I read an interesting question and i was wondering what you all thought about it.

Hypothetically, if a monkey were to type randomly on a typewriter for infinity, would at some point the complete, exact works of Shakespeare be typed?

Yes. Infinite time is a VERY long time. In this particular example, every conceivable text which could be produced on the typewriter <i>would</i> be produced eventually.

And who is going to feed the monkey during all this time? Better get a factory making paper, ribbons, typewriters and even trash baskets on the job, their going to need a lot of them.

Talk about a project for all time.

Oh, I forgot, welcome to sciforums, Bigtraine.

thanks for the welcome wet1

I agree with you James but I have heard people argue against it. I was hoping someone would offer a counter arguement that made some sense.

To me the odds would be extremely small that you would get such in any meaningful and reasonable time. Remember you are talking infinity here. So you basically come up with all things are possible however unlikely, as you can not rule out what chance will give you given enough time.

Here's an argument I just came up with:

Suppose your monkeys produced the complete text of <i>Hamlet</i> at some stage, somehow, except that they got one letter wrong somewhere (e.g. they typed "To be or not tx be..."). Given an infinite amount of time, having typed the whole thing at least once except for that one letter, what are the chances of typing the whole thing correctly? That one letter can only be replaced by about 100 other characters from your typewriter keyboard, and there would be an infinite time in which to repeat all the rest of the text and get that one letter right. Pretty high chance of doing it, given that they've done the whole thing before except for that letter, wouldn't you say?

Now suppose they had typed the whole thing except for two wrong letters. A similar argument shows that it is plausible that they could type the whole thing correctly given enough time. So, what if they typed the whole except for three letters? Four? Five? Ten? One thousand? Ten thousand?

Eventually, we ask the question: given that they haven't typed <i>any</i> of <i>Hamlet</i> yet, will they ever get it right? I'd say the answer is clearly "yes", since they have ample time to correct all their "mistakes".

How's that?

Butterfly collectors have been finding the alpabet on their wings for centuries.

My theory is that somewhere there is a butterfly with E=mc² on one wing and on the other wing is the formula to unite all four forces.

So I'd have to say yes.

Think about this:

If the universe is infinite there is another Earth with another you doing the exact same thing. In fact, there are an infinite number of you's!

Hmmm... I have heard his (and similar questions) before.
How about: if you were to write down the entire decimal code for the number pi (3,14159....) would there be a region in the code of exactly 245376328 executive times nuber 7, with a 2 in front and followed by a 9?
It's just too hard to imagine.

shrike:

That's not necessarily true. The processes which led to you being here now need not occur anywhere or anywhen else, even in an infinite universe.


Merlijn:

The answer to your question is no. It is quite possible that after a certain number of decimal places pi has a decimal pattern which looks like this:

...2456010010001000010000010000001....,

in which case a given pattern of the type you mentioned need never appear in pi, yet the decimal expansion would still never repeat.

I say no. How can an infinite amount of randomness produce a pattern? There can be no pattern, or it's not random.

Alpha,

The pattern in this case is one which <i>we</i> recognise after the event. It is in no way pre-destined or anything like that. To get to it, we'd have to wade through an infinite amount of text which had no discernable pattern.

Hmm, I think I see your point, but there is a flaw.
Since the sequence is infinite, there is an infinite number of combinations of characters that the monkey can type without ever typing anything meaningful to us. While it is possible it may type something that long that makes sense, it is (dare I say) infinitely unlikely.

Infinity, by definition, means that every possible permutation will occur - no matter how unlikely they are. Obviously, there is a possibility of the Monkey typing the works of Shakespeare by chance, even if the probabilty 1 in 1,000,000,000,000,000,000 (or whatever it is). So, given infinity, I dont see how the monkey could NOT type the works of Shakespeare.

Aaaah ... maybe because someone ran out of bananas?

Welcome to Sciforums.

Take care ;)

<i>Since the sequence is infinite, there is an infinite number of combinations of characters that the monkey can type without ever typing anything meaningful to us.</i>

Yes, but for any finite section of that infinite sequence, such as one the length of <i>Hamlet</i>, there is only a finite number of possible texts the monkey could type. One of those is the text of <i>Hamlet</i>. The chances of typing that text by chance are very small, but since the monkey has an infinite number of "tries" at it, it is guaranteed to do it sooner or later, provided it is indeed typing completely randomly.

Infinity, by definition, means that every possible permutation will occur - no matter how unlikely they are.Wrong. Check the [url="http://dicitonary.com"]dictionary[/ul].

Yes, but for any finite section of that infinite sequence, such as one the length of Hamlet, there is only a finite number of possible texts the monkey could type.True.
One of those is the text of Hamlet.Not necessarily.
The chances of typing that text by chance are very small, but since the monkey has an infinite number of "tries" at it, it is guaranteed to do it sooner or later, provided it is indeed typing completely randomly.In an infinite sequence of random characters, there are an infinite number of finite sections you could check for Hamlet. Since the entire sequence is infinitely long, you could check an infinite number of finite sections and never find anything. If you try every possible section in order, you may never get to where the section is. If you try every section at random, you may never land on it, or you may land on part of it, but not all of it. The chance of it being typed out in full is infinitely small, and so is the chance of you finding it even if it is there.

alpha,

It must be there, somewhere. Finding it is a completely different issue.

Back to the original post, of course i think it is possible. Practically speaking, a monkey would not be typing for infinity et cetera, but how could it NOT be possible?

Hypothetically, if a monkey were to type randomly on a typewriter for infinity, would at some point the complete, exact works of Shakespeare be typed?


that was the question right? And it states, if A monkey. one. dearies. A monkey can't live forever.
Maybe if it were a whole bunch of monkies. but then wouldnt that screw up the probability.

I think we are all aware that monkeys don't live forever, strawberryfyre, but thanks for the reminder.

One monkey or a hundred or a million makes no difference. More monkeys just means parallel processing. Stuff will be typed a bit faster, that's all. The same "final" document will still be produced, in the long (infinite) run.

What original great works of art wouldn´t the monkeys randomly write while pursuing the perfect copy of a book you can find anywhere.... :D

So you want the mathematical perspective on the "monkeys typing"
scenario? Keep in mind that this is going to be an entirely
theoretical answer. As you can imagine, there are some serious
practical problems with having an actual infinite number of monkeys
typing on an infinite number of typewriters (e.g. where would you put
them? what would you feed them?), but since we're mathematicians we
can gleefully ignore such considerations.

The cheap and easy answer to your question is, "yeah, they'll crank
out Shakespeare's works... eventually." This is assuming they really
are typing at random. The monkeys with typewriters I have personally
observed (mostly of the "young human/little sister" variety) tend to
bang on the same keys repeatedly, so it's hard to imagine them
actually turning out Shakespeare. But again, this is math so we will
ignore the real world.

As large as Shakespeare's collected works are, they are still finite.
If you type at random, eventually some six-jillion-letter combination
you type will end up being the collected works of Shakespeare.

An easier way to think about this is picking lottery numbers. Imagine
you are filthy rich and decide to buy a bunch of lottery tickets in an
effort to win Powerball. Since you are filthy rich, you can afford to
buy six jillion lottery tickets with every possible combination of
numbers that could come up, and thus you would be guaranteed to win
the lottery. It's the same concept with monkeys typing.

