Monkey Typing For Infinite = Shakespeare?

Lazar · **#121** by **Lazar** » Sep 09, 2010 1:46 pm

*Silly me Roger Cook already posted this. Consider this an expansion for simple folk like me. :grin:

Good overview of the probability behind this question at talk stats (post four) here by on of the mods.

Dason@Talkstats wrote:.......[snip]

Take for example coin flipping. If we flip a perfectly fair coin we would expect that after an infinite number of flips the number of heads and the number of tails should be equal. Now listen closely.... In that infinite sequence it is almost sure that you will see a sequence of heads as long as K. What is K you ask? Who cares, it can be any natural number. Let's say K is 8 for the time being. If we randomly group the infinite sequence into blocks of 8 then each block of 8 has probability 1/2^8 of being all heads. Now if we use the naive approach and only look at the successive blocks of 8 you can ask what the probability is that you won't have one of those blocks be a complete run of heads. The probability that any block won't be a run of 8 heads is 1-(1/2^8). So the probability that the neither of the first two blocks is the a sequence of heads is (1 - (1/2^8)^2. The probability that none of the first m blocks contains that sequence of heads is (1-(1/2^8)^M. You can see that 1-1/2^8 < 1 so if we take it to higher and higher powers it becomes smaller and smaller. (Note that I'm being really conservative in that the sequence of 8 heads could have been somewhere that didn't fit perfectly into one of our blocks but this route makes the calculation easier and gets the point across just as well). So taking M to infinity means that the probability that we don't see this sequence goes to 0. But my block size of 8 was completely arbitrary. This works for any block size. So if we have an infinite sequence the probability that there isn't a sequence of heads at least 23423136572352352345 long goes to 0. I hope you can see how this directly relates back to the problem you "disproved".

HughMcB · **#122** by **HughMcB** » Sep 09, 2010 2:29 pm

How can he be so specific that a "23423136572352352345 long goes to 0"? :scratch:

Lazar · **#123** by **Lazar** » Sep 09, 2010 2:39 pm

HughMcB wrote:How can he be so specific that a "23423136572352352345 long goes to 0"?

pick any number. Could have easily been 354865748354573486534657450 or what ever.

HAJiME · **#124** by **HAJiME** » Sep 10, 2010 10:48 am

Okay, firstly, I hate maths.

But probability I really do not get.

If I flip a coin however many times, It's fairly unlikely that exactly half of those times it will land on heads. Maybe it could be argued that if it was flipped in a controlled environment with a coin which weighs equal on both sides it would? Still do not believe it, has anyone proven that it is the case?

Take for example coin flipping. If we flip a perfectly fair coin we would expect that after an infinite number of flips the number of heads and the number of tails should be equal.

Can someone explain how it is possible to flip a coin 'an infinite number of times', and what exactly half of infinite is? If these questions cannot be answered, how do we know probability works...? Has anyone actually even proven that it works in the real world? If it's not possible in the real world, what is it's purpose? What's the point?

I always used to hate maths (and still do) because I fail to see it's working in the real world. I've never, ever, ever had to use anything bar very basic maths in my daily life... And even use of very basic maths is done on the calculator on my phone. I didn't want to do anything with maths for a career, so why on earth was I being taught all this useless bullshit?

Eduard · **#125** by **Eduard** » Sep 10, 2010 12:03 pm

HAJiME wrote:
Can someone explain how it is possible to flip a coin 'an infinite number of times', and what exactly half of infinite is? If these questions cannot be answered, how do we know probability works...? Has anyone actually even proven that it works in the real world? If it's not possible in the real world, what is it's purpose? What's the point?

I always used to hate maths (and still do) because I fail to see it's working in the real world. I've never, ever, ever had to use anything bar very basic maths in my daily life... And even use of very basic maths is done on the calculator on my phone. I didn't want to do anything with maths for a career, so why on earth was I being taught all this useless bullshit?

NNOOOOOO!

I can understand your dislike for maths, but its practical implications in the real world are abundant! Complicated math (I'm assuming anything more complicated than basic arithmetic) is everywhere: your cellphone, your computer, car, tv everywhere.

Math is not useless, it's helped in defining gravity, sending people to the moon. It's allowed us to formulate and solve incredibly complex or computive intensive problems we can't solve with simple arithmetic.

