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2nd August 2018, 09:12 AM  #201 
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Nah, he's arguing for him being a Boltzmann Brain. The thing is, even if this is resolved since he's refusing to consider alternatives it's invalid. Really once you start asking this question it's not "which of these two hotels am I in" because it's not just two options. If I'm taking Boltzmann Brains seriously I have an infinite number of other things it could be as well. So not only are there infinite rooms in the hotels but there's infinite hotels.

2nd August 2018, 08:08 PM  #202 
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2nd August 2018, 11:49 PM  #203 
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2nd August 2018, 11:51 PM  #204 
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That works when you know there are only 2 choices, in the general case my comment was about you wouldn't know there are only 2 hotels to choose from. This goes back to the where this thread came from, trying to claim there is a 50/50 chance of us being a Boltzmann brain or the "external" world exists. The point being is that we don't know those are the only two possible choices.

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3rd August 2018, 12:21 AM  #205 
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"That information" became discountable when it was revealed that each hotel had an infinite number of rooms and we realized that we had NO information about how the rooms were grouped nor how a room was selected.
That information was the first thing given in the OP. 
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3rd August 2018, 01:29 AM  #206 
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The problem is that you don't know the distribution of types of rooms if the total is infinite. That's the bit I didn't get at first. The statement "There are an infinite number of rooms, of which one in a billion is opulent and the rest are dingy" is in fact meaningless starting from the words "of which"; the apparent proportion can be changed in an infinite set just by ordering it differently.
If the question had been phrased as "The receptionist in the Motel 6 places every billionth customer in an opulent room, and the rest in dingy rooms; in the Ritz it's the other way round," then the question would have made sense (and, in fact, I read it as if it had been phrased that way). But that requires knowledge of how one came to be in the room, which the original question is trying to exclude. Dave 
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3rd August 2018, 02:26 AM  #207 
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3rd August 2018, 03:58 AM  #208 
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3rd August 2018, 05:29 AM  #209 
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Can you see an invisible pink unicorn if it's standing in front of you in broad daylight, is about the same size as a horse, and your eyesight is 20/20?
What is the difference between a hotel with an infinite number of rooms and an invisible pink unicorn? What is the probability of god existing if an infinite force can move an immovable object? Etc. 
3rd August 2018, 05:37 AM  #210 
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We should never consider the implications of infinities, the square root of a negative number, negative numbers themselves, irrational numbers (how can you have a fraction that can't be expressed as a ratio of whole numbers???) zero, etc.

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3rd August 2018, 07:00 AM  #211 
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In the general case you choose the maximum entropy distribution given certain constraints (those constraints would be the information you have, so in case you have no information there are no constraints).
Given no constraints, in the case of a finite universe the maximum entropy distribution is uniform, each outcome is equally likely. This gets us P(Ritz) = P(Motel 6) = 0.5. In the case of an uncountably infinite universe the maximum entropy distribution is also uniform. In the case of a countably infinite universe though there is no maximum entropy distribution.[*] Hence why we can't assign a distribution over the rooms, but we can assign a distribution over the hotels. In other words, the problem is not that we have no information but that the specific case of "no information + countably infinite universe" doesn't give us a maximum entropy distribution. Compare with W.D. Clinger's argument that we could get a uniform distribution if the rooms were finite or uncountably infinite but not if they are countably infinite. Slightly more precise would be that we're not necessarily looking for a uniform distribution as such but for the maximum entropy distribution, which just happens to be uniform under no constraints (though might not be uniform if there are constraints). * For 3 points: Prove that there does not exist a maximum entropy distribution (under no constraints) over a countably infinite universe. 
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3rd August 2018, 06:45 PM  #212 
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caveman1917 has raised a couple of interesting points, which I do not fully understand but would like to understand.
The technical parts of this post are encapsulated within spoilers. That much I understand. Here I question whether entropy is even defined over arbitrary uncountable state spaces. I think we need to assume we can take integrals. Even if we restrict our attention to Lebesgueintegrable state spaces and probability distributions, I suspect the maximal entropy distribution is not welldefined, if it exists at all. 
3rd August 2018, 09:04 PM  #213 
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I was indeed using uncountable in the sense of the cardinality of the reals, not cardinalities above it. Specifically I meant uncountable universe where "universe" means base set of a probability space, so we can assume integrability of p etc. It is true that that none of those things are defined over an uncountable set of hotel rooms but I was assuming the existence of a map from the set of hotel rooms to some space which has the required properties, such as a real interval.
For the entropy, just replacing the sum with an integral doesn't work, it stops being invariant under coordinate transformations. A better form is S = ∫ p(x) log(p(x)/m(x)) dx where m plays the role of the measure, a function proportional to the limiting density of discrete points when extending the discrete case to the continuous case. See Jaynes 1957 for the specific procedure of taking such limit. Note here that we could write the entropy for the finite case similarly as S = ∑_{i} p_{i} log (p_{i}/m_{i}) where m_{i} is a constant which factors out. But we don't actually need the entropy, we're just looking for the maximum entropy distribution so all we really need is an ordering of distributions according to entropy. Which gives us for the discrete case K(p, q) = ∑_{i} p_{i} log (p_{i}/q_{i}) and for the continuous case K(p, q) = ∫ p(x) log(p(x)/q(x)) dx It's interesting to note here that the absolute entropy is a special case of the relative entropy with q being uniform under some conditions. In the discrete case let q_{i} = m_{i} / ∑_{i} m_{i} where q is uniform since m is constant. In the continuous case let q(x) = m(x) / ∫ m(x) dx where m(x) is the measure. Both of these only work under the conditions that in the discrete case the set is finite and in the continuous case that the total measure is finite. In both these cases q is the uniform distribution, which is the maximum entropy distribution under no constraints, so we can interpret the absolute entropy of a distribution as the relative entropy to the leastconstrained maximum entropy distribution.
Quote:
It's interesting that it is specifically the cases with an infinite measure (interpreting the m from above as the measure in both the discrete and continuous case) where neither the absolute entropy can be seen as a relative entropy nor where an unconstrained maximum entropy distribution exists.
Quote:

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3rd August 2018, 10:12 PM  #214 
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Huh, I guess that in a way it is then true that the problem here is infinity, but not in the sense of cardinality but in the sense of measure. Infinite measure + no information = problem (no maximum entropy distribution), in both the discrete and the continuous case.

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4th August 2018, 03:41 AM  #215 
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I appreciate your explanation. Thank you.
For those who may be wondering what all this technical stuff means for Fudbucker's original question and for any arguments that might be based on that or similar questions: It means game theory and Bayes' Theorem simply cannot be applied to infinite hotel rooms and similar problems in the absence of information or (typically unjustified) assumptions about the a priori probabilities or probability measure. As caveman1917 summarized: 
4th August 2018, 04:00 AM  #216 
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Glad to see you agree with what I posted!

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4th August 2018, 04:55 AM  #217 
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I think I need to respond to this, lest someone take my lack of response as support for Darat's claim.
Darat's claim that game theory cannot deal with infinities was incorrect. Game theory is applicable when we have relevant information. As has been pointed out several times, it was at first possible to interpret Fudbucker's original question as asking us to assume certain a priori probabilities, which would have allowed a straightforward solution using game theory despite the infinity of rooms. In subsequent posts, Fudbucker appears to have clarified his question in ways that suggest to me he did not intend for us to assume the a priori probabilities that would have made his question answerable. It has been suggested that Fudbucker did so because he wanted to make a gametheoretic argument in some other thread where a priori probabilities were clearly not available. I have no opinion on that. 
4th August 2018, 07:09 AM  #218 
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4th August 2018, 09:29 AM  #219 
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I'm wondering how much the idea can be generalized. If m and n are sigmafinite measures on some measurable space, with m absolutely continuous with respect to n, then the RadonNikodym derivative exists and we can define a similar relative entropy
K(m, n) = ∫ log(dm/dn) dm which, if it exists, gives us an ordering again. So it seems that it can be generalized to any collection of sigmafinite measures which are absolutely continuous with respect to each other. ETA: But both the counting measure over the integers and the Lebesgue measure over the reals are sigmafinite, and any probability measure p is going to be absolutely continuous with respect to them, so I must be missing something here as to why it won't work in those cases. ETA2: Of course it works just fine in those cases as well, it just happens to be the case that there is no maximal element in our ordered set of distributions. Silly me. 
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4th August 2018, 11:15 AM  #220 
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I think I'm starting to wrap my head around it. Let M = (X, ∑) be some measurable space. Earlier on I was talking about the unconstrained maximum entropy distribution p which was correct for probability distributions since p(X) = 1 by definition of a probability space, and is hence not a constraint as such. If we however consider a general measure space then the total measure needn't be equal to 1 and hence p(X) = 1 becomes a constraint.
Let μ be a sigmafinite base measure on M, which will usually be the counting measure in the discrete case and the Lebesgue measure in the continuous case, but it could be any measure. Let's extend the notion of a maximum entropy distribution to a maximum entropy measure as per my previous post. Let S be the collection of sigmafinite measures on M including μ which are pairwise absolutely continuous with respect to each other. Then, assuming the necessary integrals exist, we do get an unconstrained maximum entropy measure ν ∈ S  up to proportionality  in all cases. The problem is then that if μ(X) is infinite then so will ν(X) and it hence can't be normalized to a probability measure. So the unconstrained maximum entropy measure does always exist, and is equal  up to proportionality  to the counting measure or Lebesgue measure as the case may be. Or differently, it is the constraint that ν(X) = k for some strictly positive k which stops the maximum entropy measure from existing if μ(X) is infinite, but that is a necessary constraint in order to be able to normalize ν to a probability measure. ETA: interesting that this seems to give us a justification for why we conventionally use the counting measure or Lebesgue measure, they happen to be the maximum entropy measure. 
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4th August 2018, 01:53 PM  #221 
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Enjoyed reading this thread and seeing everyone's perspectives.
