Either this problem is not completely stated, there are important things left out, or the answers are gibberish. In any case, I have no idea what is going on even after reading the answers. I can collude with prisoners ahead of time but I can’t communicate with them. This presumably means I can’t smile or wink or grab or stare meaningfully or longingly at them. I can’t tell them what my guess is.
So I find myself in a room full of people, some with red hats and some with black hats, in what is essentially a still picture, for all intents and purposes equivalent to freezing the room full of other people with their red and black hats and I can then examine this fixed scene. How does this even conceivably impact what I would guess the color is of the hat on my head?
There are important things left out. Specifically, this is a math problem and putting it in terms of prisoners or humans is intended to be ignored. You should of course just know that when someone is a human in a math puzzle, what they really mean is a hypercomputer or oracle that can return correct answers for uncomputable algorithms. Here’s another example:
There is a prisoner in a cell, the same as the prisoners in the question above. Each day the warden hands him an infinitely complex algorithm, and he is released if and only if he can correctly tell the warden whether it halts or not. On what day is he released?
The problem is correct as stated, and solutions above by RichardKennaway and Oscar_Cunningham are correct. I think you may have missed that the prisoners are all distinguishable, a.k.a. they are numbered 1,2,3,.… Or you are confused about the win condition; we don’t have to guarantee that any particular prisoner guesses correctly, just that only finitely many guess incorrectly.
Sub-puzzle: prove definitively that if the prisoners are not distinguishable, then there is no winning strategy.
In the winning strategy, do fewer than half the prisoners guess wrong? Do more prisoners guess correctly than incorrectly? I’m trying to get a handle on whether it is worth my while to try to penetrate the jargon in the “correct solutions.”
What do you mean by half the prisoners? Let’s start there.
How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head? In particular, is that probability more than, less than, or equal to 0.5?
While we are in the neighborhood, if there is a prisoner who is more likely to get the answer correctly than not, if you could tell me what ihis step by step process of forming his answer is, in detail similar to “if he is prisoner n, he guesses his hat color is the opposite of that of prisoner n^2+1” or some such recipe that a Turing machine or a non-mathematician human could follow.
How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head?
So what do you mean to choose a prisoner at random when one has infinitely many prisoners?
Whatever mwengler’s answer is, your answer is going to have to be “in that case the set you asked about isn’t measurable, and so I can’t assign it a probability”.
Maybe this way then. I set down somewhere in the universe in a location that I don’t reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.
Of that 100 prisoners, I count the number of prisoners who identified their hat color correctly.
My question is what is the probability that I have counted 50 or fewer correct answers? Is it greater than, less than, or equal to the probability that I have counted 51 or more correct answers?
Thanks to you and JoshuaZ for trying to help me here.
Maybe this way then. I set down somewhere in the universe in a location that I don’t reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.
So how have you set up the prisoners in the universe in advance and how do you decide on the location you set down?
Is it safe to say that this problem, this result, has no applicability to any similar problem involving a merely finite amount of prisoners, say a mere googol of them?
Yes. But I do think that thinking critically about the assumptions you are making, in particular that you can meaningfully talk about what it means to pick a random individual in a uniform fashion, is worthwhile for understanding a fair bit of probability and related issues which are relevant in broader in contexts.
As to critically understanding what it means to pick a random individual in a uniform fashion, yes it is worth understanding what one means, as a tremendous amount of mischief is done in the name of randomness. With a finite number of prisoners, I would actually not need to pick prisoners randomly in order to gather statistics. If I would simply count up all the prisoners who got the hat color correct and all that got it wrong, and I do the experiment say 100 times and plot a histogram of the results and then decide whether the results deviate from a normal distribution with mean 50% by enough to make it practically interesting.
But since you say this result has no applicability to a finite population of prisoners, I am assuming there is nothing here that would push the result away from 50:50?
Is the mathematician’s world really so insular,that someone from outside asking some questions about how a problem relates to concepts he understands gets downvoted?
Or are you pretending that unless you skate with ease over three different kinds of infinities, their differences and similarities, and the paradoxical results of probability problems with infinity in the numerator and denominator, that you are just a time wasting intruder on an otherwise valuable conversation?
Or did my questions, which I’d love to know the answer to, come across as a veiled negative comment?
Your questions were easily answered by looking up the definitions of the terms “finitely many” and “countably infinite”.
Are you aware that having a mathematician tell you a question is easily answered without actually answering it is actually the punch line to a joke? Closest I can find on the web is the 2nd one on this page.
You asked why you were downvoted. I told you why; you asked a question that showed you hadn’t made even a cursory attempt to understand the terms in the question.
The answers, in case you still haven’t put in the minimal effort required, are
Yes, a finite portion of an infinite set is infinitely less than half.
Yes, all but a finite number of an infinite set is infinitely more than a finite number.
This is not fancy jargon. These are terms anyone who has taken highschool calculus would know.
Either this problem is not completely stated, there are important things left out, or the answers are gibberish. In any case, I have no idea what is going on even after reading the answers. I can collude with prisoners ahead of time but I can’t communicate with them. This presumably means I can’t smile or wink or grab or stare meaningfully or longingly at them. I can’t tell them what my guess is.