As for probability, well I only had calculus and stats to my third year in university, but to put it in basic terms (I might be wrong): if we can prove a mathematical theorem on paper it's true. In real life, at best we can do experiments to replicate and test these theorems, but they're always true. I'm talking about how probability theory shows that a coin toss gives use a 50/50 chance for heads or tails. Similarly, we can test all sorts of other math theories and find it to be true, like pi = 3.14... and Pythagoras' theorem with triangles. Without a lot of these things big industries we have today won't exist as they do.

Btw, if you divide infinity by 2 you still have infinity. I'll leave it to guys who are way more clued up to inform us on how infinity works in comparison to really really unimaginaly big numbers.

HAJiME · **#126** by **HAJiME** » Sep 10, 2010 8:07 pm

Eduard wrote:
HAJiME wrote:
Can someone explain how it is possible to flip a coin 'an infinite number of times', and what exactly half of infinite is? If these questions cannot be answered, how do we know probability works...? Has anyone actually even proven that it works in the real world? If it's not possible in the real world, what is it's purpose? What's the point?

I always used to hate maths (and still do) because I fail to see it's working in the real world. I've never, ever, ever had to use anything bar very basic maths in my daily life... And even use of very basic maths is done on the calculator on my phone. I didn't want to do anything with maths for a career, so why on earth was I being taught all this useless bullshit?

NNOOOOOO!

I can understand your dislike for maths, but its practical implications in the real world are abundant! Complicated math (I'm assuming anything more complicated than basic arithmetic) is everywhere: your cellphone, your computer, car, tv everywhere.

Math is not useless, it's helped in defining gravity, sending people to the moon. It's allowed us to formulate and solve incredibly complex or computive intensive problems we can't solve with simple arithmetic.

The problem is, I was FORCED to retake Math GCSE to gain at least a C grade when I took my A-Levels (I realise that if you're not British none of that made sense, but basically they wouldn't let me continue studying unless I retook maths)... When I wanted to take arts. Why? It's utterly ludicrous. I do not have to use complicated mathematics in my daily life and never will. It's irrelevant how important it is to society. No?

As for probability, well I only had calculus and stats to my third year in university, but to put it in basic terms (I might be wrong): if we can prove a mathematical theorem on paper it's true. In real life, at best we can do experiments to replicate and test these theorems, but they're always true. I'm talking about how probability theory shows that a coin toss gives use a 50/50 chance for heads or tails. Similarly, we can test all sorts of other math theories and find it to be true, like pi = 3.14... and Pythagoras' theorem with triangles. Without a lot of these things big industries we have today won't exist as they do.

Btw, if you divide infinity by 2 you still have infinity. I'll leave it to guys who are way more clued up to inform us on how infinity works in comparison to really really unimaginaly big numbers.

Does not compute. :C

It's true... because it's true?

I want to see evidence that in the real world a coin toss is really a 50/50 chance. If it's not applicable to the real world, why is it relevant at all?

Kid A · **#127** by **Kid A** » Sep 11, 2010 9:40 am

Lazar wrote:*Silly me Roger Cook already posted this. Consider this an expansion for simple folk like me.

Good overview of the probability behind this question at talk stats (post four) here by on of the mods.

Dason@Talkstats wrote:.......[snip]

Take for example coin flipping. If we flip a perfectly fair coin we would expect that after an infinite number of flips the number of heads and the number of tails should be equal. Now listen closely.... In that infinite sequence it is almost sure that you will see a sequence of heads as long as K. What is K you ask? Who cares, it can be any natural number. Let's say K is 8 for the time being. If we randomly group the infinite sequence into blocks of 8 then each block of 8 has probability 1/2^8 of being all heads. Now if we use the naive approach and only look at the successive blocks of 8 you can ask what the probability is that you won't have one of those blocks be a complete run of heads. The probability that any block won't be a run of 8 heads is 1-(1/2^8). So the probability that the neither of the first two blocks is the a sequence of heads is (1 - (1/2^8)^2. The probability that none of the first m blocks contains that sequence of heads is (1-(1/2^8)^M. You can see that 1-1/2^8 < 1 so if we take it to higher and higher powers it becomes smaller and smaller. (Note that I'm being really conservative in that the sequence of 8 heads could have been somewhere that didn't fit perfectly into one of our blocks but this route makes the calculation easier and gets the point across just as well). So taking M to infinity means that the probability that we don't see this sequence goes to 0. But my block size of 8 was completely arbitrary. This works for any block size. So if we have an infinite sequence the probability that there isn't a sequence of heads at least 23423136572352352345 long goes to 0. I hope you can see how this directly relates back to the problem you "disproved".