If I've learned anything from these types of problems, it's that whenever you see wordings such as "you wake up in uncertain conditions/with amnesia, what is the probability of xxxx".......then run for the hills! (Sleeping Beauty problem, anyone?) Most of these the problems seem to be illdefined, and the OP problem appears to fall into this category. Particularly, as noted in Post #147 by W.D.Clinger, the sentence "In the Motel 6 set, for every opulent room, there are a billion dingy ones." is ambiguous. It needs to be made more clear. It could be saying the order of the countably infinite* rooms goes opulent, followed by a billion dingy, followed by opulent, followed by a billion dingy, etc, which is not to be confused with merely saying "for every opulent room, there are a billion dingy ones". In the case of sequential ordering, it makes more sense to conclude P(opulentMotel 6) could be something like (1/1,000,000,001), since if we assume that somehow you could find yourself in a particular equallylikely random room, with a set room number, then you could be certain that the set of room numbers containing your room number, and the billion following numbers (which should be equally likely under this assumption, however impossible this assumption is), contains exactly one opulent number translating the infinite to a finite case, so to speak. Or bypassing all this, perhaps the highlighted phrase above, as W.D.Clinger noted, could simply be defined as expressing the conditional probabilities in question. *Some are talking about this being a hidden assumption, and indeed it is, but I would argue from the language of the problem, this appears warranted. "Infinite hotel" brings to mind infinite hallways of sequential rooms, or infinite floors, etc. If someone is in a particular room, there would be a set "next" room next door (there is no next real number after 1). But anyway we could just add this as an assumption....yet another reason why vague physical impossibilities dont translate well to mathematical problems. Perhaps the Motel 6 owner could have a policy to simply put all the guests in opulent rooms, in which case P(opulentMotel 6) could be 1, but this of course would be additional information unknown to the guest from the given info, and so wouldnt factor into their credence. The question as given in the OP cant be answered clearly because it is not welldefined, but I suppose the best way to estimate probabilities involved would be to somehow repeat the experiment over and over again (just pretend it could be), and record the relative frequency of times people are right/wrong when saying Motel 6/Ritz, after waking up in an opulent room (empirical definition of probability). 
4th August 2018, 02:06 PM  #222 
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We can simply define
"A has the same size as B" as "there exists a onetoone correspondence between the elements of A and B". Mathematically, then, it would make sense to talk about infinite sets having the same size. I think someone brought it up earlier, but the above definition fits nicely with our notion of "same size" of finite sets. If there are two collections of books, each containing 7 books, I can conclude these collections are the same size by pairing off books (exactly) from each collection. Or I can take the longer route and count the number of books in each (which is really just using the onetoone correspondence notion between the books and the counting numbers, or say my fingers, but you are doing it twice) 
4th August 2018, 05:54 PM  #223 
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Welcome to the ISF, Dabbler.
You have stated that definition exactly as it is stated and used by logicians and mathematicians. When we want to be stuffy or need to be careful, as when there is some other notion of size under discussion (such as measure), we say two sets have the same cardinality or are in onetoone correspondence. Most of the time, though, we just say they have the same size. If we are willing to assume the axiom of choice, we can prove a law of trichotomy for that notion of cardinality: either two sets A and B are the same size, or one of the sets is strictly smaller than the other (i.e. A has the same cardinality as a proper subset of B but does not have the same cardinality as B). If we don't want to assume the axiom of choice or any equivalent of it, then we have to face the possibility that A and B may be incomparable in size, much as you can have two unequal sets where neither is a subset of the other. 
4th August 2018, 09:23 PM  #224 
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9th August 2018, 02:06 PM  #225 
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9th August 2018, 07:43 PM  #226 
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Although it is not fashionable today, frequentist statistics and the frequentist interpretation of probability explicitly assume there is some (real or imagined) infinite set of events/experiments/measurements. Infinity and probability are not incompatible. By definition, if the true ratios are 1:1e9 and 1e9:1 as stated in the original post then those are the ratios that one would observe as the 1 person in one of an infinite number of hotel rooms was repeated a very large number of times (approaching infinity).