The grittiest detail in this problem is that the answer is only yes if
we are talking about an infinite number of trials; that is, having an
infinite number of monkeys or letting one monkey pound away for an
infinite amount of time. If we are restricted to a finite number of
monkeys and a finite amount of time, then the answer is no. It is
entirely possible that in a finite amount of time a finite number of
monkeys may type out nothing but pages upon pages of meaningless
drivel. It's also possible (although unlikely) that one monkey may get
it right the first time.

A good way to think of this is to imagine rolling a six-sided die
numerous times and waiting for a six to come up. It may come up on the
first roll. It's possible that you could keep rolling and rolling
millions of times without a six coming up, although you would expect
it to come up within six rolls, since there is a 1/6 chance of a 6
turning up on each roll.

Let's do an actual example. Since the collected works of Shakespeare
are a pretty lofty goal, let's just see about how long we would expect
it to take for a monkey to crank out one of Shakespeare's sonnets, for
example the following:


Look in thy glass and tell the face thou viewest -48
Now is the time that face should form another -45
Whose fresh repair if now thou not renewest -43
Thou dost beguile the world unbless some mother -47
For where is she so fair whose uneard womb -42
Disdains the tillage of thy husbandry -37
Or who is he so fond will be the tomb -37
Of his self love to stop posterity -34
Thou art thy mothers glass and she in thee -42
Calls back the lovely April of her prime -40
So thou through windows of thine age shall see -46
Despite of wrinkles this thy golden time -40
But if thou live rememberd not to be -36
Die single and thine image dies with thee -41


In the above sonnet I removed all punctuation, just leaving the
letters and spacing--we can't expect too much; they're only monkeys,
right? If my letter count is correct, this leaves 572 letters and
spaces. To further simplify, we won't worry about carriage returns,
capital letters, or any other such stuff.

Anyhow, say we give a monkey a special typewriter that has 27 keys
(26 keys for the letters of the alphabet along with a space bar).
We let the monkey type 572 characters at a time, pull the sheet out,
and see if it's the sonnet. If not, we keep going.

We'll do some calculations on the fly here to see how long this
process will take. Got a calculator handy? First of all let's find out
how many 572-letter possibilities there are for the monkey to type.
We have 572 characters, and 27 choices for each character, so there
will be 27^572 possibilities (that's 27 times itself 572 times).
Punching this into my calculator... er... okay, on second thought
better use a computer....I get the following number of possibilities:

54963337845610993936930485313680443448879261941985 32520694117049056247
25684243954820588519270755936792132632239916490954 44601504350463483987
50256101041408646085049085341195267896083992229861 17684072414622768253
62149083044273958125194745460868312880102366397357 83766919573127540345
25750895660448104139321160600317628945055249884512 85440971813773606694
01639464734676689707119196898634602719367508376097 98272198814318196353
50867707235286031854386928555038640076056898115339 68043988986405766599
46346269826532711524739691906555343297647268049242 35126863461599117918
74530078058908290711145228946720656232179617918122 04851353664903930975
35654199381688528812727552134080728906214345304165 60019423439471934830
84885587282853385530453996615799028022689403488087 63480359167736446637
8909091744053824079947245708112252748079248200721

It's a big number, about 5*10^818.

Let's say our monkey can type about 120 characters per minute. Then
the monkey will be cranking out one of these about every five minutes,
12 every hour, 288 per day, and 105120 of them per year. Divide that
big number by 105120 and you get that it would take that monkey about
5*10^813 years to type out that sonnet.

Now say we get 10^813 (that's ten followed by 813 zeros) monkeys
working on the job. With that many monkeys working 24 hours a day,
typing at random, one of them is likely to crank out the sonnet we are
looking for within five years. If the monkeys are particularly
unlucky, you may have to let them run an infinite amount of time
before they crank out the desired sonnet, but chances are with this
many monkeys on the job you will get results in five years.

To make a long story short, if you have only a finite number of
outcomes and you take an infinite number of trials, you will end up
getting the outcome you are looking for.

Well, forget about making a long story short, I'll give you one more
mind-blowing example. A typical digitized picture on your computer
screen is 640 pixels long by 480 pixels wide, for a total of
307200 pixels. Using only 256 different colors, you can get decent
resolution. Now if you take 256^307200 (256 times itself 307200 times)
you get... well, a pretty big number, but a finite number nonetheless.
That's the number of different images you can have of that particular
size. Any picture you would scan into a computer at that size and
resolution will necessarily be one of those images. Therefore,
contained in those images are the images of the faces of every human
being who ever lived along with the images of the faces of every
person yet to be born.

Deep stuff, eh? I'll leave you with that thought.

Think about this:

If the universe is infinite there is another Earth with another you doing the exact same thing. In fact, there are an infinite number of you's!

Doesn't work.

The size of the universe does not increase the chances. For your trick to work, you should be asking for an infinite number of planetary systems, or an infinite number of Earths.

You have several hundred monkeys typing on typewriters, but they are in an infinitely large room... see where I'm going?


Regarding the original post: Just as it is difficult to imagine how small the probability of hammering out Shakespeare's complete works, so is it impossible for us to grasp the enormity of the infinite. The latter overpowers the former every time.

Well, forget about making a long story short, I'll give you one more
mind-blowing example. A typical digitized picture on your computer
screen is 640 pixels long by 480 pixels wide, for a total of
307200 pixels. Using only 256 different colors, you can get decent
resolution. Now if you take 256^307200 (256 times itself 307200 times)
you get... well, a pretty big number, but a finite number nonetheless.
That's the number of different images you can have of that particular
size. Any picture you would scan into a computer at that size and
resolution will necessarily be one of those images. Therefore,
contained in those images are the images of the faces of every human
being who ever lived along with the images of the faces of every
person yet to be born.

Deep stuff, eh? I'll leave you with that thought.

Not that impressive, really. Sure... the number is able to be abstractly represented using power notation, but most readers might not realize how ridiculously large that number is. By comparison, there are about 10^80 particles in the known universe. The number you came up with isn't amazing for how small and encompassing it is (to hold all possible pictures at that resolution and color depth), rather it is startling for how gargantuan it is.

I would be more impressed to know, abstractly, that there are a finite number of possible images at any resolution... seeing how big that number is should give more doubt to the monkey/shakespeare skeptics (of which I am not one).

Doesn't work.

The size of the universe does not increase the chances. For your trick to work, you should be asking for an infinite number of planetary systems, or an infinite number of Earths.
Even if there are an infinite number of earths, it doesn't mean that any two of them would have to be the same, or that any conceivable scenario must exist on one of them. You can find infinitely many numbers between 2 and 3 without having any of them be the same number, and without having any of them be greater than 4.

True a finite and abstract number would be more interesting but to find that, well you have a one in 27^105210 chance of doing that alone. But more about the topic. It took me days to figure out that long number and I would like to know if there ia a real calculator that can do it in mere seconds like simpler equations. I can't find one and my thesis is about infinity and it's finite proportions. LARGE PROCESSING CALCULATORS ARE NEEDED!!!!!!!:eek:

A monkey is a sentient being - there is no guarantee that it won't just repeat what it has previously typed or simply avoid typing certain letters at all.

You could only guarantee random output with a fairly powerful computer which, given an infinite amount of time, could create every way in which the alphabet and punctuation marks can be arranged. At some point it would churn out the sentences I have just finished typing.

