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?
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?