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Old 30th July 2018, 08:25 AM   #41
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If you look carefully at all the classic Hilbert Hotel thought experiments, you'll notice that in each case the solution prescribes a specific method for assigning guests to rooms. There is not one where "assign a guest to a random room with equal probability." There's a reason for that.

There are ways to assign a guest to a random room with unequal probability. That's obvious; for instance you could roll a 20-sided die numbered 1-20 to get a number n, and assign the guest to the nth available room. That's very limited. You could instead roll a million-sided die for each available room in sequence, and if the roll comes up 1 you assign the guest to that room, otherwise keep rolling. That gives a finite probability that the guest could end up in any of the infinite number of unoccupied rooms, though the probabilities get extremely low for higher numbered rooms and are nowhere close to equally distributed through the infinite set of rooms.

So the question you must ask when you wake up in the hotel room, is how were you assigned to that particular room? You don't know, of course, but you can be certain that it was not at random with equal distribution. Which makes the fact that the number of rooms is infinite moot, because whatever procedure by which you were assigned a specific room, it could not have had equal access to all the rooms. Therefore, assuming the special rooms in each hotel are approximately equally distributed, DaveRogers and MetalPig's reasoning applies.
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Old 30th July 2018, 08:25 AM   #42
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Quote:
No, they're reverse.No, they're reverse.
Why do you think so.

Suppose both hotels number their rooms as you suggest, but put the opulent rooms in corridor A and the dingy rooms in corridor B.

Are there more rooms in corridor A than corridor B?
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Old 30th July 2018, 08:26 AM   #43
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Originally Posted by ddt View Post
Yes, to be precise: those numbers are Aleph-naught.


No, they're reverse.


Because calculations with infinites don't work the same way as with finite numbers.

Maybe this helps.

Both the Ritz and Motel-6 have their rooms numbered 1, 2, 3, etc., ad infinitum, using all the natural numbers. In the Ritz, all rooms of which the number ends with 9 zeros are dingy, the other rooms opulent. In Motel-6, all rooms of which the number ends with 9 zeros are opulent, the rest dingy. (*) The room numbers are only displayed on the outside; there's nothing inside the room that tells you its number.

(1) As soon as you wake up, someone phones you and tells your room number is lower/equal to 10^12. What is the probability now that you are in the Ritz?

(2) As soon as you wake up, someone phones you and tells your room number is lower/equal to 10^15. What is the probability now that you are in the Ritz?

(3) Repeat, with 10^18.

(4) Repeat, with 10^21.

Et cetera.

(*) In the interest of giving a simple description, I've slightly changed the ratios from 1 billion to 1 to 999,999,999 to 1, but that shouldn't distract from the point.
Bump.

Robin, do your homework.
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Old 30th July 2018, 08:38 AM   #44
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Originally Posted by jrhowell View Post
I don't know if I am right mathematically, but if all I had to go on is that one piece of information (opulent room) and I had to stake my life on one choice or the other then I would pick the Ritz.

That's a poor way of calculating probability. You're basically equating aversion to risk to probability. They're two different things. For example, if I had a penny to bet on which hotel I was in (and I was in an opulent room), I'd bet I was at the Motel 6. I could stand to lose a penny but would like the chance to win $10,0000,000.00. If, however, I had to bet ten million dollars which hotel I was in, I'd pick the Ritz because I really don't want to lose ten million dollars. In both cases, the probabilities are the same. But in each case, my bet changes based on my aversion to risk.

It's not probability at that point, it's psychology.
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Old 30th July 2018, 08:40 AM   #45
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Originally Posted by SOdhner View Post

We can't really talk about picking a room at random because we can't calculate probability with infinity.
We can't? Why not? That's literally what I'm asking.
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Old 30th July 2018, 08:41 AM   #46
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Originally Posted by Robin View Post
Why do you think so.

Suppose both hotels number their rooms as you suggest, but put the opulent rooms in corridor A and the dingy rooms in corridor B.

Are there more rooms in corridor A than corridor B?
Scenario 1.
I have an infinite hotel with ten corridors, A thru J. Corridor A contains the rooms with numbers 1, 11, 21, 31, etc., i.e., all numbers ending with digit 1. Corridor B the rooms with numbers 2, 12, 22, 32, etc. All through to corridor J which contains rooms numbers 10, 20, 30, etc.

Are there more rooms in corridor B than in corridor A?

Scenario 2.
I have an infinite hotel with two corridors, Y and Z. Corridor Y contains all rooms with numbers ending in 1, i.e., 1, 11, 21, etc. So it's the same, essentially, as corridor A in scenario 1. Corridor Z contains all other rooms.

Are there more rooms in corridor Z than in Y?
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Old 30th July 2018, 08:47 AM   #47
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Originally Posted by Robin View Post
Why do you think so.

