Not necessarily. Each monkey is independent, right? So if we think about the first letter, it’s either going to be, idk, A, the correct letter, or B, any wrong letter. Any monkey that types B is never going to get there. Now each money independently chooses between them. With each second monkey, the chances in aggregate get smaller and smaller than we only see B, but… It’s never a 0 chance that the monkey hits B. If there’s only two keys, it’s always 50/50. And it could through freak chance turn out that they all hit B… Forever. There is never a guarantee that you will get even a single correct letter… Even with infinite monkeys.
I get that it seems like infinity has to include every possible outcome, because the limit of P(at least one monkey typing A) as the number of monkeys goes to infinity is 1… But a limit is not a value. The probability never reaches 1 even with infinite monkeys.
Infinite monkeys. Any probability greater than zero times infinity is infinity. You will see an infinite number of monkeys hitting A and an infinite number hitting B. If there were a finite number of monkeys, you would be correct, but that is not the case.
No, that’s not how probability works. “Any probability times infinity is infinity” doesn’t even mean anything. Probabilities are between 0 and 1, and you do not multiply them by fixed factors. Infinite probability has no meaning.
I explained the infinity monkeys in another comment more clearly than I did above -here you go.
I could have worded that better. Any probability with a non-zero chance of occurring will occur an infinite number of times given an infinite sequence.
To address the comment you linked, I understand what you’re saying, but you’re putting a lot of emphasis on something that might as well be impossible. In an infinite sequence of coin flips, the probability of any specific outcome - like all heads - is exactly zero. This doesn’t mean it’s strictly impossible in a logical sense; rather, in the language of probability, it’s so improbable that it effectively “never happens” within the probability space we’re working with. Theoretically, sure, you’re correct, but realistically speaking, it’s statistically irrelevant.
Eh, I don’t think it’s irrelevant, I think it’s interesting! I mean, consider a new infinite monkey experiment. Take the usual setup - infinite monkeys, infinite time. Now once you have your output… Do it again, an infinite number of times. Now suddenly those near impossibilities (the almost surely Impossibles) become more probable.
I also think it’s interesting to consider how many infinite sequences there are which do/do not contain hamlet. This one I’m still mulling over… Are there more which do, or more which don’t? That is a bit beyond my theoretical understanding of infinity to answer, I think. But it might be an interesting topic to read about.
In terms of the question, “Are there more infinite sequences that contain Hamlet or more that don’t?”- in the context of true randomness and truly infinite sequence, this feels like almost a trick question. Almost every truly random infinite sequence will contain Hamlet an infinite number of times, along with every other possible finite sequence (e.g., Moby Dick, War and Peace, you name it). In fact, the probability of a random infinite sequence not containing Hamlet is effectively zero.
Where it becomes truly interesting is if you have an infinite number of infinite sequences. Now you’d certainly get instances of those “effectively zero” cases, but only in ratios within infinity itself, haha. I guess that’s probably what you were getting at?
I thought that at first… But then for every infinite series with exactly one hamlet in it, there’s an infinite series where one character is wrong. And there’s another one where a different character is wrong… And so on and so on. Even if the series contains an infinite number of hamlets, you can replace one character in each in a huge number of ways! It starts to seem like there are more options with almost Hamlet than there are specifically with Hamlet.
In fact, I begin to wonder if almost any constraint reducing the search space in the infinite set of such infinite sequences, you will inevitably have fewer items within the search space than without… Since you can usually construct multiple non-matching candidates from any matching one.
But… Honestly I’m not sure how much any of that matters in infinite contexts. Since they are impossible it boggles my mind trying to imagine it.
The birthday problem fits into this somehow, but I can’t quite get there right now. Something like an inverse birthday problem to illustrate how, even though the probability of two monkeys typing the same letter rises quickly as more monkeys are added to the mix, and at a certain point (n+1, where n is “possible keystrokes”) it is inevitable that at least two monkeys will key identically, the inverse isn’t true.
If you have 732 people in a room, there’s no guarantee that any one of them was born on August 12th.
There’s another one that describes this better but it escapes me.
That’s not true. Infinite doesn’t mean “all”. There are an infinite amount of numbers between 0 and 1, but none of them are 2. There’s a high statistical probability, sure, but it’s not necessarily 100%.
