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31st July 2018, 06:56 AM  #161 
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Yeah that's what I thought, but W.D.Clinger said that's wrong.
It's reasonable, but from the OP's point of view it's not helpful. Despite how he worded his post he isn't really picturing something where anything would need to be randomly selected. So you're not wrong but to the Fudbucker it's kinda a red herring. That, on the other hand, is exactly what he needs. Thank you! 
31st July 2018, 07:04 AM  #162 
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I think, having paid attention and learned some bits that I'd previously missed, I can now answer that one.
Suppose the rooms are all given positive integer numbers, all rooms with red numbers except those whose number is an integer multiple of 1,000,000,000, which have blue numbers. We then assign then to guests in the following order: Each odd numbered guest is assigned to the empty room with the lowest red number. Each even numbered guest is assigned to the empty room with the lowest blue number. The number of rooms is infinite, so we can continue this system infinitely. The odds of any guest being in a blue room are 50:50. Now let's go back to the start, but this time we assign rooms to guests in simple numerical order. The odds of any guest being in a blue room are 1:999,999,999. What we can't do is say we'll assign rooms to guests completely at random, because it's impossible to choose a random number from an infinite set. So we need a method that already fixes some probability distribution, and the probability distribution embodied in the method determines the probability of being in a particular subset of rooms, rather than any intrinsic mathematical properties of the infinite set or sets of rooms involved. Dave 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

31st July 2018, 07:06 AM  #163 
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Again I think we're all talking around each other. "What is the probability of a single event happening within an infinite set?" does not have an answer.
Again it's the fundamental problem of the Supertask, if a series of infinite events happens within a finite period of time, what is the final state of the sequences? It's asking us to imagine a never ending sequences of events and asking us to predict what happens at the end of it. It's the paradox of a sequence that has a final state, but no ending state. And that applies to probability as well. Probability is a descriptor term for (and yes I'm laymaning the hell out of this...) "How many times will X happen in Y timeframe?" If you a roll a fair die in a sequence of 6 times over and over, the probability is going to average out for each number rolled to about 1 in 6. If you shuffle a deck of cards over and over and over the probability of any particular card being the top card is eventually going to average out to about 1 in 52. So it's not that questions of probability in an infinite set are right or wrong, it's that the very concept of probability as we normally use it (I'm sure there is some splinter of probability mathematics that deals with infinity) doesn't apply here. In an infinite set "How many times does a specific event occur" can't be answered and "How many overall events period occur?" also can't be so how and why does probabiliy even factor into this? I guess an infinite sequence of the exact same thing over and over could still be said to have a probability of 1 over 1 (As in if I have a countably infinite deck of just the Ace of Spades card over and over it's both intellectual honest and technically correct to say my odds of drawing an Ace of Spades from the deck is 100%) but put any variation in that at all and that goes out the window. 
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31st July 2018, 07:22 AM  #164 
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And what if the guest isn't assigned a room at all, but are allowed to choose their own?
The hotel receptions, lifts and corridors look alike, so no clues there. But if a guest is handed a master key and invited to pick any room? They might go to the closest (my choice, as it's then much easier to nip outside for a smoke), or choose to go to the 23987th floor, hop on a motorbike and drive for hours to a distant room. Does the nature of that room give them any clue about which hotel they're in? It strikes me that it does, though this takes us right back to the beginning of the discussion. It certainly doesn't prove anything, but the odds of being deceived by sheer bad luck are extreme. 
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31st July 2018, 07:36 AM  #165 
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31st July 2018, 08:07 AM  #166 
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I suspect you are a sandwich, metaphorically speaking. Donn And a shot rang out. Now Space is doing time... Ben Burch You built the toilet  don't complain when people crap in it. _Kid Eager Never underestimate the power of the Random Number God. More of evolutionary history is His doing than people think.  Dinwar 

31st July 2018, 08:08 AM  #167 
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I suspect you are a sandwich, metaphorically speaking. Donn And a shot rang out. Now Space is doing time... Ben Burch You built the toilet  don't complain when people crap in it. _Kid Eager Never underestimate the power of the Random Number God. More of evolutionary history is His doing than people think.  Dinwar 

31st July 2018, 02:42 PM  #168 
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One of the problems here is that cardinality is independent of order, yet people implicitly use a specific ordering when coming to conclusions such as "there are less primes than nonprimes". For example I could reorder the set of positive integers as such: take the first 10 primes, then take the first nonprime, then take the next 10 primes, then take the next nonprime, and so on. If we then apply that shoddy reasoning we would conclude the opposite: "there are more primes than nonprimes".

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31st July 2018, 03:31 PM  #169 
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31st July 2018, 03:58 PM  #170 
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31st July 2018, 05:25 PM  #171 
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A zømbie once bit my sister... 