54963337845610993936930485313680443448879261941985 32520694117049056247
25684243954820588519270755936792132632239916490954 44601504350463483987
50256101041408646085049085341195267896083992229861 17684072414622768253
62149083044273958125194745460868312880102366397357 83766919573127540345
25750895660448104139321160600317628945055249884512 85440971813773606694
01639464734676689707119196898634602719367508376097 98272198814318196353
50867707235286031854386928555038640076056898115339 68043988986405766599
46346269826532711524739691906555343297647268049242 35126863461599117918
74530078058908290711145228946720656232179617918122 04851353664903930975
35654199381688528812727552134080728906214345304165 60019423439471934830
84885587282853385530453996615799028022689403488087 63480359167736446637
8909091744053824079947245708112252748079248200721I think I see a pattern in that number...perhaps because of all "88"'s?

I say no. How can an infinite amount of randomness produce a pattern? There can be no pattern, or it's not random.

random itself can be considered a pattern. Consistantly Inconsistant.

random itself can be considered a pattern. Consistantly Inconsistant.Yes, and I think that there might be numbers with fractions that go over not all numbers from 1 to 9 also there might be numbers where there always is a random pattern, since of the way the sequence is repeated.

You can find infinitely many numbers between 2 and 3 without having any of them be the same number, and without having any of them be greater than 4.

This isn't true. The only way you can have infinitely many numbers between 2 and 3 is if you keep changing the level of precision of the group of numbers. That is, you can only do this by adding decimal points as you go along.

Any level of precision you choose for your set, you will find a finite number in that set. By defining your set as something dynamic (i.e. the rules keep changing), you are cheating.

This myth is still taught in college classrooms, and it is the maddening source of the Zeno paradox, which shouldn't be a paradox at all.

This isn't true. The only way you can have infinitely many numbers between 2 and 3 is if you keep changing the level of precision of the group of numbers. That is, you can only do this by adding decimal points as you go along.

Any level of precision you choose for your set, you will find a finite number in that set. By defining your set as something dynamic (i.e. the rules keep changing), you are cheating.

This myth is still taught in college classrooms, and it is the maddening source of the Zeno paradox, which shouldn't be a paradox at all.

I'm incensed that nobody is arguing with me on this. :bugeye:

This isn't true. The only way you can have infinitely many numbers between 2 and 3 is if you keep changing the level of precision of the group of numbers. That is, you can only do this by adding decimal points as you go along.

Any level of precision you choose for your set, you will find a finite number in that set. By defining your set as something dynamic (i.e. the rules keep changing), you are cheating.

This myth is still taught in college classrooms, and it is the maddening source of the Zeno paradox, which shouldn't be a paradox at all.

If you choose specific numbers to start and stop at YES you could argue there is finite space between but the reality is there is infinity between in fractions.

Because the number is irrational doesn't mean it does not exist.....You can continuously fracture an object infinitely and that is an example of infinity.

As far as CHEATING by continuously fracturing a measurement infinitely smaller and smaller I don’t agree as that is no more cheating than picking and choosing when the measurement of an object starts and ends with set increments of numbers.

Fracturing an object using SET increments is a practical way of measuring an object for a purpose which is why its taught in school and always used in all measurements.

I’m not necessarily arguing anything other than Infinity is real it does exist and it exists everywhere and the implications of that are extremely interesting.

As far as measuring an object for a practical purpose you HAVE to use set increment measurement otherwise you would never get anything done. Outside of practical purpose driven measurement you can fracture/divide any object infinitely.

Outside of practical purpose driven measurement you can fracture/divide any object infinitely.

No you can not. In order to do so, your unit of measurement would have to be zero, which is impossible.

Again, you are falling for the never-ending process of choosing finer and finer units of measurement and pretending that this process will ever reach zero. It will not, therefore you always have a finite number of divisions in any segment.

The never-ending nature of irrational numbers does not dent the argument. You can not change the "last" number of each irrational number to make it a distinct number, as there is no "last" number.

Let's take the segment of the number line between zero and one. Divide it in half, and you have a smaller number of .5 and 2 segments. Divide in half again and you have .25 and 4 segments. You can do this forever and the first number will never reach zero and the second number will never "reach" infinity. This, despite the fact that the first number will forever grow in decimal precision and the second number will never stop doubling.

No you can not. In order to do so, your unit of measurement would have to be zero, which is impossible.

Again, you are falling for the never-ending process of choosing finer and finer units of measurement and pretending that this process will ever reach zero. It will not, therefore you always have a finite number of divisions in any segment.

The never-ending nature of irrational numbers does not dent the argument. You can not change the "last" number of each irrational number to make it a distinct number, as there is no "last" number.

Let's take the segment of the number line between zero and one. Divide it in half, and you have a smaller number of .5 and 2 segments. Divide in half again and you have .25 and 4 segments. You can do this forever and the first number will never reach zero and the second number will never "reach" infinity. This, despite the fact that the first number will forever grow in decimal precision and the second number will never stop doubling.

If you could cut an object of any size in half over and over again.....how and when do you propose that it would ever reach ZERO? when you cant see it anymore? When your instruments are no longer capable of cutting such a small object in half? When the object reches its fundamental atomical paticles that are considered the "smallest" possible units? (That we know of)

Its a very perplexing concept to understand but I think the human desire/need to MEASURE everything is what is making infinity so incredibly hard to understand.

When you realise because of infinity, there is no difference between the earth in the universe and a grain of sand in the desert, it makes measuring everything rather.....pointless unless of course your trying to build a condo. Then measurment becomes the "practical" thing to do.

Im obviously not taking the scientific approach to this understanding as it would take far too long. My approach is philosophical.

hense....the forum location.

Let's take the segment of the number line between zero and one. Divide it in half, and you have a smaller number of .5 and 2 segments. Divide in half again and you have .25 and 4 segments. You can do this forever and the first number will never reach zero and the second number will never "reach" infinity. This, despite the fact that the first number will forever grow in decimal precision and the second number will never stop doubling.

I like this thought experiment. You are right that they will never be zero or infinity. Funny thing is, the number of divisions you perform on this segment can be infinity!!!

I like this thought experiment. You are right that they will never be zero or infinity. Funny thing is, the number of divisions you perform on this segment can be infinity!!!

Nope. You can do it FOREVER. Which is an eternity. But you will never reach zero or infinity, which is what you would have to do to prove your point.

As I said a few posts back, it is the eternal nature of the process of creating ever finer standard units of measurement which create the illusion that there are an infinite number of divisions in a line segment. As soon as you pick your degree of precision, you both define your unit of measurement and tell me how many of them are in the segment. If you keep sliding your hand to the right, down a growing line of decimal places, and looking up at me with a clever grin, you are fooling yourself, but not me.

This resolves Xeno's paradox and also this simpler but more confounding one: I just took my finger and ran it along a ruler, passing through an infinite number of points.

The answer is that points do not exist. They are the unit size=zero, which will never be reached. A figment of our imagination with theoretical uses only. I didn't run through an infinite number of anything.

Yes. Infinite time is a VERY long time. In this particular example, every conceivable text which could be produced on the typewriter <i>would</i> be produced eventually.

Logic flaw. Aren't you assuming every possible "random" sequence of text, to be equally probable? Doesn't that like, almost never happen in nature?

And No. Neither a monkey nor a typewriter would last that long.

I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times.

I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times.

Agreed. We wouldn't get the Complete Works of Shakespeare in the proper order and with exact punctuation, we would get an infinite number of copies of them. And an infinite number of copies with every play spelled out in reverse. And an infinite number of copies with the plays arranged in every order possible. Even copies with scenes from one play inserted into another.