It is necessarily 100%. That’s the whole idea behind infinity. There is a 0% chance of rolling a “2” because it’s outside the bounds of the question. Theres a 0% chance of the monkeys typing in chinese too.
No, it isn’t, that’s a misunderstanding of how independent random variables behave. Even with an infinite number of trials, in this case there is never a guarantee of a particular outcome.
Consider a coin flip, 50/50 chance of either getting heads or tails on each flip. Lets say we do an infinite number of flips, one by one, so that we end up with an infinite ordered set of outcomes, like so: {H, T, T, H, … }. Now, consider the probability of getting a particular arrangement of heads/tails in this infinite list, like the one I wrote before. You can’t calculate a probability for each arrangement - there are an infinite number - but it should be clear that each arrangement is equally likely, right? Because {H, …} is just as likely as {T, …}, same with {H, H, …} and {H, T, … } and so on and so on. In other words the probabilty of getting all heads on infinite coin flips is the same as the probability of getting any other combination.
In the same way, the infinite monkeys are doing ‘coin flips’ involving more than 2 options. Lets just assume they have 26 keys, one for each letter, and assume they hit each of them with equal probability. In the same way, for an individual monkey the probability of going {a, a, a, a, a, a, …, a} is the same as the probability of the same sequence with hamlet somewhere (in a particular position that is - the probabilities are only equal when we consider exactly one arrangement). What might make it more intuitively clear is that even after an infinite number of trials you only have one sequence of letters (or set of sequences, with infinite monkeys). It’s clear that there are other possible sequences - like only the letter a - and these all have a non 0 chance of having arisen given a different infinite set of monkeys for a different infinite time period.
It’s easy to be misled here! If we return to the coin flip example, the probability of flipping at least 1 head after infinite coin flips approaches 1. The limit of P(at least one H) as the number of flips approaches infinity is 1. But this is a limit! You never reach the limit, even considering infinite situations.
if it’s infinite monkeys then an infinite amount of them do it correct on the first try
That’s assuming they’re typing truly randomly. Which is a fair assumption.
Not necessarily. Each monkey is independent, right? So if we think about the first letter, it’s either going to be, idk, A, the correct letter, or B, any wrong letter. Any monkey that types B is never going to get there. Now each money independently chooses between them. With each second monkey, the chances in aggregate get smaller and smaller than we only see B, but… It’s never a 0 chance that the monkey hits B. If there’s only two keys, it’s always 50/50. And it could through freak chance turn out that they all hit B… Forever. There is never a guarantee that you will get even a single correct letter… Even with infinite monkeys.
I get that it seems like infinity has to include every possible outcome, because the limit of P(at least one monkey typing A) as the number of monkeys goes to infinity is 1… But a limit is not a value. The probability never reaches 1 even with infinite monkeys.
Infinite monkeys. Any probability greater than zero times infinity is infinity. You will see an infinite number of monkeys hitting A and an infinite number hitting B. If there were a finite number of monkeys, you would be correct, but that is not the case.
No, that’s not how probability works. “Any probability times infinity is infinity” doesn’t even mean anything. Probabilities are between 0 and 1, and you do not multiply them by fixed factors. Infinite probability has no meaning.
I explained the infinity monkeys in another comment more clearly than I did above -here you go.
I could have worded that better. Any probability with a non-zero chance of occurring will occur an infinite number of times given an infinite sequence.
To address the comment you linked, I understand what you’re saying, but you’re putting a lot of emphasis on something that might as well be impossible. In an infinite sequence of coin flips, the probability of any specific outcome - like all heads - is exactly zero. This doesn’t mean it’s strictly impossible in a logical sense; rather, in the language of probability, it’s so improbable that it effectively “never happens” within the probability space we’re working with. Theoretically, sure, you’re correct, but realistically speaking, it’s statistically irrelevant.
Eh, I don’t think it’s irrelevant, I think it’s interesting! I mean, consider a new infinite monkey experiment. Take the usual setup - infinite monkeys, infinite time. Now once you have your output… Do it again, an infinite number of times. Now suddenly those near impossibilities (the almost surely Impossibles) become more probable.