Suppose both hotels number their rooms as you suggest, but put the opulent rooms in corridor A and the dingy rooms in corridor B.

Are there more rooms in corridor A than corridor B?
There are an infinite number of rooms in both. The concept "more" doesn't apply in the sense you're trying to use it.

Now, though, suppose the rooms are divided up into a billion corridors. What's the chance that your room is on one specific corridor - say, corridor number 201,637,225?

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Old 30th July 2018, 08:49 AM   #48
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Originally Posted by Dave Rogers View Post
What is the probability that a randomly chosen positive integer will be divisible by ten?

That probability is zero, because there's no such thing as a randomly chosen integer, if you mean randomly chosen with even distribution from the entire set of positive integers.

Any actual method of randomly choosing an integer, such as randomly choosing with equal distribution from a specific range, or choosing with an asymptotically decreasing probability (unequal distribution) from (potentially) the entire set, will be selecting from among a greater number of accessible or preferentially available non multiples of 10 than multiples of 10. In that case, assuming the selection method is not specifically skewed toward multiples of 10, the probability will be 1 in 10.
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Old 30th July 2018, 08:51 AM   #49
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Originally Posted by ddt View Post
Bump.

Robin, do your homework.
Are you avoiding answering my question about the corridors?

If you are pressing for an answer to the two questions you pose, the answer is I don't know. It really depends on the configuration of the hotel.

That is how probability and infinity works - it depends on how the entities are lined up or how they are chosen, or counted.

Maybe you should do your homework.
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Old 30th July 2018, 08:52 AM   #50
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Suppose I have an urn with an infinite number of marbles. For every blue marble there are ten red ones. Are my odds of my picking a blue marble 1 in 10?
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Old 30th July 2018, 08:55 AM   #51
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Originally Posted by Dave Rogers View Post
There are an infinite number of rooms in both. The concept "more" doesn't apply in the sense you're trying to use it.
It applies perfectly well, I don't know why you think it doesn't. There are equally many rooms in both.
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Old 30th July 2018, 08:56 AM   #52
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Originally Posted by Fudbucker View Post
Suppose I have an urn with an infinite number of marbles. For every blue marble there are ten red ones. Are my odds of my picking a blue marble 1 in 10?
Why don't you tell me and show me the calculation. Are there more red than blue marbles?
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Old 30th July 2018, 08:56 AM   #53
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These are both countable infinite sets, are they not? Therefore, that makes them equal. So how can the probability between two equal sets be anything other than 50/50?
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Old 30th July 2018, 08:58 AM   #54
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Originally Posted by Robin View Post
Why don't you tell me and show me the calculation. Are there more red than blue marbles?
I don't know the answer. I assumed that since there were an infinite amount of both types of marble, the odds are 50/50, but if you work out a Bayesian calculus, I think you should get 1 in 10. I'm too rushed to do the calculus.
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Old 30th July 2018, 09:02 AM   #55
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Originally Posted by Robin View Post
It applies perfectly well, I don't know why you think it doesn't. There are equally many rooms in both.
It doesnt in the sense that, if A>B, then A-B>0, if A=B then A-B=0, and if A<B, then A-B<0. If A=B=aleph-zero, A-B is neither positive nor negative; it is undefined. One cannot therefore assign equalities or inequalities.

Now, how about you answer my question?

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Old 30th July 2018, 09:03 AM   #56
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Originally Posted by Fudbucker View Post
Suppose I have an urn with an infinite number of marbles. For every blue marble there are ten red ones. Are my odds of my picking a blue marble 1 in 10?
Please describe the method you will use to select a marble (and don't just say "at random").
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Old 30th July 2018, 09:05 AM   #57
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Originally Posted by Robin View Post
Why don't you tell me and show me the calculation. Are there more red than blue marbles?
Jesus Christ. I can't speak for Fudbucker, but if I knew the calculation I probably wouldn't be asking the question.

If you know the answer, could you please give the calculation here, and explain it for me? Thanks in advance.
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Old 30th July 2018, 09:06 AM   #58
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Originally Posted by Robin View Post
Are you avoiding answering my question about the corridors?
No, I don't, I purposely gave you a counter question. And I was first in asking you. It seems you studiously avoided answering my post while my answered all other posts.

Originally Posted by Robin View Post
If you are pressing for an answer to the two questions you pose, the answer is I don't know. It really depends on the configuration of the hotel.

That is how probability and infinity works - it depends on how the entities are lined up or how they are chosen, or counted.

