understanding probability theory... Infinite monkey theorem
- Thread starter sao123
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theres nothing too difficult here, each time a moneky hits a key at complete random there is a chance it could be the right key, say a 1/32 chance or whatever (not sure how many total letters and punctuation marks there are), so if there are a million letters, then thats jsut (1/32)^1,000,000 chacne you get that. Obvisouly thats an insanely small number, but multiply it by infinity and its a certainty. Its just another stupid mind game seeing as there arent an infinite number of monkeys, but it does prove the point that even an incredably unlikely event will happen if there are an even larger number of chances. Like a trillion bacteria over 1000 generations evolving a resistance to a drug, the chance of a complely random mutation creating a drug reistance is really small, but when you have 10^12 chances to get it right you just might.
Well, if you have a monkey on a keyboard for an infinite amount of time, eventually it will hit the keys in a sequence that recreates (insert famous work here). In the timeframe of infinity, any event that even has the slightest probability of happening will - if it never happened, then there would be no way to attach a probability to it.
Do you understand that a monkey hitting keys at random and independently will hit the key A eventually with probability 1?
DrPizza
Administrator Elite Member Goat Whisperer
Simply tell your professor that you disagree...
When he asks why, show, via a quick calculation, that even though you'll allow for an infinite amount of monkeys, and an infinite amount of time, the professor said nothing about an infinite amount of ink for the printers. The probability is so low that if every keystroke represented 1 particle in the universe, you'd run out of particles long before it happened.
Not to mention that the keyboard will wear out!
Seriously, though, I see your point, and I agree with you to a certain degree. I'd love nothing more than to see some sort of Eureka proof which somehow proves that statement incorrect. (But, I still acknowledge that the statement is true... at this point.)
When he asks why, show, via a quick calculation, that even though you'll allow for an infinite amount of monkeys, and an infinite amount of time, the professor said nothing about an infinite amount of ink for the printers. The probability is so low that if every keystroke represented 1 particle in the universe, you'd run out of particles long before it happened.
Not to mention that the keyboard will wear out!Seriously, though, I see your point, and I agree with you to a certain degree. I'd love nothing more than to see some sort of Eureka proof which somehow proves that statement incorrect. (But, I still acknowledge that the statement is true... at this point.)
- May 27, 2002
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Originally posted by: DrPizza
Simply tell your professor that you disagree...
When he asks why, show, via a quick calculation, that even though you'll allow for an infinite amount of monkeys, and an infinite amount of time, the professor said nothing about an infinite amount of ink for the printers. The probability is so low that if every keystroke represented 1 particle in the universe, you'd run out of particles long before it happened.
Seriously, though, I see your point, and I agree with you to a certain degree. I'd love nothing more than to see some sort of Eureka proof which somehow proves that statement incorrect. (But, I still acknowledge that the statement is true... at this point.)
If I only had a professor to ask... sadly I did not think of this six years ago.
Actually I found this, because it was one of todays featured articles on wikipedia.
now back to my thoughts...
Assuming a 104 key standard keyboard...
the chance of not typing the word banana in a block of 6 letters is 1 - 1 / 104^6.
the chance of not typing the word banana in the first Z blocks of 6 letters is
(1 - 1 / 104 ^ 6) ^ Z Az Z grows... the probability of not typing the string get smaller, thus eventually guaranteeing the string will be typed...
so far so good.
but for strings approaching very large values, would cause the second term to shrink exponentially, thus leading to an infinity vs infinity concept.
(1 - 1 / 104 ^ ? ) ^ ? which this should either lead to an indeterminate value or 0. Either way, the theory should not hold in this case.
(forgive me I lost the alt code for infinity)
You have an upper bound with your formula, which is what you need. The probability of a monkey typing a particular countably infinite sequence eventually is zero. William Shakespeare's works are of finite length, not infinite.
almost impossible != impossible
almost impossible = improbable but possible, and certain to occur at least once with an infinite number of trials
Flip a coin millions of times. Even a coin that is quite unfair is still going to give you at least one heads. Roll a dice millions of times. You're going to get a streak of three one's in a row at least once. Sure, writing the complete works of Shakespeare is very unlikely, but given the conditions of the problem, the infinte number of trials would dictate that it's possible.
I'm not holding my breath, though.
really, I think people are overcomplicating these things. If you are willing to accept that there are no laws of physics violated by this occurance, and that there is some increabibly small chance that it can happen then you should understand that when you multiply that chance by infinity the occurance is a sure thing to happen. Now, of course it is just a stupid little statement that will never happen in real life, but the point is still valid.
The real application as I stated previously is that it shows how even a very unlikely occurance can and will occur if there are an equally large number of chacnes for it to occur. Just look at some famous engineering disasters, and often you will see an series of very unlikely mishaps leeding up to the eventual disastor, but when you consider all the millions of places there are for things to go wrong, its inevitable that sometimes they will. Like the cance of lightning hitting a dry area in the middle of a forest and starting a forest fire is very low, but every year there are forest fires, or people getting struck by lighting works too. I mean hell, even look at the chacnes of a single celled sperm meeting a single egg cell is incredibly unlikely, but try a few million of the suckers and it led to every single one of us being here today...
Think about Murphys law "if it is possible for something to go wrong then eventuall it will go wrong", this is where I see the biggest application of this principle. Now many people might consider this to be a pessimistic outlook on life, but it is an extremely valid observation that if ther is a finite probability of failure then eventually failure will occur. I'm only an engineering student, and already I have learned that whatever you design it has to be designed with considerable tolerances left in place, and even with that there will always be some idiot who manages to screw it up. I'm sure many of us here have had to write programs where the error checking portion was larger then the actualy program itself, certainly I have argued in the past with teachers about what type of complete idiot would type numbers into a text box asking for their name, or type the number 5 billion into one asking for their age, but you HAVE to assumme that someone will, because eventually it WILL happen.