I also think it’s interesting to consider how many infinite sequences there are which do/do not contain hamlet. This one I’m still mulling over… Are there more which do, or more which don’t? That is a bit beyond my theoretical understanding of infinity to answer, I think. But it might be an interesting topic to read about.
Fair enough, I suppose it is interesting!
In terms of the question, “Are there more infinite sequences that contain Hamlet or more that don’t?”- in the context of true randomness and truly infinite sequence, this feels like almost a trick question. Almost every truly random infinite sequence will contain Hamlet an infinite number of times, along with every other possible finite sequence (e.g., Moby Dick, War and Peace, you name it). In fact, the probability of a random infinite sequence not containing Hamlet is effectively zero.
Where it becomes truly interesting is if you have an infinite number of infinite sequences. Now you’d certainly get instances of those “effectively zero” cases, but only in ratios within infinity itself, haha. I guess that’s probably what you were getting at?
I thought that at first… But then for every infinite series with exactly one hamlet in it, there’s an infinite series where one character is wrong. And there’s another one where a different character is wrong… And so on and so on. Even if the series contains an infinite number of hamlets, you can replace one character in each in a huge number of ways! It starts to seem like there are more options with almost Hamlet than there are specifically with Hamlet.
In fact, I begin to wonder if almost any constraint reducing the search space in the infinite set of such infinite sequences, you will inevitably have fewer items within the search space than without… Since you can usually construct multiple non-matching candidates from any matching one.
But… Honestly I’m not sure how much any of that matters in infinite contexts. Since they are impossible it boggles my mind trying to imagine it.
The birthday problem fits into this somehow, but I can’t quite get there right now. Something like an inverse birthday problem to illustrate how, even though the probability of two monkeys typing the same letter rises quickly as more monkeys are added to the mix, and at a certain point (n+1, where n is “possible keystrokes”) it is inevitable that at least two monkeys will key identically, the inverse isn’t true.
If you have 732 people in a room, there’s no guarantee that any one of them was born on August 12th.
There’s another one that describes this better but it escapes me.
That’s not true. Infinite doesn’t mean “all”. There are an infinite amount of numbers between 0 and 1, but none of them are 2. There’s a high statistical probability, sure, but it’s not necessarily 100%.
It is necessarily 100%. That’s the whole idea behind infinity. There is a 0% chance of rolling a “2” because it’s outside the bounds of the question. Theres a 0% chance of the monkeys typing in chinese too.
No, it isn’t, that’s a misunderstanding of how independent random variables behave. Even with an infinite number of trials, in this case there is never a guarantee of a particular outcome.
Consider a coin flip, 50/50 chance of either getting heads or tails on each flip. Lets say we do an infinite number of flips, one by one, so that we end up with an infinite ordered set of outcomes, like so: {H, T, T, H, … }. Now, consider the probability of getting a particular arrangement of heads/tails in this infinite list, like the one I wrote before. You can’t calculate a probability for each arrangement - there are an infinite number - but it should be clear that each arrangement is equally likely, right? Because {H, …} is just as likely as {T, …}, same with {H, H, …} and {H, T, … } and so on and so on. In other words the probabilty of getting all heads on infinite coin flips is the same as the probability of getting any other combination.
In the same way, the infinite monkeys are doing ‘coin flips’ involving more than 2 options. Lets just assume they have 26 keys, one for each letter, and assume they hit each of them with equal probability. In the same way, for an individual monkey the probability of going {a, a, a, a, a, a, …, a} is the same as the probability of the same sequence with hamlet somewhere (in a particular position that is - the probabilities are only equal when we consider exactly one arrangement). What might make it more intuitively clear is that even after an infinite number of trials you only have one sequence of letters (or set of sequences, with infinite monkeys). It’s clear that there are other possible sequences - like only the letter a - and these all have a non 0 chance of having arisen given a different infinite set of monkeys for a different infinite time period.
It’s easy to be misled here! If we return to the coin flip example, the probability of flipping at least 1 head after infinite coin flips approaches 1. The limit of P(at least one H) as the number of flips approaches infinity is 1. But this is a limit! You never reach the limit, even considering infinite situations.