So I find myself in a room full of people, some with red hats and some with black hats, in what is essentially a still picture, for all intents and purposes equivalent to freezing the room full of other people with their red and black hats and I can then examine this fixed scene. How does this even conceivably impact what I would guess the color is of the hat on my head?
Fix!
There are important things left out. Specifically, this is a math problem and putting it in terms of prisoners or humans is intended to be ignored. You should of course just know that when someone is a human in a math puzzle, what they really mean is a hypercomputer or oracle that can return correct answers for uncomputable algorithms. Here’s another example:
There is a prisoner in a cell, the same as the prisoners in the question above. Each day the warden hands him an infinitely complex algorithm, and he is released if and only if he can correctly tell the warden whether it halts or not. On what day is he released?
Gur svefg qnl.
The problem is correct as stated, and solutions above by RichardKennaway and Oscar_Cunningham are correct. I think you may have missed that the prisoners are all distinguishable, a.k.a. they are numbered 1,2,3,.… Or you are confused about the win condition; we don’t have to guarantee that any particular prisoner guesses correctly, just that only finitely many guess incorrectly.
Sub-puzzle: prove definitively that if the prisoners are not distinguishable, then there is no winning strategy.
In the winning strategy, do fewer than half the prisoners guess wrong? Do more prisoners guess correctly than incorrectly? I’m trying to get a handle on whether it is worth my while to try to penetrate the jargon in the “correct solutions.”
What do you mean by half the prisoners? Let’s start there.
How about I choose a prisoner at random from among all the prisoners in the problem. What is the probability that the prisoner I have chosen has correctly stated the color of the hat on his head? In particular, is that probability more than, less than, or equal to 0.5?
While we are in the neighborhood, if there is a prisoner who is more likely to get the answer correctly than not, if you could tell me what ihis step by step process of forming his answer is, in detail similar to “if he is prisoner n, he guesses his hat color is the opposite of that of prisoner n^2+1” or some such recipe that a Turing machine or a non-mathematician human could follow.
Thanks in advance
So what do you mean to choose a prisoner at random when one has infinitely many prisoners?
Whatever mwengler’s answer is, your answer is going to have to be “in that case the set you asked about isn’t measurable, and so I can’t assign it a probability”.
Maybe this way then. I set down somewhere in the universe in a location that I don’t reveal to you ahead of time and then identify the 100 prisoners that are closest to my position. If the 100th and 101st furthest prisoners from me are exactly equally distant form me I set down somewhere else in the universe and I keep moving until I find a location where I can identify the 100 closest prisoners to my current position.
Of that 100 prisoners, I count the number of prisoners who identified their hat color correctly.
My question is what is the probability that I have counted 50 or fewer correct answers? Is it greater than, less than, or equal to the probability that I have counted 51 or more correct answers?
Thanks to you and JoshuaZ for trying to help me here.
So how have you set up the prisoners in the universe in advance and how do you decide on the location you set down?
Is it safe to say that this problem, this result, has no applicability to any similar problem involving a merely finite amount of prisoners, say a mere googol of them?
Yes. But I do think that thinking critically about the assumptions you are making, in particular that you can meaningfully talk about what it means to pick a random individual in a uniform fashion, is worthwhile for understanding a fair bit of probability and related issues which are relevant in broader in contexts.
As to critically understanding what it means to pick a random individual in a uniform fashion, yes it is worth understanding what one means, as a tremendous amount of mischief is done in the name of randomness. With a finite number of prisoners, I would actually not need to pick prisoners randomly in order to gather statistics. If I would simply count up all the prisoners who got the hat color correct and all that got it wrong, and I do the experiment say 100 times and plot a histogram of the results and then decide whether the results deviate from a normal distribution with mean 50% by enough to make it practically interesting.
But since you say this result has no applicability to a finite population of prisoners, I am assuming there is nothing here that would push the result away from 50:50?
Is the mathematician’s world really so insular,that someone from outside asking some questions about how a problem relates to concepts he understands gets downvoted?
Or are you pretending that unless you skate with ease over three different kinds of infinities, their differences and similarities, and the paradoxical results of probability problems with infinity in the numerator and denominator, that you are just a time wasting intruder on an otherwise valuable conversation?
Or did my questions, which I’d love to know the answer to, come across as a veiled negative comment?
Your questions were easily answered by looking up the definitions of the terms “finitely many” and “countably infinite”.
Are you aware that having a mathematician tell you a question is easily answered without actually answering it is actually the punch line to a joke? Closest I can find on the web is the 2nd one on this page.
You asked why you were downvoted. I told you why; you asked a question that showed you hadn’t made even a cursory attempt to understand the terms in the question.
The answers, in case you still haven’t put in the minimal effort required, are
Yes, a finite portion of an infinite set is infinitely less than half.
Yes, all but a finite number of an infinite set is infinitely more than a finite number.
This is not fancy jargon. These are terms anyone who has taken highschool calculus would know.
Infinity is weird.