Think about the nature of infinity. If the "editor" in our analogy has not yet gotten the document he is after, he has forever to wait for it. Once he gets what he needs, he has forever to wait on the next one. Is the single transcription of a lone letter at the very end of the Complete Works enough to dissuade them from their Sisyphean task? Of course not, they have an infinite number of stabs to go.

I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times.Alas, not true.
As there are an infinite number of finite-length sequences, you can never type them all out even once, let alone an infinite number of times.

You couldn't even finish typing out all the finite-length sequences of just the letter "a" (e.g. a, aa, aaa, aaaa, aaaaa,... ) due to there being an infinite number of them. Even if you used "aaaaaaaaaa" to be 1x 10"a" sequence and 10x1"a" sequence at the same time (and all permutations in between), you still wouldn't achieve it.
You'd be left tapping away at "a" for eternity, never to succeed.


Agreed. We wouldn't get the Complete Works of Shakespeare in the proper order and with exact punctuation, we would get an infinite number of copies of them. And an infinite number of copies with every play spelled out in reverse. And an infinite number of copies with the plays arranged in every order possible. Even copies with scenes from one play inserted into another.I'd agree with this, because there aren't an infinite number of combinations for Shakespeare's complete works, but very much a finite number.
So in an infinite time we would get an infinite number of this particular finite-length combination.

But this is different to saying we would get every possible finite-length sequence, as there are an infinite number of those.

But this is different to saying we would get every possible finite-length sequence, as there are an infinite number of those.

I disagree. I would argue that we could get every finite-length sequence, as well as every sequence of any chosen length.

This is the same distinction that I bring up above on the nature of the infinitesimal. As soon as you choose your finite length, we can state that every sequence will be arrived at an infinite number of times. If one tries to continue an eternal process of adding one more to that length, they will keep avoiding the inevitable, and thereby fool themselves into thinking they have escaped the simple fact:

No matter which length you settle on, we can agree that all sequences are produced an infinite number of times. Every time you lengthen the sequence by one, this still holds. It will always hold. That is an eternity of correctness, not an eternity of defeat.

Don't mistake the ETERNAL process of increasing the length with the INFINITE. You will never reach the infinite, even if you go on FOREVER in this manner. And at every step along the way the fact remains that you have a finite number in your sequence, and all possible outcomes will be produced.

I disagree. I would argue that we could get every finite-length sequence, as well as every sequence of any chosen length.But how do you reconcile that to there being an INFINITE number of finite-length sequences. And there are an infinite number of them.

I agree that once you have chosen your finite length you will get an infinite number of them... but it is in choosing ALL finite lengths that is problematic when there are an INFINITE number of them.

If you think there are a finite number of finite-length sequences, what is the largest finite-length sequence?

Bear in mind that I agree with your comments regarding Shakespeare's work: it is specifically with Nin''s comment: "I'll go even further to say that every possible finite sequence of letters will have been typed out an infinite amount of times. " that I disagree.


To argue it another way:
In an infinite pile of pebbles, what is the largest finite number of pebbles you can have?
The answer is that there isn't one.
Or do you disagree?

If you agree, then how can you think you can reach the end of the infinite number of finite-length sequences to be able to repeat it, let alone an infinite number of times?




This is the same distinction that I bring up above on the nature of the infinitesimal....It is a flawed comparison, though.
The bounded range, no matter how infinitessimally small the interval, is very different when compared to an unbounded range.
A bounded range can be leapt across with simplicity by something that is larger than the range.
The unbounded can not - as there is nothing larger.

But how do you reconcile that to there being an INFINITE number of finite-length sequences. And there are an infinite number of them.

The same way I imagine a stack of an infinite number of Collected Works of Shakespeare.

The stack of infinite number of sequences of ever-growing size are a stack of FINITE length sequences, all of which would result in every combination possible.

Are you ever going to add one more to the length of a sequence and confound the monkeys? Nope. So you have an infinite collection of an infinite collection. All of which obey a simple principle of statistics.

Nope. You can do it FOREVER. Which is an eternity. But you will never reach zero or infinity, which is what you would have to do to prove your point.

I had agreed with you that you will never reach zero or infinity.

My other sentence was worded wrongly:

I meant to say

Funny thing is, the number of divisions you perform on this segment can be infinite!!!

rather than

Funny thing is, the number of divisions you perform on this segment can be infinity!!!

and you had agreed with this by stating that
You can do it FOREVER which essentially means the same thing

I had agreed with you that you will never reach zero or infinity.

My other sentence was worded wrongly:

I meant to say

Funny thing is, the number of divisions you perform on this segment can be infinite!!!

rather than

Funny thing is, the number of divisions you perform on this segment can be infinity!!!

and you had agreed with this by stating that which essentially means the same thing

Gotcha. The power of a single letter, eh my fiend? ;)

Alas, not true.
As there are an infinite number of finite-length sequences, you can never type them all out even once, let alone an infinite number of times.

So which sequences won't be typed over an infinite amount of time and why? This implies that only a finite number of sequences will be written over an eternity which is false.

The complete works of Shakespear are a finite sequence of characters.

Starting from the big bang it took about 16 billion years to burn hydrogen down into heavier elements, form a planet, evolve monkeys, have them develop writing and completely at random write the complete works of Shakespear.

Unfortunately they did it before developing the typewriter so as soon as the universe breaks back down into quarks we will have to start over and try again.

Stupid monkies.

So which sequences won't be typed over an infinite amount of time and why? This implies that only a finite number of sequences will be written over an eternity which is false.There are an infinite number of finite sequences... so one monkey could never type them all out an infinite number of times.
One monkey can only type out a finite number of things an infinite amount of times, or an infinite number of things a finite number of times.

To write out an infinite number of things an infinite number of times would take an infinite number of such monkeys.


How long is the longest finite sequence?
Ironically it is infinitely long (i.e. there are an infinite number of finite sequences - with the longest being infinitely long).

If one monkey starts by writing the longest finite sequence, it will take an infinite amount of time. He will be unable to write anything else.

If he wants to write it an infinite amount of times, he could start, for example, writing the first letter of the sequence an infinite amount of times... but this lone monkey is then again unable to do anything else, such as move on to the secong letter in the sequence.


A lone monkey, given an infinite amount of time, can either write a finite sequence an infinite amount of times, or an infinite sequence a finite amount of times. He can not write an infinite sequence an infinite amount of times.


If you think differently I would say it is because you are assuming that because you have written "longest finite sequence" it must therefore be finite, and assume that it is thus not infinite in length.

If it is not infinite in length, how long is it?
Answer me that, and I might be able to see your / swivel's side of the argument.

How long is the longest finite sequence?
Ironically it is infinitely long (i.e. there are an infinite number of finite sequences - with the longest being infinitely long).


That can't be correct. The longest FINITE number can't be INFINITE. You are mistaking, once again, the eternal process of writing out finite numbers for the existence of an infinite number of them. True, you can continue making finite sequences as long as you like, but you never get to say that any of them go on forever.

Since the monkeys type for an infinite length of time, they easily keep up with the process, typing out every possible sequence an infinite number of times.

Going further, if you have an infinite number of monkeys that type randomly, but also for a random length of time, every sequence will be typed out perfectly, from beginning to end.

And if you have an infinite number of monkeys with infinite time, each sequence of INFINITE length will also be typed out faithfully.

There is nothing those smelly bastards can't do if you give them the power of infinity.