Maybe you should do your homework.
The questions I asked don't depend on the configuration of the hotel.
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Old 30th July 2018, 09:07 AM   #59
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Originally Posted by jrhowell View Post
Please describe the method you will use to select a marble (and don't just say "at random").
Reaching in and pulling a marble out without looking. In other words, randomly.
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Old 30th July 2018, 09:07 AM   #60
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Originally Posted by theprestige View Post
We can't? Why not? That's literally what I'm asking.

Approximately how many digits would you expect a randomly selected integer to have?

There's no answer to that question that makes sense.

Every integer has a finite number of digits. Yet, for every integer no matter how many digits it has, there are an infinite number of higher integers and only a finite number of lower ones. So every integer is vanishingly tiny relative to the overall set.
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Old 30th July 2018, 09:15 AM   #61
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Originally Posted by Fudbucker View Post
Reaching in and pulling a marble out without looking. In other words, randomly.
So you are actually selecting from the set of marbles that are within reach of the opening?
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Old 30th July 2018, 09:18 AM   #62
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Originally Posted by Fudbucker View Post
Reaching in and pulling a marble out without looking. In other words, randomly.

It's not that simple.

If a marble is something you can reach in and pull one out separately from the others, then it has a finite size. If there are an infinite number of them, then the urn must have infinite volume to contain them, and only a finite subset of the marbles can be actually within your reach at any given moment.

We can try to make it possible to reach any (all) of the marbles. To avoid having to dig or dive, we might elongate the urn into a sort of trough instead, with a walkway beside it, extending off into infinite distance. Then you can walk along it to your heart's content, and stop anywhere and reach in and pick out a marble.

But how far are you willing to walk? When you pick a marble you can only have walked a finite distance. (If you walk an infinite distance you never actually get around to picking a marble, so that doesn't help.) That means you're still only selecting from the finite subset of marbles within that distance of your starting point.
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Old 30th July 2018, 09:28 AM   #63
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Can't I just reorder the rooms to get any ratio I like of opulent and dingy rooms? Infinite series are strange beasts.
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Old 30th July 2018, 09:34 AM   #64
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Originally Posted by shuttlt View Post
Can't I just reorder the rooms to get any ratio I like of opulent and dingy rooms? Infinite series are strange beasts.
Yes, I rather think that's the point Myriad's making, and which I see rather more clearly now - thanks for the insight.

So how would the problem look when re-framed like this?

You wake up in one of the two infinite hotels. Each has a billion corridors, but only one of the corridors in the Motel-6 contains opulent rooms; the remainder are all small and dingy. On the other hand, only one of the corridors in the Ritz contains small and dingy rooms; all the others are opulent. You've heard that the receptionist, when asked to assign a guest a room, rolls a billion-sided fair die to choose a corridor, then assigns the guest to the next free room in that corridor. You look around you and see that your room is opulent. What is the likelihood that you are in the Ritz?

Myriad, does that look like a better formulation? Fudbucker, does it still relate to your original point?

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Old 30th July 2018, 09:40 AM   #65
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Originally Posted by Robin View Post
Why do you think so.

Suppose both hotels number their rooms as you suggest, but put the opulent rooms in corridor A and the dingy rooms in corridor B.

Are there more rooms in corridor A than corridor B?

There are the same number of rooms in both corridors.

But how was which corridor the room you ended up in selected? If you start by choosing a room (randomly or otherwise) and then go to whichever corridor your room is in, then the corridors make no difference.

If you explicitly choose a corridor, then you're making an explicitly biased selection of rooms. This would be comparable to using one of the necessarily non-uniform random integer selection methods I described a few posts ago, then flipping a coin and if the coin is heads, multiplying the selection by ten. That would make the probability of selecting a multiple of ten greater than 0.5, instead of 0.1.

But if you're tweaking the selection probabilities you can't draw any general conclusions. If you always pick 7 when the coin is heads, it doesn't mean there are as many integers as there are integers equal to 7.
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Old 30th July 2018, 09:46 AM   #66
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Originally Posted by Dave Rogers View Post
Myriad, does that look like a better formulation?

Yes indeed.

Or there could only be two corridors, and the receptionist rolls the billion-sided die and puts you in the exceptional corridor (dingy in the Ritz, opulent in the Motel 6) if and only if it comes up 1. At that point the room numbering scheme would no longer matter.
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Old 30th July 2018, 09:52 AM   #67
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Originally Posted by Dave Rogers View Post
Fudbucker, does it still relate to your original point?
I'm going to say no. While we don't want to derail into a discussion about it here, he's asking because he wants to make a point about Boltzmann Brains which are hypothetical brains that randomly appear in an infinite universe of entropic boredom. Since they would have infinite space and time to appear in he wants to argue that there would be an infinite amount that act just like his brain and so even if there are also an infinite amount of actual physical people it's at worst a 50/50 chance that you're a disembodied brain in space.

