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30th July 2018, 01:18 PM  #121 
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Fudrucker (and this is 100%, no snark serious. We butt heads a lot so I want to make it clear this is sincere)
Vsauce has a wonderful series of videos along these topics that are well worth your time. They cover a lot of what you are talking about. Vsauce: How to Count Past Infinity https://www.youtube.com/watch?v=SrU9YDoXE88 Vsauce: The Supertask: https://www.youtube.com/watch?v=ffUnNaQTfZE 
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30th July 2018, 01:19 PM  #122 
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30th July 2018, 01:20 PM  #123 
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There being the same number of primes as integers seems wrong intuitively, but mathematically Fudbucker is correct and your impression "from a logical perspective" is wrong. (Technically you're actually correct that not all infinite sets are the same size, but that's because there are uncountably infinite sets and other cardinalities beyond the "countably infinite" cardinality of the integers. Regardless, all countably infinite sets, such as the integers, the primes, the rational numbers, and the multiples of a billion, are the same size.) 
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30th July 2018, 01:20 PM  #124 
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30th July 2018, 01:24 PM  #125 
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30th July 2018, 01:29 PM  #126 
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The physical scenarios of picking marbles or cards of whatever are just for illustration, to try to present an intuitive understanding of the underlying mathematical issues. Disregarding physical scenarios doesn't fix those underlying mathematical issues. Dividing by zero is invalid not only because it overflows the registers of your computer, or because if you started putting putting groups of zero apples into a basket you'd never finish filling it, but because division by zero is undefined mathematically. The underlying issue in the case of describing a random member of a countably infinite set is the one given by A Mathematician. I meant to include a link; here it is. 
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30th July 2018, 01:33 PM  #127 
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Ah, but there are no paradoxes  the existence of one would itself be a paradox and it would disappear in a puff of logic before it even came into existence.
Damn fascinating though. What I'm taking from the discussion is that as soon as the words infinity and probability appear in the same puzzle we reach for a large scotch and enjoy the ride? 
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30th July 2018, 01:39 PM  #128 
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30th July 2018, 01:42 PM  #129 
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30th July 2018, 01:43 PM  #130 
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My question is, how do #2 and #3 work out, mathematically. Not metaphorically, or analogically, or physicallyunphysical thoughtexperimentally. What. Is. The. Actual. Math.
A related, but less relevant question I have is, why is everyone so excited about #4, anyway? It seems entirely tangential to the actual question being asked in the OP. Is this a Peter Principle thing? Everybody is rising to the level of the question they can actually answer, instead of the question that was actually asked? 
30th July 2018, 01:47 PM  #131 
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30th July 2018, 01:47 PM  #132 
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#2 I'd argue they are the same size because the method of numbering them is arbitrary...
I have an infinite countable set, let's say a list of all the cardinal numbers. I count 1, 2, 3, 4, etc into infinity. I count 2, 4, 6, 8, etc into infinity. I count 1, 2, 3, 5, 8, 13, 21, 34 etc into infinity. I count 10, 20, 30, 40, 50, etc into infinity. Nothing's changed between those various sets, I've just labeled them differently. 
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30th July 2018, 01:54 PM  #133 
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I can explain #2, hopefully to your satisfaction.
If we have two countably infinite sets, say all positive whole numbers and all positive even whole numbers, then both are infinite because numbers are abstract and therefore never run out. So, set X (all positive whole numbers) is infinite. Set Y (all positive even whole numbers) is infinite. Infinite means you never run out. If we count the numbers in set X we get... I can't seem to properly make an Aleph here. A_{0} is the closest I'm going to bother with, but that "A" is actually an Aleph, okay? Now I count the numbers in set Y. I also get A_{0}. Therefore neither one is larger than the other. For #3 I can't explain it so well except to say if neither is larger than the other it seems hard to argue the ratio matters. The second it's not infinite, no matter how large it is, we're back to normal math. It is. 
30th July 2018, 02:01 PM  #134 
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30th July 2018, 02:06 PM  #135 
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30th July 2018, 02:23 PM  #136 
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I think it's different if you're talking about theoretically or if you're talking about actually randomly selecting one.
As a general theoretical thing, we can say that since they're either even or odd it's 50/50. If you're talking about choosing one at random then since the actual method of choosing is essentially impossible with an infinite set you get all hung up with #4 in that list of questions. I think. 
30th July 2018, 02:38 PM  #137 
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30th July 2018, 02:43 PM  #138 
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Doesn't that mean that as a general theoretical thing, we can say that the OP's likelibilities are 1/6bn and 6bn/1, respectively?