That can't be correct. The longest FINITE number can't be INFINITE.Then how long is the longest finite number?

Let's just start with that question:
How long is the longest finite number?

Imagine it another way... imagine an infinitely large cake. What is the biggest chunk of this cake you can take and still leave some left.



You are mistaking, once again, the eternal process of writing out finite numbers for the existence of an infinite number of them. True, you can continue making finite sequences as long as you like, but you never get to say that any of them go on forever.No, I'm not.
Your confusion lies in you assuming that "the longest finite number" is finite in length - but it isn't. There is no "longest finite number":

Proof:
Imagine a finite number with X digits.
There exists a longer number with X+1 digits, correct?

Now imagine that the "longest finite number" has Y digits, and we know that Y > X.

However, there exists a longer number than the one with Y digits, and it has Y+1 digits.

There is therefore NO "longest finite number" - 'cos as soon as you state what it is, someone can add an additional digit - ad infinitum.

Therefore there are an infinite number of finite numbers, and a single monkey can thus NOT type them all out an infinite number of times.


Where does my logic break down?
I have shown yours to break down in your assumption that the longest finite number is finite.



As for the rest of your post, I have no issue with an infinite number of monkeys doing what you propose, but the OP is regarding a single monkey, or at most a finite number of monkeys.
What you claim is logical for an infinite number of monkeys - just not a finite number.

I read an interesting question and i was wondering what you all thought about it.

Hypothetically, if a monkey were to type randomly on a typewriter for infinity, would at some point the complete, exact works of Shakespeare be typed?


Yes, because you would have an infinite amount of time to let the dirty little monkeys ''do their thing.'' :D

That can't be correct. The longest FINITE number can't be INFINITE. You are mistaking, once again, the eternal process of writing out finite numbers for the existence of an infinite number of them. True, you can continue making finite sequences as long as you like, but you never get to say that any of them go on forever.

Since the monkeys type for an infinite length of time, they easily keep up with the process, typing out every possible sequence an infinite number of times.

Going further, if you have an infinite number of monkeys that type randomly, but also for a random length of time, every sequence will be typed out perfectly, from beginning to end.

And if you have an infinite number of monkeys with infinite time, each sequence of INFINITE length will also be typed out faithfully.

There is nothing those smelly bastards can't do if you give them the power of infinity.

Wow, this thread is wrecking my head!!!

Proof:
Imagine a finite number with X digits.
There exists a longer number with X+1 digits, correct?

Now imagine that the "longest finite number" has Y digits, and we know that Y > X.

However, there exists a longer number than the one with Y digits, and it has Y+1 digits.



There is a problem with your proof. You already defined 'Y' as the "longest finite number". You can not then add '1' to this number two steps later.

And I understand the difference between my scenarios and the OP's. I'm just having fun rolling the ideas around. I don't see this as a debate to get testy over. I'm certainly not worked up.

This all boils down to your contention that there is a FINITE number with INFINITE size. I call bollocks. There is no END to the number of FINITE numbers, but each and every one is FINITE. The HYPOTHETICAL nature of their eternal mystery is what we call the infinite.

Really, it is quite obvious that your reasoning fails since it is paradoxical. There is no finite number of infinite length. Please observe the lunacy of this statement.

Again: There are an infinite number of finite-length numbers. There is no end to them. But each one has the length of x+1. None of them will ever NOT be calculable by that simple formula.

There is a problem with your proof. You already defined 'Y' as the "longest finite number". You can not then add '1' to this number two steps later.

This all boils down to your contention that there is a FINITE number with INFINITE size. I call bollocks. There is no END to the number of FINITE numbers, but each and every one is FINITE. The HYPOTHETICAL nature of their eternal mystery is what we call the infinite.

Really, it is quite obvious that your reasoning fails since it is paradoxical.Firstly if you re-read closely you'll note that I state "There is no 'longest finite number'". You apparently seem to concur. :shrug:
All I have been doing, which you so aptly highlight as well, is raise the paradoxical nature of the assumption that there is a "largest finite number", and in doing so it makes the conclusions invalid.

The very phrase "largest finite number" is meaningless.
It has no place in logic, since logically it does not exist.
Thus to use its existence as an assumption - or a claim - makes the conclusion / claim meaningless.

You are trying to make a set of all finite numbers, when such a set can not exist precisely because of its paradoxical nature.
To then say that someone can type out this set an infinite times is, thus, meaningless.


There is no finite number of infinite length. Please observe the lunacy of this statement.

Again: There are an infinite number of finite-length numbers. There is no end to them. But each one has the length of x+1. None of them will ever NOT be calculable by that simple formula.
Excellent - moving on then, you admit that there are an infinite number of finite-length numbers.

Okay, next question... how long will it take one monkey to type out the infinite number of finite-length numbers just once, given that, as you put it "there is no end to them"?

And next you'll be able to answer how one monkey, while spending an infinite amount of time trying to type out all the finite-length numbers, can then write them each an infinite amount of times?



And I understand the difference between my scenarios and the OP's. I'm just having fun rolling the ideas around. I don't see this as a debate to get testy over. I'm certainly not worked up.Glad to hear it.
Just as long as you stick to the single monkey for the purposes of discussion with me - as I concur with your assessment of the type-writing ability of an infinite number of monkeys given an infinite amount of time.
It is just this one solitary monkey that appears to be causing the issue, and his claim that, given enough time, he can do the same thing as the infinite number of his simian-colleagues can do. And, as you so succinctly put it earlier, I call bollocks on that. ;)

There is no longest finite number. For any squence of numbers of n length there is a sequence n+1.

Note n and n+1 are unbounded, but not inifinate. An infinate sequence can't be enumerated.

If there is a finite probability of an occurance then it has that probability for each trail. I.E. if the odds are 1 in 6 then every throw they are 1 in 6 no matter what has happened in previous throws. So even though the odds are 1 in 6 and you make 6 throws there is no garranty you will get a hit. There is no garranty you will get a hit with a hundred throws either, but it is likely you will.

With a finite probablility task, like the complete works of Shakespear, and an unbounded time frame, you are basicly saying the monkies will keep pecking away until they succeed. In this case it took about 16 billions years.

Then how long is the longest finite number?

Let's just start with that question:
How long is the longest finite number?

Imagine it another way... imagine an infinitely large cake. What is the biggest chunk of this cake you can take and still leave some left.


No, I'm not.
Your confusion lies in you assuming that "the longest finite number" is finite in length - but it isn't. There is no "longest finite number":

Proof:
Imagine a finite number with X digits.
There exists a longer number with X+1 digits, correct?

Now imagine that the "longest finite number" has Y digits, and we know that Y > X.

However, there exists a longer number than the one with Y digits, and it has Y+1 digits.

There is therefore NO "longest finite number" - 'cos as soon as you state what it is, someone can add an additional digit - ad infinitum.

Therefore there are an infinite number of finite numbers, and a single monkey can thus NOT type them all out an infinite number of times.


Where does my logic break down?
I have shown yours to break down in your assumption that the longest finite number is finite.



As for the rest of your post, I have no issue with an infinite number of monkeys doing what you propose, but the OP is regarding a single monkey, or at most a finite number of monkeys.
What you claim is logical for an infinite number of monkeys - just not a finite number.

Can't you use this same this same logic on anything? There's always a longer finite number of times you can type 1, therefore you can't type 1 an infinite amount of times over an infinite amount of time. Sorry if I missed something, I only skimmed through your responses, if you already addressed this then go ahead and ignore me (busy atm).

