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Old 31st July 2018, 06:56 AM   #161
SOdhner
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Originally Posted by psionl0 View Post
You can't calculate probabilities where infinities are concerned because dividing infinity by infinity doesn't make sense.
Yeah that's what I thought, but W.D.Clinger said that's wrong.

Originally Posted by Dave Rogers View Post
Is that reasonable?
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.

Originally Posted by Roboramma View Post
That, on the other hand, is exactly what he needs. Thank you!
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Old 31st July 2018, 07:04 AM   #162
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Originally Posted by SOdhner View Post
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.
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.

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Old 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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Old 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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Old 31st July 2018, 07:36 AM   #165
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Originally Posted by Dave Rogers View Post
We then assign then to guests in the following order:
That can't be it because the scenario I gave that he was referring to did not involve anyone being assigned anywhere.
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Old 31st July 2018, 08:07 AM   #166
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Originally Posted by Fudbucker View Post
The probability is undefined??? Then what do we say? Do we just shrug in these kinds of cases?
Thread is way past my late response
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Old 31st July 2018, 08:08 AM   #167
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Originally Posted by Fudbucker View Post
So then what are the odds of me holding the ace of spades, if someone picked the card from an infinite number of decks and handed it to me?
Splunge
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Old 31st July 2018, 02:42 PM   #168
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Originally Posted by Myriad View Post
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.
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 non-primes". For example I could reorder the set of positive integers as such: take the first 10 primes, then take the first non-prime, then take the next 10 primes, then take the next non-prime, and so on. If we then apply that shoddy reasoning we would conclude the opposite: "there are more primes than non-primes".
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Old 31st July 2018, 03:31 PM   #169
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Originally Posted by Dave Rogers View Post
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
Now, how does that apply to the subjective probability that you are in the Ritz? Suppose someone puts a gun to your head and says, "Make a guess which hotel you're in". What's the optimum strategy? 50/50?

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Old 31st July 2018, 03:58 PM   #170
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Originally Posted by Fudbucker View Post
Now, how does that apply to the subjective probability that you are in the Ritz? Suppose someone puts a gun to your head and says, "Make a guess which hotel you're in". What's the optimum strategy? 50/50?
depends how dingy your room is.
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Old 31st July 2018, 05:25 PM   #171
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Originally Posted by Fudbucker View Post
Suppose someone puts a gun to your head and says, "Make a guess which hotel you're in".

The Motel 6, because the Ritz has better security.
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Old 31st July 2018, 05:38 PM   #172
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Originally Posted by Myriad View Post
The Motel 6, because the Ritz has better security.
Winner!
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Old 31st July 2018, 05:59 PM   #173
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Non-joke 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's-branch 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 three-star 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 philosopher-psychopath 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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Old 31st July 2018, 06:23 PM   #174
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Originally Posted by Myriad View Post
Non-joke 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's-branch 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 three-star 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 philosopher-psychopath 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.
That's just a version of Pascal's Wager.
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Old 31st July 2018, 06:30 PM   #175
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Originally Posted by Fudbucker View Post
That's just a version of Pascal's Wager.
You're flailing at infinite sets, trying to cobble together some sort of coherent argument out of math you don't understand.

Pascal's wager is probably the most advanced analysis in the thread so far. Let's see if you can do any better.

Or at least more useful.

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Old 31st July 2018, 07:13 PM   #176
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Originally Posted by Myriad View Post
Non-joke 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's-branch 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 three-star 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 philosopher-psychopath 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.
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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Old 31st July 2018, 07:16 PM   #177
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Originally Posted by theprestige View Post
You're flailing at infinite sets, trying to cobble together some sort of coherent argument out of math you don't understand.

Pascal's wager is probably the most advanced analysis in the thread so far. Let's see if you can do any better.

Or at least more useful.
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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Old 31st July 2018, 07:35 PM   #178
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Originally Posted by Fudbucker View Post
That's just a version of Pascal's Wager.

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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Old 31st July 2018, 07:39 PM   #179
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Originally Posted by Roboramma View Post
So you seem to have assumed a 50/50 probability in your analysis.
That's the best assumption to make when you have no information about the odds whatsoever. You are in one hotel or another and have no way of determining which one is more probable.
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Old 1st August 2018, 08:03 AM   #180
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Originally Posted by Myriad View Post
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!)
Well, it wasn't Pascal's Wager, really. I would guess I'm in the Ritz.
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Old 1st August 2018, 08:10 AM   #181
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Originally Posted by Fudbucker View Post
Now, how does that apply to the subjective probability that you are in the Ritz?
Not sure what you mean by "subjective probability." It means that there is no information on which to base a probability calculation.

Originally Posted by Fudbucker View Post
Suppose someone puts a gun to your head and says, "Make a guess which hotel you're in". What's the optimum strategy? 50/50?
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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Old 1st August 2018, 08:22 AM   #182
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Originally Posted by Dave Rogers View Post
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
Well, what are your odds of falling into a black hole? Pretty unlikely, right? But it's still interesting thinking about what would happen.
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Old 1st August 2018, 08:34 AM   #183
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Originally Posted by Fudbucker View Post
Well, what are your odds of falling into a black hole? Pretty unlikely, right?
Agreed.

