The defeater is each key needs to be statistically as likely as any other key to be pressed next, i.e. statistically independent events. For example after a monkey pressed S they are then just as likely to press K as W. If there is any reason they prefer a key or sequence you don’t get a normal distribution and they probably will never create any of Shakespeare’s works.
haha no, if you have a magic infinite sized room with an infinite number of magic, immortal monkeys, with an infinite number of typewriters with infinite paper and ink, you don’t even need to park the monkeys in front the fucking keyboard, you will instantaneously have all the works of Shakespeare and every other book ever published and every book never published, and probably an infinite number of volumes of books that reveal every secret of the universe. (The hard part will be finding them.)
Instantly.
Just by having the means for anything random to happen to those keyboards on an infinite scale. The thought experiment isn’t designed to make you believe that anything is possible as much as it’s designed to show you the absurdity of infinity as a concept.
But they still would be limited to only what monkeys can actually do with typewriters given enough time or monkeys to do everything a monkey will do with a typewriter.
Infinity only allows anything that can happen to happen no matter how unlikely to happen, but it doesn’t allow something that has 0% likelihood to happen like a monkey turning into a cup to happen. If there are any 0% probability events necessary for the task then it wouldn’t happen regardless of the number of monkeys or given time.
But they still would be limited to only what monkeys can actually do with typewriters given enough time or monkeys to do everything a monkey will do with a typewriter.
Not arguing this at all, I think a lot of people get hung up on this though because they don’t actually know what’s “possible or impossible” in our universe, which may not in fact have a good answer. All that aside, it’s just a thought experiment to reveal the inherent problems with working with infinities, because the number of “possible” things that can happen are quite radical.
Yeah I think we’re on the same page there, I was just pointing out a limitation of the thought experiment that draws attention to the fact that infinity only allows what’s improbable possible and doesn’t make the impossible possible. But yeah it doesn’t undermine the idea that introducing infinities gives unintuitive results.
I agree, and I think it’s an absolutely fascinating area to study, because it does touch on some very important questions about our universe. We still don’t know if on the most fundamental levels, if our universe is constrained in some way, or if given enough time everything can change including those constants. I think about this a lot, but there are a surprising number of people who can’t grasp the ideas and problems, so apologies if I came on strong, I just want to make sure we’re all talking about the same things.
Yeah I think the recentness of formalizing infinities into math with Newton’s and Leibnez’s calculus (infinite series, limits approaching infinity) in the 1600s and Cantor’s sets (cardinality of infinite sets) in the late 1800s speaks to the difficulty of even conceptualizing the problems they introduce and the rigor needed to handle them
What? That’s not what independence means. They need to be independent, yes, because otherwise you might get into weird corner cases where the probably doesn’t converge to 1, but they don’t have to be equally likely. In fact, weighing the odds based on how often letters are used by Shakespeare should lower the expected timeframe. Heck, Shakespeare doesn’t use “J”, why would that key even be relevant? Where in the world do normal distributions even come into this? How does this comment have 4 upvotes? What am I missing here?
Those are some of the conditions necessary for the probability calculation to result in a non zero chance of writing the works of Shakespeare. From the article:
Consider the probability of typing the word banana on a typewriter with 50 keys. Suppose that the keys are pressed independently and uniformly at random, meaning that each key has an equal chance of being pressed regardless of what keys had been pressed previously. The chance that the first letter typed is ‘b’ is 1/50, and the chance that the second letter typed is ‘a’ is also 1/50, and so on. Therefore, the probability of the first six letters spelling banana is:
The result is less than one in 15 billion, but not zero.
But if they weren’t independent, say every time a monkey hits b their lack of fine motor skills causes them to also hit yhb all together, then even infinite monkeys with infinite time wouldn’t be able to type banana. Or if after hitting b they keep hitting b and ignore all the other keys they would never type banana. Evenly distributed just makes sure they can hit every key, it can take some unevenness like you mentioned j and some other letters come up very rarely. But if they never hit a or e you’re never going to get Hamlet.
No, I agree that independence is necessary, not just because of “always”, but because if, as a crude example, your odds of hitting B halve each time you hit A, an infinite number of tries isn’t guaranteed to give you Shakespeare, even if the odds aren’t technically 0. My problem was that what you originally described wasn’t independence, it’s uniformity, which isn’t a prerequisite. And it’s up to 9 upvotes now so I don’t know what’s going on.
In a literal and real universe, if you had real magic and could organize this in reality without it collapsing into a singularity instantly, then not only would you indeed get Hamlet instantly, you will also get monkey poop sculptures in every possible configuration, including models of us typing these words on PC monitors made entirely of monkey feces. They would just be fecal scultpures because, as you said, a monkey can’t poop actual electronics so pooping an entire, working internet with two people chatting is impossible… probably. You would get typewriters thrown around randomly until some collapse into black holes, until some accidentally form whole, working machines that do a vast array of functions, until some mix with the poop sculptures and create models of whole worlds. If you want to limit it to a mathematical thought experiment, then yes you can make points about key distribution and so on, you can make limitations that will constrain the outcomes, but in the real world all we really have to create those limitations is the speed of light and other fundamental conservation forces, which would constrain contradictory outcomes in an infinite model.