The real application as I stated previously is that it shows how even a very unlikely occurance can and will occur if there are an equally large number of chacnes for it to occur. Just look at some famous engineering disasters, and often you will see an series of very unlikely mishaps leeding up to the eventual disastor, but when you consider all the millions of places there are for things to go wrong, its inevitable that sometimes they will. Like the cance of lightning hitting a dry area in the middle of a forest and starting a forest fire is very low, but every year there are forest fires, or people getting struck by lighting works too. I mean hell, even look at the chacnes of a single celled sperm meeting a single egg cell is incredibly unlikely, but try a few million of the suckers and it led to every single one of us being here today...
Think about Murphys law "if it is possible for something to go wrong then eventuall it will go wrong", this is where I see the biggest application of this principle. Now many people might consider this to be a pessimistic outlook on life, but it is an extremely valid observation that if ther is a finite probability of failure then eventually failure will occur. I'm only an engineering student, and already I have learned that whatever you design it has to be designed with considerable tolerances left in place, and even with that there will always be some idiot who manages to screw it up. I'm sure many of us here have had to write programs where the error checking portion was larger then the actualy program itself, certainly I have argued in the past with teachers about what type of complete idiot would type numbers into a text box asking for their name, or type the number 5 billion into one asking for their age, but you HAVE to assumme that someone will, because eventually it WILL happen.
- May 27, 2002
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does this theory then necessarily rule out all occurances of singularly occuring random phenomenon?
Lets say event X happen within the first countable infinity of chances. Using the fact that event X happened, this determines that the probability of event X must be greater than zero. By removing that countable space, I still have an uncountable remaining infinity of space left. I can reapply that theory to say that it must happen again.
This seems to null and void statements like lightning never strikes the same place twice...indeed it must strike that place an infinite many times.
Equivolly, life must exist on multiple planets... and not just life, an exact mirror of earth and humans must exist somewhere.
This seems to specifically state that all random events must happen infinitely number of times.
That seems to contradict the whole concept of random probability.
Lets say event X happen within the first countable infinity of chances. Using the fact that event X happened, this determines that the probability of event X must be greater than zero. By removing that countable space, I still have an uncountable remaining infinity of space left. I can reapply that theory to say that it must happen again.
This seems to null and void statements like lightning never strikes the same place twice...indeed it must strike that place an infinite many times.
Equivolly, life must exist on multiple planets... and not just life, an exact mirror of earth and humans must exist somewhere.
This seems to specifically state that all random events must happen infinitely number of times.
That seems to contradict the whole concept of random probability.
Yeah, I know. I just remembered someone posting a study here on Anandtech a while back where monkeys were actually given a typewriter. They did hit the keys but they showed a preference for certain letters.
comeon people, its not suppsoed to frikin represent the real world, it has nothing to do with typewritters or monkey, or anything like that. All it says is that if something has a finite probability and an infinite number of chacnes to occur then it will occur. The whole monkey on a typewritter thing is just trying to make a funny example of a mathmatical formula. I dunno, maybe some people here think its clever to point out that mokeys would jsut throw poo at the keyboard, or that hte typewritter would break, but that is just missing the entire point.
It rules out a singly occuring event whose probability of occurence at each period is independent of other occurences.
yes, the monkey-keyboard analogy is an attempt to make the concept readily understandable. But what you cannot miss is the fact that there is A monkey and A keyboard. You must have something acting apon something else for this to be true. Just because life is present here does not mean that life MUST be present elsewhere, given an infinite amount of time.
I can put a bird in a cage and feed him. In due time, he will crap on the newspaper. I can also set up a bird cage with no bird, but no matter how long I wait, there will never be crap on the newspaper. This is of course ignoring the idea that given enough time a bird just might evolve in the cage, and then would mess the paper (if you subscribe to the theory of evolution). But that brings the argument full circle; from what will the bird evolve if nothing is present in the cage? Given an infinite amount of chances, will the amino acids form themselves from the vacuum?
I can put a bird in a cage and feed him. In due time, he will crap on the newspaper. I can also set up a bird cage with no bird, but no matter how long I wait, there will never be crap on the newspaper. This is of course ignoring the idea that given enough time a bird just might evolve in the cage, and then would mess the paper (if you subscribe to the theory of evolution). But that brings the argument full circle; from what will the bird evolve if nothing is present in the cage? Given an infinite amount of chances, will the amino acids form themselves from the vacuum?
DrPizza
Administrator Elite Member Goat Whisperer
Of course, if you subscribe to the laws of quantum mechanics, there is a finite probability that the bird in your neighbors house will spontaneously disappear from the cage there and reappear in your cage (in which case it would immediately crap on the newspaper. At least, I think it would. I know that if I spontaneously did that, I'd crap my pants, assuming they were still with me.)
Can't we test this, just for fun?
IIRC, Java has a random number generator. If a programmer could develop a random letter generator I'm sure someone would be willing to run it on their computer.
Perhaps a contest in Distributed Computing on who gets Macbeth first?
IIRC, Java has a random number generator. If a programmer could develop a random letter generator I'm sure someone would be willing to run it on their computer.
Perhaps a contest in Distributed Computing on who gets Macbeth first?
- May 27, 2002
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