So what are the implications of that for the Shakespeare example in simple-folk terms?

Fooglmog · **#128** by **Fooglmog** » Sep 11, 2010 10:13 am

It depends on how you conduct the experiment.

If you have a monkey produce a document in which every character is chosen entirely at random, then repeat if the document is not an exact match to the works of Shakespeare: It's possible the monkey would produce the works of Shakespeare, but it's very unlikely. This is because the result set is also infinite. Infinity theory clearly demonstrates that an infinite number of attempts will not yield every result if the potential result set is also infinite.

If you follow the above procedure, but stop the monkey after it's exceeded the number of characters in Shakespeare: It's certain the monkey would produce the works of Shakespeare. This is because with a limited result set, an infinite number of attempts will always result in every outcome.

If you step outside of established infinity theory and treat infinity like an integer (which is what most mathematical approaches seem to have done in this thread) you can easily manipulate circumstances to produce any of the given possibilities. These are, however, what I view as the best two answers.

StatAnalyst · **#129** by **StatAnalyst** » Oct 02, 2011 9:59 pm

My favorite answer:

by Audley Strange » Aug 26, 2010 11:37 am
"Not if the monkey just pressed the caps lock key for all eternity."

The actual Infinite Monkey Theorem assumes that every key has an equal probability of being pressed. Hmmmm, it seems to me that given an infinite number of infinitely large possible permutations (since any key can be pressed an infinite number of times during our infinite period), there are infinitely many possible key combinations that might be infinitely repeated, thus never resulting in the complete works of Shakespeare. Another option is something like an irrational decimal that is infinite and non-repetitive. Although I suppose one could find as subsets of such a set, the desired patterns. In fact, I guess a data set that never repeats is a good example of the theorem in effect since it, by definition, would have to eventually contain all information. Well, in that specific case the outputs are restricted to a ten element dataset, so maybe not. Wait a minute, now I'm just crazy; Whatever . . . The converse result that might be realized using the logic of the former example, is our infinite monkey might produce the complete works of Shakespeare, perfectly typed in chronological order an infinite number of times.

Well, that's enough silliness for now.

cavarka9 · **#130** by **cavarka9** » Oct 15, 2011 12:41 pm

lets just add something to this infinite monkey guesses.

consider that there are infinite monkeys typing infinitely,randomly, will they get the value of say pi.They only have ten symbols and the dot symbol.

Then, will any of those monkeys keep typing the value of pi?. If it does, then we might as well say that the function of the monkey over those keys results in pi.?
I am not sure, does randomness mean nothing or does it mean everything that there is?.

cavarka9 · **#131** by **cavarka9** » Oct 16, 2011 5:44 pm

cavarka9 wrote:lets just add something to this infinite monkey guesses.

consider that there are infinite monkeys typing infinitely,randomly, will they get the value of say pi.They only have ten symbols and the dot symbol.

Then, will any of those monkeys keep typing the value of pi?. If it does, then we might as well say that the function of the monkey over those keys results in pi.?
I am not sure, does randomness mean nothing or does it mean everything that there is?.

anyone?

home_ · **#132** by **home_** » Oct 16, 2011 9:05 pm

cavarka9 wrote:consider that there are infinite monkeys typing infinitely,randomly, will they get the value of say pi.They only have ten symbols and the dot symbol.

Depends on what do you mean by "infinity". If you mean countable infinity (=the same as natural numbers), then the anwser is: probability that one of the monkeys will get the value of Pi is 0, but this doesn't mean that it can't happen. But if you mean continuum infinite number of monkeys (the same as real numbers), then the anwser is: probability that some monkey will get number Pi is equal to 1, but that doesn't mean that it will surely happen.

cavarka9 wrote:I am not sure, does randomness mean nothing or does it mean everything that there is?.