And doesn't that mean that if we are given a room, we can reason mathematically about which set it more likely belongs to?
Quote:
Quote:

30th July 2018, 02:52 PM  #139 
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I just want to say that that is about the weirdest statistics answer I have ever seen. Robin, have you ever had a statistics class... ever? As someone pointed out above, your answer is quite literally the opposite of what the word probability means. I'm reminded of a statistics fail that is usually brought up early in statistics 101 type courses. There is a bag of 100 marbles. 99 are black. 1 is white. A student is asked: "What are the odds of picking a black marble." The student says: "Well, it is either going to be black or white. So the odds are 50/50." 
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30th July 2018, 02:53 PM  #140 
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The answer to 2 is well established in set theory. The only meaningful measure of the "size" of an infinite set is whether or not its members can be put in one to one correspondence with the members of some other set. (The same applies to finite sets, if we interpret the act of counting the members as equivalent to putting the members into one to one correspondence with the successive positive integers. So a set's cardinality (roughly, "number of members") is 5 if its members can be put into one to one correspondence with the set {1, 2, 3, 4, 5} or if its members can be put into one to one correspondence with some other set whose cardinality has been determined to be 5.)
We cannot count the members of infinite sets in any meaningful way, but we can attempt to put the members into one to one correspondence with the positive integers just as we do for finite sets. We can do that with the positive even numbers, for example, by pairing each even number m with the corresponding integer m/2. We can do it with the sequence of perfect squares {1, 4, 9, 25, 36, ...} by pairing off each square with its integer square root. We can do it with the primes by defining the sequence of primes as prime(n) where n is the position of that prime in the sequence, and pairing off each prime(n) with n. We can even do it with the rational numbers, by putting the rational numbers into a sequence that is guaranteed not to miss any, such as {1/1, 1/2, 2/2, 1/3, 2/3, 3/3, 1/4, 2/4, ...) and pairing each rational with its position in that sequence. And of course we can do it with the multiple of a billion by pairing off 1 with 1 billion, 2 with 2 billion, and so forth. It's important that such pairings go both ways; that is, no member of either set is left over unpaired. That's the case in all the examples above. Once such a one to one correspondence is shown for some infinite set, we say that set is countably infinite or just "countable" (which seems sloppy because being infinite would appear to mean it's not actually countable, but mathematicians use that term anyhow and "uncountable" has a different specific meaning) or that it has "aleph null" cardinality. But that's all we can say about its "size" at that point. We shouldn't (and mathematicians don't) say that the set of primes or the set of multiples of a billion is "smaller" than the set of integers even though seems as though that should be the case. They're all countably infinite, they all have the same cardinality, so they're all same "size" to the extent that the word "size" has any meaning at all in this context. There are sets that cannot be put into one to one correspondence with the positive integers. These include the real numbers and the set of all subsets of a countably infinite set. That's when the term "uncountable" or "uncountably infinite" come in, as well as the higher "aleph" symbols (aleph one, aleph two, etc.) 
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30th July 2018, 02:55 PM  #141 
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Back to the actual thread. Here's my answer which I am only pretty sure is correct:
If it is a countable infinity then yes the ratio still holds up and you can use probability to safely guess that you are in the Ritz because you awoke in an opulent room. And from the way you describe it, it would be a countable infinity. Now, if for some reason it is an uncountable infinity then it is quite possible that you are correct that ratios (and therefore probability) might become meaningless. Though I might be wrong about that, and here is why: Saying something like "In the Motel 6 set, for every opulent room, there are a billion dingy ones. In the Ritz set, it's the opposite." would never be said (knowable) about an uncountable infinity. So that pretty much cancels out it being one. 
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30th July 2018, 03:00 PM  #142 
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The answer to 3 is also well established in set theory. Assuming that the "things" in question are countably infinite subsets of the countably infinite set in question. Such as, the ratio of aces of spades to all cards in an infinite set of decks of cards.
It is no. 
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30th July 2018, 03:13 PM  #143 
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30th July 2018, 03:55 PM  #144 
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Yep. I left out some technical details. What I showed is Cantor's ordering for the countability of the set of positive ordered pairs. The same basic scheme is used for the rationals, but skipping reducible fractions and also including negative fractions. It still works. 
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30th July 2018, 04:54 PM  #145 
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Almost everything that has been said in this thread is wrong. (Early on, Dave Rogers said a number of things that are not wrong. If there is anyone else whose posts have been more right than wrong, I apologize for having failed to notice that as I skimmed through the thread.)
The mathematically rigorous theory of probability is based on measure theory, not cardinality. There are good reasons for that. One of the most common mistakes has been trying to reason about cardinality instead of measure. 