Originally Posted by Fudbucker View Post
But it's still interesting thinking about what would happen.
In the case of the black hole, yes. In the case of trying to imagine what information I would pretend to have if I had no information, no, it's really not very interesting at all.

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Old 1st August 2018, 10:55 AM   #184
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Originally Posted by Dave Rogers View Post
Agreed.



In the case of the black hole, yes. In the case of trying to imagine what information I would pretend to have if I had no information, no, it's really not very interesting at all.

Dave
Different strokes. I'm curious what game theory says my best move is, in the scenario I outlined.
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Old 2nd August 2018, 01:03 AM   #185
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Originally Posted by Fudbucker View Post
Different strokes. I'm curious what game theory says my best move is, in the scenario I outlined.
You might as well ask, if someone puts a gun to your head and asks you whether fleems are bigger or smaller than ploots, what's your best strategy?

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Old 2nd August 2018, 01:55 AM   #186
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Originally Posted by psionl0 View Post
That's the best assumption to make when you have no information about the odds whatsoever. You are in one hotel or another and have no way of determining which one is more probable.
Slightly disagree with how you put that. The assumption shouldn't be that's it 50/50 when you have no information, you shouldn't make any assumptions as to the probabilities as it is for you a total guess. You simply don't know.
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Old 2nd August 2018, 01:57 AM   #187
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Originally Posted by Fudbucker View Post
Different strokes. I'm curious what game theory says my best move is, in the scenario I outlined.
Which "game theory"? I don't know of any that deal with infinities.
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Old 2nd August 2018, 01:57 AM   #188
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Originally Posted by Fudbucker View Post
Well, what are your odds of falling into a black hole? Pretty unlikely, right? But it's still interesting thinking about what would happen.
Today - zero.
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Old 2nd August 2018, 02:06 AM   #189
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Originally Posted by Darat View Post
Today - zero.
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.

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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
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Old 2nd August 2018, 02:54 AM   #190
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Originally Posted by Dave Rogers View Post
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.
The highlighted is where you go wrong here.
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Old 2nd August 2018, 03:08 AM   #191
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Originally Posted by Roboramma View Post
The highlighted is where you go wrong here.
How likely is it, then?

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
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Old 2nd August 2018, 03:47 AM   #192
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Originally Posted by Fudbucker View Post
I'm curious what game theory says my best move is, in the scenario I outlined.
Originally Posted by Darat View Post
Which "game theory"? I don't know of any that deal with infinities.
Game theory has no problem with infinities.

Even in the simple case of a complete-information two-person zero-sum 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.)

Originally Posted by Darat View Post
Slightly disagree with how you put that. The assumption shouldn't be that's it 50/50 when you have no information, you shouldn't make any assumptions as to the probabilities as it is for you a total guess. You simply don't know.
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 first second post within this thread, it is possible to interpret Fudbucker's original question (but not SOdhner's reformulation of that question) as a question that assumes there are an uncountable infinity of rooms, each of them equally likely, with opulent rooms a billion times more (or less) likely than dingy rooms, depending on the name of the hotel. (As yet another implicit assumption, we would also have to assume the hotels are equally likely.) Under that interpretation, applying game theory to the problem is quite trivial, and the question was answered immediately by the third and fourth posts in this thread.

In a subsequent post, however, Fudbucker sought to clarify a follow-up 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
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Old 2nd August 2018, 04:07 AM   #193
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Originally Posted by W.D.Clinger View Post
The problem is the lack of a priori information about the probabilities of the rooms.
Short of flat-out knowing the answer I'm having trouble imagining what other information you could have in that scenario.
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Old 2nd August 2018, 07:06 AM   #194
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Originally Posted by Dave Rogers View Post
How likely is it, then?

Dave
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 1/1012 /year.



*This is different from interactions with the supermassive black hole at the center of the galaxy.
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Old 2nd August 2018, 07:12 AM   #195
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Originally Posted by Fudbucker View Post
Different strokes. I'm curious what game theory says my best move is, in the scenario I outlined.
If you want the game theory answer, why are you complaining about Pascal's Wager?
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Old 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.
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Old 2nd August 2018, 07:34 AM   #197
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Absolutely.
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Old 2nd August 2018, 07:51 AM   #198
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Originally Posted by Darat View Post
Slightly disagree with how you put that. The assumption shouldn't be that's it 50/50 when you have no information, you shouldn't make any assumptions as to the probabilities as it is for you a total guess. You simply don't know.
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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Old 2nd August 2018, 08:27 AM   #199
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Originally Posted by caveman1917 View Post
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.
I have a sneaking suspicion Fudbucker is going to use this thread to argue that there's a 50/50 chance of there being a Creator...
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Old 2nd August 2018, 09:01 AM   #200
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Originally Posted by Roboramma View Post
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.
Context matters. I'm fairly confident Fudbucker is working on an Answer of the Gaps that meets a rhetorical goal he's pursuing in other threads.
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