31st July 2018, 05:38 PM  #172 
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31st July 2018, 05:59 PM  #173 
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Nonjoke answer:
"I have no way to know which is the right answer. However, if I guess I'm in the Motel 6, and I'm correct (and therefore survive), I'll still be at a Motel 6, with a crappy Motel 6 outdoor pool and a crappy Denny'sbranch restaurant, even if my room happens to be opulent. If I guess I'm in the Ritz, and I survive, I'll be at the Ritz, with great room service and a full spa and a threestar restaurant, even if my room happens to be dingy. "If I guess wrong, then it doesn't matter which hotel I'm at, because I'll be dead. Unless the armed philosopherpsychopath is bluffing, in which case it doesn't matter what I guess. "Therefore, I guess I'm at the Ritz." In other words, having no way to determine (even applying probability theory) which answer is correct, I choose the answer that is most useful. Now, apply that same principle to the question of whether you are an organism that arose from, exists in, and interacts with a real world; or a Boltzmann brain existing only for the present flash of a moment with false memories of a life and false perceptions of a livable environment in an otherwise patternless void. 
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A zømbie once bit my sister... 

31st July 2018, 06:23 PM  #174 
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31st July 2018, 06:30 PM  #175 
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31st July 2018, 07:13 PM  #176 
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Inherent in this idea is the assumption that the two are equally probable and thus your choice of which to guess won't impact on the probability of your survival.
If it doesn't impact on the probability of your survival, the spa starts to matter. If it does impact on your survival the spa doesn't seem so important anymore. So you seem to have assumed a 50/50 probability in your analysis. 
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31st July 2018, 07:16 PM  #177 
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Not having a more useful answer doesn't suggest that other answers are good. As I said earlier the measure problem really is a problem and it's one that unsolved and important to cosmology. Worldclass physicists are basically saying "we don't know how to deal with this", but the fact that Fudbucker doesn't have an answer to it is somehow a strike against him?
Personally I think it counts in his favour that he doesn't think he's found a solution. 
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31st July 2018, 07:35 PM  #178 
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You introduced the guy who demands a correct response to an opaque dilemma on threat of violence. What possible continuation of your scenario would not resemble Pascal's Wager? Try this one: "Here's a sealed lead case that contains either $1000 or $2000. I'll tell you nothing that would help you measure which possibility is more likely. You may not examine the case in any way. You get one guess what the amount in the case is. If and only if your guess is correct, you win the money in the case." What would you guess, and why? (Make sure your answer isn't just a version of Pascal's Wager!) 
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A zømbie once bit my sister... 

31st July 2018, 07:39 PM  #179 
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"The process by which banks create money is so simple that the mind is repelled. Where something so important is involved, a deeper mystery seems only decent."  Galbraith, 1975 

1st August 2018, 08:03 AM  #180 
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1st August 2018, 08:10 AM  #181 
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Not sure what you mean by "subjective probability." It means that there is no information on which to base a probability calculation.
I have no idea, but since the probability of being in a hotel with an infinite number of rooms in which someone puts a gun to my head and asks me to guess which hotel I'm in is zero, I'm not too concerned at trying to figure one out. But what would be utterly futile is to try to conclude, from whatever strategy I chose, any knowledge whatsoever about which hotel I happened to be in. This is why, in general, it's a more fruitful line of argument to ask "How did this situation come about" than to try to base one's understanding of the situation on a priori probability calculations based on a complete absence of data, knowledge or understanding. Dave 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

1st August 2018, 08:22 AM  #182 
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1st August 2018, 08:34 AM  #183 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

1st August 2018, 10:55 AM  #184 
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2nd August 2018, 01:03 AM  #185 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

2nd August 2018, 01:55 AM  #186 
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I wish I knew how to quit you 

2nd August 2018, 01:57 AM  #187 
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I wish I knew how to quit you 

2nd August 2018, 01:57 AM  #188 
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I wish I knew how to quit you 

2nd August 2018, 02:06 AM  #189 
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You don't know that. There might be a black hole currently approaching Earth at near lightspeed, such that information about its existence hasn't reached us yet and won't until mere hours before impact. But of course we have no idea how likely such an event is, so according to the unknown=50% school of probability theory you actually have a 50% chance of falling into a black hole today.
Dave 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

2nd August 2018, 02:54 AM  #190 
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"... when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together." Isaac Asimov 

2nd August 2018, 03:08 AM  #191 
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Me: So what you're saying is that, if the load carrying ability of the lower structure is reduced to the point where it can no longer support the load above it, it will collapse without a jolt, right? Tony Szamboti: That is right 

