore a million typewriters until one monkey takes a typewriter, at which time they will all want that typewriter.

[RightWhale]

Would they be able to produce Hamlet any quicker?

Ah, but they did produce Hamlet.

[lawdude]

"...a million monkeys typing randomly on a million keyboards for a million years will eventually produce the collected works of Shakespeare."

Sorry but I believe the number was an INFINITE number. As the probability asymtotically approaches 1, it leaves evident the probability that there will be a one letter error in the writings!

[mlo]

Typical of the usual evolution debunking arguments, it doesn't understand evolution, or DNA, enough to do a good job.

Nobody requires DNA formation to be a random process. It is a straw man argument. If it isn't a random process the calculation of odds is pointless.

[justshutupandtakeit]

Simpson's episode.

Monkeys chained to typewriters in basement typing madly away. Mr. Burns walks in takes page from a monkey's typewriter reads "It was the best of times it was the blurst of times." He yells at and smacks monkey who squeals.

[The Shootist]

Funny how Creationists accept, what do they call it, micro-evolution. Especially since micro-evolution is the same thing as Evolution. (btw, "micro-evolution" is a term made up by people who don't know what they are talking about, ie. idiots)

Funny how DNA changes daily. Chemical, biological and radiological stressors have a funny way of changing DNA.

Funny how one species of finch became 36 species of finch in less than 10000 years.

Funny how one species of honey sucker became 26 species of honey sucker, in less than 12000 years.

As far as the math goes its like anything else, garbage in, garbage out.

You can't tell a flat-earther that it is round. Message ends.

[Texas_Jarhead]

So I'm reading this thread and following the comments fairly well. And then I come across yours. You said, "There's no fitness check. If we posit an environment with selective pressures, where a sentence that is more like the final sentence is favored over a sentence that is less like the final product, then we will very, very quickly arrive at the final sentence..." Ok, so the process isn't so random then because some intelligence is aware of the desired outcome and is preserving and building on data from previous attempts. Is that it.

You continue, "The error in the original post is the assumption that every attempt is a fresh attempt, where we throw everything out from the last run and make a random stab in the dark...". If order is derived by chance from nothing then mustn’t we assume that each try is completely unique and in no way connected with any other attempt? Isn’t this very meaning of randomness?

I no statistician or mathematician but how can a process be both random yet retain and build upon previous data?

Thanks in advance and FReepOn

[Doomonyou]

Ah, but they did produce Hamlet.

Ah, but that is not the question.

[Capitalist Eric]

The beauty of religious mania is that it has the power to explain everything. Once God (or Satan) is accepted as the first cause of everything which happens in the mortal world, nothing is left to chance... Or change. Once such incantatory phrases as "we see now through a looking glass darkly" and "mysterious are the ways in which He chooses His wonders to perform" are mastered, logic can be happily tossed out the window. Religious mania is one of the few infallible ways of responding to the world's vagaries, because it totally eliminates pure accident. To the true religious fanatic, it's all on purpose.
---S. King

[Texas_Jarhead]

Postscript.

How can a random process accrue 'data' to achieve some eventual state when said state is supposed to be an unknown? In ‘proof’ above, the monkey does not know what to keep or throw away nor does he realize that there is a desired outcome or understand what that outcome should be.

I'm just trying to accurately relate my query.