Again, the idea is to understand that infinity is an absurd idea, and we can’t work with infinities in any meaningful way.
The reference:
https://en.m.wikipedia.org/wiki/Infinite_monkey_theorem
The defeater is each key needs to be statistically as likely as any other key to be pressed next, i.e. statistically independent events. For example after a monkey pressed S they are then just as likely to press K as W. If there is any reason they prefer a key or sequence you don’t get a normal distribution and they probably will never create any of Shakespeare’s works.
You don’t need a normal distribution or statistical independence. It just requires that any given key combination remain possible.
No matter how unlikely, anything that is possible will eventually happen in an infinite time.
haha no, if you have a magic infinite sized room with an infinite number of magic, immortal monkeys, with an infinite number of typewriters with infinite paper and ink, you don’t even need to park the monkeys in front the fucking keyboard, you will instantaneously have all the works of Shakespeare and every other book ever published and every book never published, and probably an infinite number of volumes of books that reveal every secret of the universe. (The hard part will be finding them.)
Instantly.
Just by having the means for anything random to happen to those keyboards on an infinite scale. The thought experiment isn’t designed to make you believe that anything is possible as much as it’s designed to show you the absurdity of infinity as a concept.
But they still would be limited to only what monkeys can actually do with typewriters given enough time or monkeys to do everything a monkey will do with a typewriter.
Infinity only allows anything that can happen to happen no matter how unlikely to happen, but it doesn’t allow something that has 0% likelihood to happen like a monkey turning into a cup to happen. If there are any 0% probability events necessary for the task then it wouldn’t happen regardless of the number of monkeys or given time.
Not arguing this at all, I think a lot of people get hung up on this though because they don’t actually know what’s “possible or impossible” in our universe, which may not in fact have a good answer. All that aside, it’s just a thought experiment to reveal the inherent problems with working with infinities, because the number of “possible” things that can happen are quite radical.
Yeah I think we’re on the same page there, I was just pointing out a limitation of the thought experiment that draws attention to the fact that infinity only allows what’s improbable possible and doesn’t make the impossible possible. But yeah it doesn’t undermine the idea that introducing infinities gives unintuitive results.
I agree, and I think it’s an absolutely fascinating area to study, because it does touch on some very important questions about our universe. We still don’t know if on the most fundamental levels, if our universe is constrained in some way, or if given enough time everything can change including those constants. I think about this a lot, but there are a surprising number of people who can’t grasp the ideas and problems, so apologies if I came on strong, I just want to make sure we’re all talking about the same things.
Yeah I think the recentness of formalizing infinities into math with Newton’s and Leibnez’s calculus (infinite series, limits approaching infinity) in the 1600s and Cantor’s sets (cardinality of infinite sets) in the late 1800s speaks to the difficulty of even conceptualizing the problems they introduce and the rigor needed to handle them
What? That’s not what independence means. They need to be independent, yes, because otherwise you might get into weird corner cases where the probably doesn’t converge to 1, but they don’t have to be equally likely. In fact, weighing the odds based on how often letters are used by Shakespeare should lower the expected timeframe. Heck, Shakespeare doesn’t use “J”, why would that key even be relevant? Where in the world do normal distributions even come into this? How does this comment have 4 upvotes? What am I missing here?
Those are some of the conditions necessary for the probability calculation to result in a non zero chance of writing the works of Shakespeare. From the article:
But if they weren’t independent, say every time a monkey hits b their lack of fine motor skills causes them to also hit yhb all together, then even infinite monkeys with infinite time wouldn’t be able to type banana. Or if after hitting b they keep hitting b and ignore all the other keys they would never type banana. Evenly distributed just makes sure they can hit every key, it can take some unevenness like you mentioned j and some other letters come up very rarely. But if they never hit a or e you’re never going to get Hamlet.
No, I agree that independence is necessary, not just because of “always”, but because if, as a crude example, your odds of hitting B halve each time you hit A, an infinite number of tries isn’t guaranteed to give you Shakespeare, even if the odds aren’t technically 0. My problem was that what you originally described wasn’t independence, it’s uniformity, which isn’t a prerequisite. And it’s up to 9 upvotes now so I don’t know what’s going on.
In a literal and real universe, if you had real magic and could organize this in reality without it collapsing into a singularity instantly, then not only would you indeed get Hamlet instantly, you will also get monkey poop sculptures in every possible configuration, including models of us typing these words on PC monitors made entirely of monkey feces. They would just be fecal scultpures because, as you said, a monkey can’t poop actual electronics so pooping an entire, working internet with two people chatting is impossible… probably. You would get typewriters thrown around randomly until some collapse into black holes, until some accidentally form whole, working machines that do a vast array of functions, until some mix with the poop sculptures and create models of whole worlds. If you want to limit it to a mathematical thought experiment, then yes you can make points about key distribution and so on, you can make limitations that will constrain the outcomes, but in the real world all we really have to create those limitations is the speed of light and other fundamental conservation forces, which would constrain contradictory outcomes in an infinite model.
Again, the idea is to understand that infinity is an absurd idea, and we can’t work with infinities in any meaningful way.