2nd August 2018, 03:47 AM  #192 
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Game theory has no problem with infinities.
Even in the simple case of a completeinformation twoperson zerosum game in which each player has only two choices, there is an uncountably infinite set of strategies corresponding to the probabilities the players might use when making their choices. (In most such games, an optimal strategy is a mixed strategy that involves making a random selection among the available choices, using probabilities that are easily calculated from the payoff matrix. That calculation selects an optimal strategy from among the uncountable infinity of possible mixed strategies.) That's the essence of the problem. In his original post, Fudbucker referred to "a probability calculus", so Fudbucker must have been aware that his question was related to the theory of probability and randomness. Fudbucker asked whether the "infinite amount of both types of rooms in both sets" makes a probability calculus impossible. The answer to that question is: No. It is not the infinite number of rooms that prevents game theory from being applied to the problem. It is the total lack of information about the probabilities of the rooms that prevents game theory from being applied to the problem. During the subsequent discussion, after I noted that SOdhner's reformulation of the original problem statement implied there is only a countable number of rooms, SOdhner asserted that "we're obviously talking about countably infinite rather than uncountably infinite." I don't think that was so obvious. Had it been obvious, it would have been obvious to those who have done their homework that it would have been mathematically impossible for all of the rooms to be equally likely. That mathematical fact would rule out any implicit assumption that all outcomes are equally likely, which has been suggested as "the best assumption to make when you have no information about the odds whatsoever." Having eliminated the possibility that all outcomes are equally likely, examination of the question would have immediately shown that there is absolutely no information available concerning which rooms are more likely than others, let alone any information concerning the numerical probabilities of the different rooms. Furthermore, some have suggested that the question might somehow be related to the measure problem of cosmology, which involves consideration of all possible universes. So far as we know, one of the ways in which possible universes can differ is in the numerical values of certain physical constants, which are real numbers. If those constants can vary over even the tiniest range in the possible universes, then there is an uncountable infinity of possible universes. Even if there are constraints on possible universes that force the values of all physical constants to be the same in all possible universes, it seems unlikely that the forces I apply to my keyboard as I type this would have to be the same in all possible universes. As those forces are vectors over the real numbers, that degree of variability would also imply an uncountable infinite of possible universes. The theory of probability has no problem at all with an uncountable infinity of possible outcomes (as is obvious to anyone who is familiar with Gaussian (aka normal) distributions), and game theory has no problem at all with an uncountable infinity of possible strategies or outcomes. As I stated in my In a subsequent post, however, Fudbucker sought to clarify a followup example by saying "These are both countable infinite sets, are they not?" That suggests to me that Fudbucker's original question assumed there is only a countable infinity of rooms, as in SOdhner's reformulation of the question, which rules out any possibility of the rooms being equally likely. Once the possibility of that implicit assumption has been eliminated, it becomes obvious that there is not enough information about the probabilities of the rooms for game theory to be applied. Infinity is not the problem. The problem is the lack of a priori information about the probabilities of the rooms. 
Last edited by W.D.Clinger; 2nd August 2018 at 04:01 AM. Reason: see strikeout and added text in gray 

2nd August 2018, 04:07 AM  #193 
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2nd August 2018, 07:06 AM  #194 
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I'd have to do some more research to understand more, but we can certainly but upper bounds on it's likelihood.
Some ways of estimating the number of black holes in our galaxy: look for black hole interactions with other objects, such as stars. There are something like 1/1000 as many black holes as stars in the galaxy. A simple model in which black holes follow similar trajectories to stars will have 1/1000th as many interactions between stars and black holes as between stars and other stars. But your scenario is certainly less likely than that as it's positing a black hole travelling at near the speed of light. I don't know of any large objects travelling at the kind of fraction of the speed of light that you're talking about. Given that we can see 100 billion stars in our galaxy alone I'll put that probability then as less than 1/100 billion. You're also not talking about just interactions but also an actual collision. A collision with a planet is less likely than a collision with a star (at that fraction of the speed of light the only real consideration is the fact that the star is a larger target). So, of all the stars in the galaxy have we seen any that have had a collision with a black hole, either in real time or seeing the aftermath? I don't know of any*, and I think we'd have evidence even, say, 1000 years after the collision, so I'll put the probability again at less than 1/100 billion/thousand years, even without the requirement that the black hole is travelling at a large fraction of the speed of light. Ignoring the requirement that it be relativistic and not having done the research to see if my assumptions are correct, I'll revise my upper bound up a couple of orders of magnitude to 1/10^{12} /year. *This is different from interactions with the supermassive black hole at the center of the galaxy. 
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"... when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together." Isaac Asimov 

2nd August 2018, 07:12 AM  #195 
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2nd August 2018, 07:20 AM  #196 
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I suspect that if a black hole were to cross your path at relativistic velocity, the risk of falling within its event horizon is going to be the least of your worries.

2nd August 2018, 07:34 AM  #197 
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Absolutely.

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"... when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together." Isaac Asimov 

2nd August 2018, 07:51 AM  #198 
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The assumption should be 50/50 when you have no information, P(Ritz) = P(Motel 6) = 0.5, since that is the maximum entropy distribution under no constraints (ie with no information). ETA: note that this isn't a probability distribution over the possible (countably infinite) rooms  which doesn't have a maximum entropy distribution  but over which of two possible hotels you're in.

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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 

2nd August 2018, 08:27 AM  #199 
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2nd August 2018, 09:01 AM  #200 
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