There is no longest finite number. For any squence of numbers of n length there is a sequence n+1.

Agreed. There is no longest finite number. But none of them are infinitely long. The process goes on forever (an eternity) but never gets where it is heading (infinity).

That's why we can treat any "very large" finite number as the longest. Choose one. Treat it as if it is the "longest" and talk about it. Whatever you say about it will be true of the +1 number, and the one after that, and the one after that.

Sarkus is making the mistake of treating this eternal process as if it is an infinite process. It is the same as with the infinitesimal. Pick a very big number or a very small number and call it the "last one". Settle on a metric and calculate away. If someone wants to move your metric, recalculate. Then analyze of the movement of your answers (or stasis in some cases).

Agreed. There is no longest finite number. But none of them are infinitely long. The process goes on forever (an eternity) but never gets where it is heading (infinity).

That's why we can treat any "very large" finite number as the longest. Choose one. Treat it as if it is the "longest" and talk about it. Whatever you say about it will be true of the +1 number, and the one after that, and the one after that.

Sarkus is making the mistake of treating this eternal process as if it is an infinite process. It is the same as with the infinitesimal. Pick a very big number or a very small number and call it the "last one". Settle on a metric and calculate away. If someone wants to move your metric, recalculate. Then analyze of the movement of your answers (or stasis in some cases).It is you making the mistake of assuming that just because you can arbitrarily label a "very large" finite number as "the largest" and "finite" that this means you can write them all.
If you admit that there is no largest finite number - how on earth do you propose to be able to write them all out when you know, as you have admited, that as soon as you think you have done so there is another, and another, and another that are longer.
The numbers themselves might be finite - but because there are an infinite number of them you can not write them all out. It is not the numbers themselves, but the series as a whole you are seemingly ignoring.
You seem to ignore this point.

You simply can not write out an unbounded infinite (so as to differentiate it from the very different matter of bounded infinitessimals) series, even if every element of that series is finite.
And yet you seem to think you can not only do it once, but do it an infinite number of times.

Can't you use this same this same logic on anything? There's always a longer finite number of times you can type 1, therefore you can't type 1 an infinite amount of times over an infinite amount of time. Sorry if I missed something, I only skimmed through your responses, if you already addressed this then go ahead and ignore me (busy atm).You stated that given an infinite amount of time you could write out (or the monkey could type out) every possible finite sequence of letters an infinite number of times.

I pointed out to yourself, and swivel, that this is incorrect - as the series of finite sequences is unbounded, and thus infinite in length (i.e. the series is infinite in length although the individual elements are finite).

So given that a single monkey is unable to type out even once what can not exist... how could he do so an infinite number of times?

You simply can not write out an unbounded infinite (so as to differentiate it from the very different matter of bounded infinitessimals) series, even if every element of that series is finite.
And yet you seem to think you can not only do it once, but do it an infinite number of times.

I disagree. I understand your complaint fully, and I reject it as wholly incorrect.

It is true that you cannot complete an infinite set, but you can write out every member of a set bounded on one side. That is, you can complete a ray, but not a line.

I am granting the writer infinite time for the task. They have an infinite number of finite sequences to write out. Excellent! They have just the right amount of time for the job.

Will they ever be done? No. Will they ever run out of time, so that you can say, "See, swivel, they didn't finish the job"? No. Every swquence will come and every one will be written down.

Hotel rooms are often used to compare the sizes of infinite sets in order to determine which infinite sets are smaller or larger than others. In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.

In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.

One minor problem. They aren't brute force doing every possible combination of characters. Each pass is randomly generated.

One minor problem. They aren't brute force doing every possible combination of characters. Each pass is randomly generated.

Which is where they get their extra-simian strength in these examples.

I mentioned earlier that the set of (things monkey types) and (all possible sequences) are the same size, but if you distinguish duplicates from one another, the typed set is a larger infinite set than the set of all possibilities. Because the monkey is going to type each possibility an infinite number of times.

Of course, you can just group each instance together and you still have a 1-1 ratio, but it is amazing to realize that their set is much larger, not much smaller as others have argued in this thread.

Its been way too long since I did serious math to know if things collapse tidely or not

I disagree. I understand your complaint fully, and I reject it as wholly incorrect.

It is true that you cannot complete an infinite set, but you can write out every member of a set bounded on one side. That is, you can complete a ray, but not a line.

I am granting the writer infinite time for the task. They have an infinite number of finite sequences to write out. Excellent! They have just the right amount of time for the job.

Will they ever be done? No. Will they ever run out of time, so that you can say, "See, swivel, they didn't finish the job"? No. Every swquence will come and every one will be written down.

Hotel rooms are often used to compare the sizes of infinite sets in order to determine which infinite sets are smaller or larger than others. In this example, the sets are equal in size, one moment in time for each sequence, which means it is a fulfillable infinite set, bounded on one side.

No problemo.Yes - it is a problem - as it was claimed that they could not only just do the infinite task once, but an infinite number of times.
Above you seem to agree that "Every swquence will come and every one will be written down." I agree that every sequence will be written down ONCE.
But an infinite number of times each? No. You would need an infinite number of hotels... or in the scenario originally posted - an infinite number of monkeys.

Yes - it is a problem - as it was claimed that they could not only just do the infinite task once, but an infinite number of times.
Above you seem to agree that "Every swquence will come and every one will be written down." I agree that every sequence will be written down ONCE.
But an infinite number of times each? No. You would need an infinite number of hotels... or in the scenario originally posted - an infinite number of monkeys.

If you agree that every sequence will be written down once then don't you concede to my statement? If it can be written even once that means that it can be written again, however improbable. Since we're talking about infinity here probability is irrelevant, we just need to discern whether or not there is a chance. Since there is a chance, it will be written infinite times, but the more improbable ones will simply just appear less.

I read an interesting question and i was wondering what you all thought about it.

Hypothetically, if a monkey were to type randomly on a typewriter for infinity, would at some point the complete, exact works of Shakespeare be typed?

Depends on his genetic code, which is determined, and the how the environment (determined) surrounding him interacts for the rest of infinity. If he doesn't get caught in a perpetual energy loop that was uninterrupted for the rest of eternity, yes. Otherwise, no.

I got one to add, considering no perpetual energy loops can exist (keeps things in a permanent stasis). What are the chances that this present world we know exists? Well, we know it at least happened once. What are the chances it happens again? Well, considering eternity and no perpetual energy loops, it will happen. In fact, with no perpetual energy loops, it will happen not only exactly the way it did, but it will happen every way infinitely possible. It will be like ground hogs day to an infinite power.

I have a question about the monkey/Shakespeare thing. (tired so forgive the halfassedness....). To my understanding, Infinity isn't a number, and it can't be measured; while time is a unit of measurement. Hmmm, wasn't much of a question after all. Gn. School in a few hours.

If you agree that every sequence will be written down once then don't you concede to my statement? If it can be written even once that means that it can be written again, however improbable. Since we're talking about infinity here probability is irrelevant, we just need to discern whether or not there is a chance. Since there is a chance, it will be written infinite times, but the more improbable ones will simply just appear less.No - I don't concede :).
You claimed an infinite number of infinite length sequences.
I claim that the single monkey can only ever write a finite number of infinite-length sequences, or an infinite number of finite-length sequences. But not both an infinite number of infinite sequences.