10th August 2018, 12:09 AM  #227 
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Whoanellie, you need additional assumption to do either frequentist or Bayesian statistics. Usually one would assume that each sample is an independent and identically distributed Bernoulli random variable. This assumption obviates the need to put a distribution on the infinite set itself.

10th August 2018, 02:24 AM  #228 
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10th August 2018, 07:55 AM  #229 
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I am not assuming anything. I am simply asserting that probability does make sense even when one is dealing with infinities. That's how frequentist statistics works. Frequentist statistics does not require an assumption of a Bernoulli variable.

10th August 2018, 08:04 AM  #230 
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There are more than one way to think of the size of a countable infinity. The concept of natural density is relevant to this discussion.

10th August 2018, 08:09 AM  #231 
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There's an infinite number of ways to think about the size of any infinite from your friend and mine the AlephNaught to infinite nested uncountable infinite sets of infinities.
Because infinity isn't a number. It doesn't have a size. It's a concept defined by it's undefinablity. The size of infinity is like the corner of a circle. It doesn't have one. All discussions of infinity run into the Supertask Paradox in one form or another; give us an infinite concept and ask us what happens at the end of it. 
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10th August 2018, 09:17 AM  #232 
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You miunderstand frequentist probability. As psion alluded to, p = limit as n approaches infinity of p(n) does not imply that p is defined at infinity. As numerous people in this thread have explained, a proportion of a countably infinite set is meaningless.
What I was getting at is that you can construct an infinite sequence (of say hypothetical hotel rooms) Xi, i=1,2,... , by treating the Xi as iid Bernoulli(p) random variables. Then, no matter which Xi (hotel room) you choose (wake up in), there is a probability p that Xi=1 (is opulent). So constructed, the sequence in a sense emulates a set with proportion p of elements that have the characreristic of interest even though the concept of proportion is undefined. Note, too, that this construction circumvents the problem of it being impossible to impose a uniform distribution on the set. 
10th August 2018, 06:59 PM  #233 
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This is where the concept of natural density comes in handy.
see also https://www.rug.nl/research/portal/f...8949/02_c2.pdf 
11th August 2018, 01:23 AM  #234 
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Whoanellie, thanks, but the only relevant question is whether you can back up your claim rigorously. You wrote:
Please state your sample space S, your sigmaalgebra F of subsets of S, and a probability function that would enable the repeated sampling you talk about above. If you think „natural density“ is helpful with this feel free to incorporate it. Sent from my iPhone using Tapatalk 
11th August 2018, 11:16 AM  #235 
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I don't know anything about sigmaalgebraa, but I don't believe they are relevant to my point. There are different interpretations of probability and it appears that one's perspective on the original question is largely determined by the interpretation of probability that one favors. The links I've posted above were intended to back up that point.