30th July 2018, 06:13 PM  #146 
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30th July 2018, 06:55 PM  #147 
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The question is illformed. Anyone who says the probability is calculable from premises that are insufficient to perform the calculation is wrong. Anyone who says all such calculations are impossible is wrong as well.
Here are some examples, all drawn from the first 40 posts in the thread. Whether the calculation is possible depends on what is meant by the highlighted sentences. If we interpret those sentences in their most literal sense, to mean there exists a function that maps every opulent Motel 6 room to a set of one billion dingy Motel 6 rooms, with those sets presumably disjoint, and there also exists a function that maps every dingy Ritz room to a set of one billion opulent Ritz rooms, with those sets disjoint, then the question cannot be answered because the existence of those functions tells us nothing about the relevant conditional probabilities. If we interpret that sentence to mean the conditional probability of waking in a dingy room is a billion times the probability of waking in an opulent room, given that he wakes in a Motel 6 room, and that the conditional probability of waking in an opulent room is a billion times the probability of waking in a dingy room, given that he wakes in a Ritz room, then the calculation is easy and the question was answered immediately by MetalPig and Dave Rogers: Then we started to see comments like those below. That's not true. That, however, was very true. Here we see an example of a poorly posed question. It looks as though Dave Rogers is assuming there exists some probability distribution on the integers such that every integer is equally likely. That is impossible. There is no such probability distribution on the integers. If all integers had the same probability, and that probability were positive, the sum of the probabilities would be infinite rather than summing to one as required by the laws of probability. If all integers had zero probability, then the total probability would be zero, also contradicting the laws of probability, because a countable set has measure zero unless one of the elements in the set has nonzero measure as a singleton set. Hence jrhowell's question was apropos: We can choose randomly from the set of real numbers between zero and one, giving equal probability to all of those real numbers, but we can't choose randomly from the set of rational numbers between zero and one while giving equal probability to all of those rational numbers. The reason for the impossibility of giving equal probability to all rationals between zero and one is essentially the same as the reason for the impossibility of giving equal probability to all integers when choosing from the set of integers. With the reals between zero and one, however, we can give equal probability to each (zero!) and have the total probability add up to one: For each subset A of [0,1], take the probability of choosing a real number in A to be the Lebesgue measure of A. Note that the probability of choosing any particular real number in [0,1] is zero, even though the probability of choosing some real number in [0,1] is one. That's because the Lebesgue measure of any singleton set {x} is 0, but the Lebesgue measure of [0,1] is 1. The fact that an event has probability zero therefore does not mean the event is impossible. Again, this question makes no sense unless Dave Rogers explains the probability distribution he has in mind. There is no probability distribution for the integers that makes every integer equally likely. Hence Robin's response was very much to the point: I could go on to the other pages, but it's too depressing. 
30th July 2018, 08:50 PM  #148 
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Please don't bother. You've criticized everyone and explained a bunch of stuff that for the most part has already been covered three or four times, but you haven't made any attempt to actually be helpful. You can tell what's being asked, do you care to address it? I'll rephrase the question
 There are two buildings with infinite rooms, and in each room is an immortal idiot named SOdhner. All the rooms are numbered with unique positive whole numbers, which aren't visible from inside the room. In one building, rooms that have a number ending in 000000000 are painted blue while all other rooms are painted red, and in the other building it's reversed. All the SOdhners know the above. So if one is, for example, in a blue room it knows it's either in the Blue building in a room that doesn't end in 000000000, or in the Red building in a room that does. In this scenario, does the color of the room it's in help them to make a educated guess? Or is it irrelevant? 
30th July 2018, 08:57 PM  #149 
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Jesus, I thought I was arrogant sometimes...

30th July 2018, 10:13 PM  #150 
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You can't calculate probabilities where infinities are concerned because dividing infinity by infinity doesn't make sense.
Consider the first building. There is nothing to stop you grouping 999999999 rooms with numbers ending in 000000000 (blue) together with 1 room that has a number not ending in 000000000 (red). Since you have an infinite number of rooms, you could have an infinite number of such groupings. NOW what is the probability that you are in a red room? ETA Another possibility could be that both hotels have a policy that the first infinity rooms will have red doors and the rest have blue doors. They may have a policy of filling all infinity of the rooms with red doors before allocating a room with a blue door. 