Your flaw is "If it can be written even once that means that it can be written again, however improbable." It will take an infinite time to write once. There is thus nothing else it can do.

I think we are all aware that monkeys don't live forever, strawberryfyre, but thanks for the reminder.

One monkey or a hundred or a million makes no difference. More monkeys just means parallel processing. Stuff will be typed a bit faster, that's all. The same "final" document will still be produced, in the long (infinite) run.

I disagree.

Just because the patterns are "random" (which is impossible, really) does not mean that eventaually every possible pattern would be produced.
Why would it?
Doesn't that presume that the monkey can not repeat himself?

If you disallowed repeating patterns, it would definitely not happen, because I am fairly certain Shakepeare used the work "King" more than once.
The only possibly way to ensure that it "will" be produced is to set restraints that A.) a "pattern" is a specific number of characters and spaces which is the resultant text (let's say you were going for Moby Dick, and Moby Dick had 234,583 characters the pattern size would be 234,583) and B.) that pattern could not repeat itself.
Without those two constraints I don't see how anything MUST be produced.

Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.
Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.

Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.
Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.

Yes, we can prove this for an infinite length of time. It would be impossible for the monkey to NOT pound out Shakespeare's entire cannon, punctuation and all.

Yes, we can prove this for an infinite length of time. It would be impossible for the monkey to NOT pound out Shakespeare's entire cannon, punctuation and all.

I have seen no proof.

"I say no. How can an infinite amount of randomness produce a pattern? There can be no pattern, or it's not random." Excellent work Alpha

Aren't strings energy densities random, yet they produce a pattern...of quantum particles...atoms...molecules...

I disagree.

Just because the patterns are "random" (which is impossible, really) does not mean that eventaually every possible pattern would be produced.
Why would it?Because of the nature of randomness and infinite time.
Given an infinite amount of time, if something does not happen then it is not possible. If there is ANY chance of it occurring, however remote, then it WILL happen.

e.g. if you have a coin where you think that the result is random, if it never landed on TAILS in an infinite amount of time then the result is NOT random. Period.
If it is truly random then as the number of instances tends to infinity, the outcomes approach the probability function of that randomness.
E.g. if you toss a coin an infinite number of times, the results will be 50:50 Heads/Tails.
If it doesn't then the assumption of the nature of the randomness needs revising.

So if you accept the assumption of randomness - e.g. a flat probability distribution across the 50 or so characters (letters + punctuation etc) - and there is a probability of hitting the keys in the right order to produce Shakespeare's complete works... then it WILL happen.



Doesn't that presume that the monkey can not repeat himself?No. Repetition is permitted and expected, but if he repeats himself so as to disclude the possibility of other combinations then his typing is NOT random.


Hell, you can't even prove that a single word MUST be produced if the monkey is typing randomly.Yes you can.
But you have to accept the assumption of randomness and understand what it means - i.e. that as the number of instances of an event tend to infinity (e.g. tapping of keyboard) then the outcome tends to the probability function of the randomness.

If we assume a flat probability function (all 100 characters of so, - letters, punctuation, including CAPS - having equal chance of being typed) then given an infinite amount of time each character would be hit 1% of the time. Not only that, but in every subdivision of the overall result you would expect the same distribution. The larger the sample, the closer to the probability function you get.
Also, if you take every-other-letter then this will have the same probability function... as will every-fifth-letter... every-100th-letter etc... because they are all infinite in length.



Certainly it is likely that he will type the word "blue" at some point, but not impossible that he will not.
On the flip side, it is certainly possible that he will type Moby Dick in it's entirety, but not likely at all.It is impossible NOT to type "blue" given an infinite time.
It is also impossible NOT to type the the complete Moby Dick, given randomness and given infinite time.

If you think it is possible that, in the entire infinite time, the monkey only ever types out the letter "A", for example, then there is no randomness.


If there is a chance of occurring, given an infinite time and the same conditions then it WILL occur.
The only things that won't are those that are impossible due to those conditions, or become impossible due to changing conditions.

Therefore if a certain combination in an infinite random sequence is possible, then it will occur.
And if a certain combination in an infinite random sequence is impossible then the sequence is not as random as you thought - and the assumptions need revising.

A picture made with randomness, fractals.

http://dev.gentoo.org/~dberkholz/wallpaper/shore_of_the_fractal_sea-1600.jpg

Thanks, Sarkus. I am out of town and hate that I can't find the time to respond to some of my favorite threads. I really enjoyed your post.

Given an infinite amount of time, if something does not happen then it is not possible. If there is ANY chance of it occurring, however remote, then it WILL happen.

Time is not known to actually be infinite. I'm also not sure that both your conclusions are supportable. They both make the assumption that all events which can happen have a discrete, independent chance of happening at every instant. That seems an unwarranted assumption.

The exercise assumes an infinite amount of time... it is thus irrelevant whether time actually is infinite or not.

And the conclusions are logically sound. You have merely taken the quote out of context (i.e. we are discussing the monkey tapping at random for an infinite time - with the assumptions of A: infinite time, and B: random tapping).
Further in my post I illucidated, stated:
If there is a chance of occurring, given an infinite time and the same conditions then it WILL occur.
The only things that won't are those that are impossible due to those conditions, or become impossible due to changing conditions.Perhaps you missed this part?

Normally after an event has occurred, the conditions will have changed so as to make a repeat of the event impossible, which is why in reality not every outcome occurs, and normally why only one outcome from all the possibilities occurs.

E.g. If you allow only one toss of a coin, each outcome (Heads/Tails) has a 50:50 chance. It lands on HEADS, but after the initial toss the conditions have changed so as not to allow a further toss.
The outcome of TAILS thus becomes impossible.

But the essence of probability is that as the number of identical events approaches infinite, the outcomes of those events approach the probability function of the event. Do you agree with this?

Therefore, if you assume the entire work of Shakespeare is 100 million characters long, and you take each string of 100 million characters as an event... and there is ANY possibility that the monkey will randomly tap 100 million characters in the right order... then given an infinite number of attempts at this 100-million character string the monkey WILL type it out correctly.

In such a scenario - if the monkey does not type out something you think is possible (however remote that possibility) then it was not actually possible, and your assumption of possibility was wrong.

I hope this clarifies.

Very well.

Though I still find it amusing that the monkeys actually wrote Shakespeare's complete works before they invented the typewriter to bang on resulting in an undefined answer to the question.

Shakespeare's works = finite number of characters (letters and spaces).

Monkey types for an eternity (towards infinity)?

Monkey types each of Shakespeare's works many times.

Given an infinite amount of time, if something does not happen then it is not possible. If there is ANY chance of it occurring, however remote, then it WILL happen.
Assertion is not proof.
I don’ t buy it.
Show me why just because it IS possible means it WILL happen.
You have not done that.


e.g. if you have a coin where you think that the result is random, if it never landed on TAILS in an infinite amount of time then the result is NOT random. Period.
First off, random implies the event was not impacted in any way by any extenuating circumstances.
We are talking about a primate (a living being who suffers from conditioned responses) banging on a mechanical device. “Truly random” is not possible.
In fact, I’m not certain that “Truly random” is possible at all, in reality.
That aside…


If it is truly random then as the number of instances tends to infinity, the outcomes approach the probability function of that randomness.
E.g. if you toss a coin an infinite number of times, the results will be 50:50 Heads/Tails.
If it doesn't then the assumption of the nature of the randomness needs revising.