There are different distributions of randomness.. But what you are asking is not distrubution function, but what kind of events can occur. The anwser is: any kind of events that are contained in your probability space.
For example (this is very informal): you can define your probability space in a way that an event is "when monkey successfully hits a button", and then you can try to find out what's the proportion of the events, that contain some specific sequence of characters. Anwser to your question of randomness in this particular case is: it means all possible sequences of characters.

cavarka9 · **#133** by **cavarka9** » Oct 17, 2011 4:06 am

home_ wrote:
cavarka9 wrote:consider that there are infinite monkeys typing infinitely,randomly, will they get the value of say pi.They only have ten symbols and the dot symbol.
Depends on what do you mean by "infinity". If you mean countable infinity (=the same as natural numbers), then the anwser is: probability that one of the monkeys will get the value of Pi is 0, but this doesn't mean that it can't happen. But if you mean continuum infinite number of monkeys (the same as real numbers), then the anwser is: probability that some monkey will get number Pi is equal to 1, but that doesn't mean that it will surely happen.

cavarka9 wrote:I am not sure, does randomness mean nothing or does it mean everything that there is?.
There are different distributions of randomness.. But what you are asking is not distrubution function, but what kind of events can occur. The anwser is: any kind of events that are contained in your probability space.
For example (this is very informal): you can define your probability space in a way that an event is "when monkey successfully hits a button", and then you can try to find out what's the proportion of the events, that contain some specific sequence of characters. Anwser to your question of randomness in this particular case is: it means all possible sequences of characters.

yes, i knew that, i was looking for similar reasoning.

But it is strange that since the characters in pi goes to infinity, that monkey is typing the value of pi perfectly. Now how can one claim this monkey to be different from a real function meant to give the exact value of pi?.

Take this argument further, we then have within infinite, every conceivable non-random distributions too. That is, the value for e, and with slight modification, every conceivable non random series which converge to something say e or pi on infinite summation gives rise to this.

How do we differentiate a function with a purpose to a non-random distribution in some monkeys among the infinite monkeys.
(i know the answer

).

home_ · **#134** by **home_** » Oct 17, 2011 11:48 am

cavarka9 wrote:Now how can one claim this monkey to be different from a real function meant to give the exact value of pi?

I'm not sure if I understand your question. Monkey typing Pi is an event in probability space, but real function is a function to real numbers. You can't say these two are the same, that's like comparing apples and pears.

cavarka9 wrote:Take this argument further, we then have within infinite, every conceivable non-random distributions too. That is, the value for e, and with slight modification, every conceivable non random series which converge to something say e or pi on infinite summation gives rise to this.

I don't know what you're talking about. Distribution is distribution, and random variable is random variable... but what is non-random distribution, that's a mystery to me.

cavarka9 wrote:How do we differentiate a function with a purpose to a non-random distribution in some monkeys among the infinite monkeys.
(i know the answer ).

You know the anwser, but I don't even understand the question.

cavarka9 · **#135** by **cavarka9** » Oct 17, 2011 3:59 pm

I just said that if we were to see something which seems to make sense like say the appearance of all the works of Shakespeare, then that is mistaken, primarily because there is no intrinsic meaning to the symbols themselves. And in probability, its about possibilities of all the symbols coming together possible and so, even if we were to get the number of pi, it really isnt pi, its just the symbols arranging themselves in a way that to us it appears as pi as we use the same symbols.

a way of analogy of a fn which keeps giving a result and the random distribution, parts of which appear as non-random, a sort of Turing-test like similarity.

home_ · **#136** by **home_** » Oct 17, 2011 5:06 pm

cavarka9 wrote:it really isnt pi, its just the symbols arranging themselves in a way that to us it appears as pi as we use the same symbols

This is philosophical question (i.e. "what Pi really is?"), sorry, I don't have anything to say about this one.

cavarka9 · **#137** by **cavarka9** » Oct 17, 2011 5:09 pm

home_ wrote:
cavarka9 wrote:it really isnt pi, its just the symbols arranging themselves in a way that to us it appears as pi as we use the same symbols
This is philosophical question (i.e. "what Pi really is?"), sorry, I don't have anything to say about this one.

there is a way to test that, take the monkey which is typing the value of pi, and restart it, because it is random infinite monkey, the chances of it repeating it would be :ask:

, any way, it wouldnt keep giving the value for pi everytime we start it over again, therefore the monkey isnt really typing pi in the first place.

Ironclad · **#138** by **Ironclad** » Oct 17, 2011 5:20 pm

May I venture, I say that we already have the answer. If you sit a single-celled organism in front of a keyboard it would take approximately 3.5 billion years for it to churn out the Complete Works of Shakespeare.

Was it worth the wait? :grin:

Monkey Typing For Infinite = Shakespeare?

Would A Monkey Typing For Infinity Produce The Works Of Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Re: Monkey Typing For Infinite = Shakespeare?

Who is online