11th August 2018, 01:33 PM  #236 
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Please see the definition of a probability space. Without a probability space your point is not in probability theory, and you have a nogo I'm afraid.
Quote:
Let N be the set of natural numbers, and let {A, B} be a partition of N such that both A and B are infinite. Then P(A) is defined as lim_{n>inf} An / (An + Bn) where Xn = {x in Nn  x in X} where Nn is the set of the first n natural numbers. The problem here is in the meaning of "first n natural numbers" (the "n>inf") part. Consider my reordering argument as per earlier in the thread. Let A be the set of of natural numbers divisible by 1e9 and B the set of natural numbers not divisible by 1e9. Suppose I now reorder N such that I first take the first number in A, then the first number in B, then the next number in A, then the next number in B, and so on. If I then take your limit above I get P(A) = 0.5. This might become clearer if we take a look at what's going on when we take that limit, we have a sequence of finite subsets of N with the limiting set being N such that each subset is a strict superset of the previous one: {1}, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}, ... > N We then have a function P from finite subsets of N to the unit interval which gives us the fraction of numbers in that subset divisible by 1e9. We apply P to the sequence above: P({1}) = 0, P({1, 2}) = 0, P({1, 2, 3}) = 0, P({1, 2, 3, 4}) = 0, ... > P(N) = 1/1e9. And then define P(N) by that limit and get a natural density of 1/1e9. But we can take such limit in many ways. Consider the following where d_{i} and nd_{i} are the ithe natural number divisible or nondivisible by 1e9 respectively: {d_{1}}, {d_{1}, nd_{1}}, {d_{1}, nd_{1}, d_{2}}, ... > N and apply P: P({d_{1}}) = 1, P({d_{1}, nd_{1}}) = 1/2, P({d_{1}, nd_{1}, d_{2}}) = 2/3, ... > P(N) = 1/2 and we get P(N) = 1/2. There are many ways to approach N with a sequence of its subsets. ETA: So that's not saying that you can't define P(N) in such a way, but there are plenty of other ways to do it which get different results. More importantly, in terms of the hotel rooms, I could change the result of the experiment by doing nothing other than switching the room numbering around a bit (ie use a different bijection between the set of rooms and the set of natural numbers, giving a different ordering on the set of rooms). 
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11th August 2018, 02:50 PM  #237 
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I read your second link (or rather, again, read the first part until I figured out what was going on and then skimmed the rest). And, as per my post above, it is indeed possible to define P(N) in various ways by changing the Kolmogorov axioms of probability or changing other conditions (which puts the solutions outside of probability theory proper). The problem isn't that it's impossible to define P(N) but that there are many ways to do it, leading to different results, with there not being clear reason to prefer one over another. And all of them have problems, for example the class of solutions you're proposing fails the reordering argument.
The experiment was someone waking up in a hotel room, looking at the wealth of the room, and then making a guess as to which hotel the subject is in. Suppose that during the night I go through hallway and do naught but switch the numbering of the rooms around, that should not change the guess made by the subject  yet in your solution(s) it does. Above Dabbler made the point that we can assume a physical succesor function of the hotel rooms, in that  given for example an infinite hallway of rooms  for each room there is, physically, a next room. But it fails the same problem, that during the night I could physically switch the rooms around, whether we consider it switching the room numbering around or physically switching the rooms around is immaterial to the point that this operation should not change the guess made by the subject when waking up and seeing the room wealth. ETA: actually, if we define Motel 6 and Ritz as per the OP (distinguished by the natural density of opulent rooms to dingy ones  assuming that, indeed, natural density was meant in the ambiguous statement) then these hotels are indistinguishable expect for the ordering of their rooms. In a way they're the same hotel, they just chose to order their rooms differently. So if the guest isn't given information as to the ordering of the rooms, which doesn't seem to be the case when only given the wealth of his or her room, then the hotels are indistinguishable to the guest and the guess should be 50/50 (or whatever the prior probabilities were). 
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11th August 2018, 04:21 PM  #238 
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Under the conditions described in the OP, there should be some optimal strategy for choosing what to say if someone puts a gun to my head and asks me which hotel I'm in. If there is no strategy, then the default strategy would be to flip a coin.

11th August 2018, 05:14 PM  #239 
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Optimal under what conditions?
Optimal solutions/strategies don't always exist. Consider, for example, the following problem: What real value for x maximizes f(x) subject to the constraint g(x) > 0? If we don't know what f(x) is, the question is unanswerable. In most cases, it is also unanswerable if we know f(x) but don't know g(x). There has been some debate concerning whether your original post asked us to assume certain prior probabilities (in which case your question is easily answered) or told us only to assume certain correspondences (in which case your question has no answer, because the existence of those correspondences implies nothing about probabilities). There has also been some speculation that you did not want us to assume prior probabilities, because prior probabilities would not be available for an argument you are alleged to want to make in another thread. If you want us to continue to consider your question, you should give us a clear statement concerning what you want us to assume. 
11th August 2018, 05:56 PM  #240 
Philosopher
Join Date: Feb 2015
Posts: 6,028

It's interesting that, the Ritz and Motel 6 being indistinguishable except for room ordering, and interpreting the room ordering as the receptionist's strategy for filling up the hotel as more and more guests arrive (giving us a specific sequence of subsets to take a limit on, namely the sequence of sets of occupied rooms as more and more guests arrive) then we can consider a similar problem.
Suppose there is only 1 hotel with an infinite hallway of rooms, for which there are two receptionists each with a different filling strategy. R_{Ritz} puts 1e9  1 guests in opulent rooms and then one guest in a dingy room, and rinse and repeat. R_{Motel 6} does the opposite. Given that you awoke in an opulent room, which receptionist was on call when you checked in? All of these are basically the same thing: receptionist's filling strategy, room ordering, sequence of subsets of hotel rooms to take limits on. 
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"Ideas are also weapons."  Subcomandante Marcos "We must devastate the avenues where the wealthy live."  Lucy Parsons "Let us therefore trust the eternal Spirit which destroys and annihilates only because it is the unfathomable and eternal source of all life. The passion for destruction is a creative passion, too!"  Mikhail Bakunin 

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