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30th July 2018, 10:21 PM  #151 
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31st July 2018, 01:16 AM  #152 
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It seems to me, having taken account of what I've learned from other posters in this thread, that the answer to the question posed in the OP is:
Without further information on the process by which you were assigned a room, we cannot determine the probability that you are in the Ritz. Is that reasonable? Dave 
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31st July 2018, 01:25 AM  #153 
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Would it be reasonable to wonder how you were assigned a room at all, given that there are an infinite number of them? If the checkin computer were pragmatic about it and just limited its choice to the first several jillion rooms then you might get one, otherwise wouldn't it take an infinite amount of time to choose?
(Apologies if this point has been made already. My scrambled brain was taking refuge in hard liquor last night ) 
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31st July 2018, 03:24 AM  #154 
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The measure problem is a real problem.
https://en.wikipedia.org/wiki/Measur...lem_(cosmology) I've only read the first page but most of the responses here seem to think that the answer is blatantly obvious. 
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31st July 2018, 04:58 AM  #155 
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I believe that is correct, provided the computer is attempting to choose between all of the rooms (or all of the available* rooms) with equal likelihood. As you say, if it limited its choice to some specific range, it would be able to choose with equal likelihood. If could also chose from all the rooms with unequal likelihood, which would never be guaranteed to complete its choice in any specific amount of time but in practice would come up with a choice rapidly in the overwhelming majority of times. *Infinite hotels always have available rooms, even if all the rooms are already filled with an infinite number of guests. However, this requires moving large or infinite numbers of guests who have already checked in into different rooms, which tends to result in infinite numbers of complaints to Corporate. So it's best to try to put newly arriving guests in currently unoccupied rooms whenever possible. 
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31st July 2018, 05:21 AM  #156 
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Yeah that's how I look at it. All countable infinites are equal because regardless of how they are numbered you can always perform a 1:1 match on every number in one set to every number in another set. It doesn't matter if I count 1,2,3,4 or just count the even or the odds or count by ten or just count the primes or whatever.
Look at it this way. I have two jars filled with a countably infinite number of balls on which are written every cardinal number from 1 up. I take one of the two jars and pull out every ball and write a 0 at the end of the number so the balls are now 10 and up counting by 10. See? I can still match 1 to 10, 2 to 20, 3 to 30 and so forth without either jar ever running out of balls. Again this is all less than academic. It's more philosophy than even math since "infinity" isn't a number you can plug into equations and calculations and expect it to make any kind of consistent sense. 
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31st July 2018, 05:39 AM  #157 
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The two "blatantly obvious" answers that as far as I can tell are actually wrong are: 1. Since the countably infinite sets are equal, the probability is 0.5. 2. Since some "proportion is maintained" between the size of the set and the size of a given subset as the set is iteratively expanded without bounds, the probability for the infinite set is determined by that proportion. Here's an example that shows why obvious answer #2 is questionable. You're making an infinite stack of cards by adding decks of 54 cards (including two jokers) to the stack. Each time you add a deck to the top of the stack, you remove the nearest joker to the bottom of the stack. Thus, each time you add a deck, you're adding a net of 53 cards including one joker to the stack. If you stop adding decks at any point (when some finite number of decks have been added so far) and shuffle the stack and draw a card, your probability of drawing a joker is 1 in 53. The proportion is maintained. But once you complete the infinite stack, your chance of drawing a joker is zero. Because every joker that is added is also removed in a later step. (For all n>=1, the two jokers added with the nth deck are removed upon the addition of decks 2n1 and 2n.) There are no jokers at all in the infinite stack. The proportion is not maintained. 
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31st July 2018, 05:43 AM  #158 
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"All the rooms are numbered with unique positive whole numbers" implies there is only a countable number of rooms, which implies there is no probability distribution that assigns equal probability to all rooms, which implies some rooms are more likely than others.
Your reformulation of the question does not provide any information concerning which of the rooms are more likely. Absent that information, the color of the room does not provide any information that would help to make an educated guess. Your reformulation was clearer than the original question, because there was a way to interpret the original question that allowed for an uncountable infinity of rooms with a probability distribution in which each room was equally likely. Your reformulation ruled out that interpretation. Good example. Yes. 
31st July 2018, 06:12 AM  #159 
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From which I think I can go on to answer the unstated question in the OP:
You cannot conclude from your existence that the nature of reality falls into one or the other of two possible subsets, notwithstanding the fact that you believe its likelihood to be overwhelmingly greater given one of the two. Dave 
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31st July 2018, 06:50 AM  #160 
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Yes, we're obviously talking about countably infinite rather than uncountably infinite.
So you're saying it's not 50/50 then? Not sure what you mean. You say some rooms are more likely than others, but then say you don't know which. You're saying the "every billionth room is blue rather than red" isn't why some rooms are more likely than others, but that's literally the only distinguishing feature. 
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