So if you accept the assumption of randomness - e.g. a flat probability distribution across the 50 or so characters (letters + punctuation etc) - and there is a probability of hitting the keys in the right order to produce Shakespeare's complete works... then it WILL happen.
The longer time you have the greater the degree of probability. I agree with that.
However, extrapolation from that to something definitely happening is where this falls apart.



If you think it is possible that, in the entire infinite time, the monkey only ever types out the letter "A", for example, then there is no randomness.
I disagree.
Randomness implies, as I said, that the outcome is not impacted by any external factors.
If you flip a coin and it lands on heads, it is STILL a 50/50 chance that it will land on heads again.
Thus, if it ever landed on heads with that 50/50 chance, then it is certainly possible for it to never land on tails, because each time that chance is STILL 50/50.
If it is possible to land on heads once, it is possible to land on heads every time.


If there is a chance of occurring, given an infinite time and the same conditions then it WILL occur.
First of all, conditions do not matter, because for it to be “Truly Random” the conditions can not affect the outcome at all.
It has to be a fresh slate each time.
Secondly, with that reasoning, you can say that it is possible for him to press A repeatedly for infinity.
There IS a chance of it occurring.


Therefore if a certain combination in an infinite random sequence is possible, then it will occur.
And if a certain combination in an infinite random sequence is impossible then the sequence is not as random as you thought - and the assumptions need revising.
Nonsense.

Is it unlikely to flip a coin once a minute for a week and it land on heads 100% of the time?
Of course.
Is it possible?
Of course.

Regardless, the expression is meaningless.

Nothing is truly random. Everything has a cause.

I don’ t buy it.

A discussion of the monkeys: http://en.wikipedia.org/wiki/Infinite_monkey_theorem

Nothing is truly random. Everything has a cause.

That claim is less supportable than god.

In fact, I’m not certain that “Truly random” is possible at all, in reality.
That aside…That matter of it being has already been discussed - and we are now talking about true randomness. Afterall, a monkey will just as likely pick the typewriter up and hurl it across the room within a few minutes of starting, not to mention be unable to live an infinite life?


Assertion is not proof.
I don’ t buy it.
Show me why just because it IS possible means it WILL happen.
You have not done that.Apologies - an oversight on my part

Take the coin-toss...
Probability of heads = 0.5, agreed (ignoring the chance of landing on its edge)...

The probability of 2 heads in a row... 0.5^2 = 0.25.
The probability of 10 heads in a row... 0.5^10 = 0.0009766 (1/1024).

As the number of heads in a row increases, the probability tends to zero.
At infinite times, the probability IS zero.


Similarly one can prove that after an infinite number of times something that has even a remote probability on a specific occurrence of the event WILL happen if an infinite number of those occurrences take place:

Take an event that has a 1 in a billion chance:
P(event) = 0.000000001
P(not-event) = 0.999999999

P(not-event) after 1 billion attempts = P(not-event)^1billion = 0.36787944

If you work it out after an 2 billion attempts, you get 0.1353353.
After 4 billion attempts: 0.0183...
After 8 billion attempts: 0.0003354...

And after an infinite number of times... : ZERO

I.e. the probability of the event not occurring after an infinite number of times is ZERO.
Therefore the event WILL happen, no matter how small the probability of it occurring on each individual occurrence.

You could take an event that has any finite chance of occurring (e.g. typing out the works of Shakespeare).
Only something that has ZERO chance will not happen given an infinite number of attempts.


First of all, conditions do not matter, because for it to be “Truly Random” the conditions can not affect the outcome at all.
It has to be a fresh slate each time.
Secondly, with that reasoning, you can say that it is possible for him to press A repeatedly for infinity.
There IS a chance of it occurring.The only way this would be possible if the tapping of the keyboard was NOT random, and the chance of tapping A on any single occurrence was 100%...

P(tapping A), in that case, would be 1.0.
The chance of it occurring every time for an infinite time would thus also be 1.0.
However, if P(tapping A) < 1.0, then the chance of it occurring an infinite number of times is ZERO.

There is certainly a chance of it occurring in a finite number of taps, but not an infinite.
See the maths above.



Nonsense.

Is it unlikely to flip a coin once a minute for a week and it land on heads 100% of the time?
Of course.
Is it possible?
Of course.Flawed argument as you are now discussing a finite time-period, not the infinite one being discussed.

Infinity basically equates to all possibilities. Your bound to get lucky 100,000 times in a row when you have eternity.

Take an event that has a 1 in a billion chance:
P(event) = 0.000000001
P(not-event) = 0.999999999

P(not-event) after 1 billion attempts = P(not-event)^1billion = 0.36787944

If you work it out after an 2 billion attempts, you get 0.1353353.
After 4 billion attempts: 0.0183...
After 8 billion attempts: 0.0003354...

And after an infinite number of times... : ZERO

I.e. the probability of the event not occurring after an infinite number of times is ZERO.
Therefore the event WILL happen, no matter how small the probability of it occurring on each individual occurrence.


Great explanation, Sarkus.

Nothing is truly random. Everything has a cause.


That claim is less supportable than god.

HUH???
There is something which isn't caused?

HUH???
There is something which isn't caused?

Matter and energy are neither created nor destroyed. They therefore are not caused.

Some quantum events are not caused.

If there ever is or was anything that wasn't caused then anything can happen anytime any number of times without any cause.

Matter and energy are neither created nor destroyed. They therefore are not caused.
So you believe that they always existed and the Big Bang is a bunch of nonsense?
Regardless, we are talking about events being caused.
Without causation, matter would be stagnant, and all energy would simply be potential energy.


Some quantum events are not caused.
This has not been proven, just postulated to account for where the Copenhagen Interpretation as fellen short of reality.

So you believe that they always existed and the Big Bang is a bunch of nonsense?

Big bang does not imply 'creation' of energy. Who is to say that energy did not come from somewhere else, converted at the time of the big bang?



Without causation, matter would be stagnant, and all energy would simply be potential energy.

What makes you think the potential itself isnt cause enough?

Big bang does not imply 'creation' of energy. Who is to say that energy did not come from somewhere else, converted at the time of the big bang?
Not me, but the Big Bang does.


What makes you think the potential itself isnt cause enough?
Because then it wouldn't be "Potential Energy".

Please explain to me a scenario in which potential energy can be the cause.
Even if there was always matter and energy, something has to make the first move, somehing had to cause the first spark, something had to be the first cause - unless the universe is as it always has been.
Either there was a first cause, or there was not - I don't see how potential energy could have been that cause.

Ok I see. Whats the kick that converts the potential to kinetic?

I dunno, how about a single quantum fluctuation and huge potential I guess.

If there ever is or was anything that wasn't caused then anything can happen anytime any number of times without any cause.

There is no reason to believe your statement is true.

That specific things happen without cause (such as the existence of the universe and the spontaneous creation and destruction of particles in vacuum decay in the presence overcritical electromagnetic fields), doesn't necessarily imply that "anything can happen anytime any number of times without any cause."

So you believe that they always existed and the Big Bang is a bunch of nonsense?

Matter and energy are neither created nor destroyed.

The big bang is a transition point from one state to another.

Matter and energy are neither created nor destroyed.

The big bang is a transition point from one state to another.

That transition, as any transition, was caused.

There is no reason to believe your statement is true.

That specific things happen without cause (such as the existence of the universe and the spontaneous creation and destruction of particles in vacuum), doesn't necessarily imply that "anything can happen anytime any number of times without any cause."

It certainly does.