= 27d567bc2d48d9f40e9ccc20ea370b15)
TsviBT comments on Open Thread March 31 - April 7 2014 - Less Wrong Discussion
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (234)
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.
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.