The hotel example given limits the hallways to a billion, but he would be arguing for an infinite number of universes as well so any limit is going to invalidate it.
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Old 30th July 2018, 09:53 AM   #68
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Originally Posted by Dave Rogers View Post
Yes, I rather think that's the point Myriad's making, and which I see rather more clearly now - thanks for the insight.

So how would the problem look when re-framed like this?

You wake up in one of the two infinite hotels. Each has a billion corridors, but only one of the corridors in the Motel-6 contains opulent rooms; the remainder are all small and dingy. On the other hand, only one of the corridors in the Ritz contains small and dingy rooms; all the others are opulent. You've heard that the receptionist, when asked to assign a guest a room, rolls a billion-sided fair die to choose a corridor, then assigns the guest to the next free room in that corridor. You look around you and see that your room is opulent. What is the likelihood that you are in the Ritz?

Myriad, does that look like a better formulation? Fudbucker, does it still relate to your original point?

Dave
That's a different question though, where the infinities don't matter at all.
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Old 30th July 2018, 10:12 AM   #69
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I think the question of how something is selected isn't going to helpful to Fudbucker, so here's a re-phrased question:

We've got the two hotels, yadda yadda yadda.

"You" are a top of the line Magic Fingers massage bed with integrated AI. There is a bed just like you in every room of both hotels. You know the design specifications of the hotels but the contractor forgot to indicate which one you're in (same contractor for both hotels, and he cut some corners).

It's opening day, the power has just come on, and you're aware of your surroundings. Nothing was randomly selected because there's one in each room. But you want to welcome a guest if one ever arrives by saying "Welcome to the _______!" and you're not certain what the best bet is.
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Old 30th July 2018, 10:57 AM   #70
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Originally Posted by jrhowell View Post
So you are actually selecting from the set of marbles that are within reach of the opening?
Yes, but I don't see how that should be relevant. Pretend I spread out a deck of cards face-down randomly, but I can only reach the ten closest to me. My odds of picking the ace of spades are still the same, even though I can't reach the other cards.
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Old 30th July 2018, 11:00 AM   #71
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Originally Posted by Myriad View Post
It's not that simple.

If a marble is something you can reach in and pull one out separately from the others, then it has a finite size. If there are an infinite number of them, then the urn must have infinite volume to contain them, and only a finite subset of the marbles can be actually within your reach at any given moment.

We can try to make it possible to reach any (all) of the marbles. To avoid having to dig or dive, we might elongate the urn into a sort of trough instead, with a walkway beside it, extending off into infinite distance. Then you can walk along it to your heart's content, and stop anywhere and reach in and pick out a marble.

But how far are you willing to walk? When you pick a marble you can only have walked a finite distance. (If you walk an infinite distance you never actually get around to picking a marble, so that doesn't help.) That means you're still only selecting from the finite subset of marbles within that distance of your starting point.
Just posit a machine that handles all the marbles in its little claws and eventually picks one at random (and nevermind the fact that it would take an infinite amount of time to do it, that's not relevant). What are the odds the machine picked a blue marble?
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Old 30th July 2018, 11:08 AM   #72
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Originally Posted by Fudbucker View Post
Yes, but I don't see how that should be relevant. Pretend I spread out a deck of cards face-down randomly, but I can only reach the ten closest to me. My odds of picking the ace of spades are still the same, even though I can't reach the other cards.
The issue is once you're talking about the distribution within a limited set you don't have to worry about infinity anymore and the question becomes both easy and irrelevant.
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Old 30th July 2018, 11:15 AM   #73
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Originally Posted by SOdhner View Post
The issue is once you're talking about the distribution within a limited set you don't have to worry about infinity anymore and the question becomes both easy and irrelevant.
The point I was making is that my inability to reach a marble or card shouldn't have a bearing on the odds of what the marble or card are. If you take 52 cards and hand me just one at random, there's a 1 in 52 chance it's the ace of spades. If you picked a card at random from ten decks and handed it to me, the odds it's the ace are 1 in 52. Same with a hundred decks, a thousand, and so on. Do the odds change when we're dealing with an infinite amount of decks?
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Old 30th July 2018, 11:22 AM   #74
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Originally Posted by Myriad View Post
Approximately how many digits would you expect a randomly selected integer to have?
I feel like your questions are predicated on a body of knowledge I don't (yet) possess.

Instead of attempting a Socratic dialogue, would you be willing to consider just explaining the basic concepts, the reasoning from those concepts, and the conclusion from that reasoning?

I can't guarantee that I'd learn something from that approach, but it will probably work better than expecting me to answer questions about stuff I don't know or understand.
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Old 30th July 2018, 11:28 AM   #75
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Originally Posted by Fudbucker View Post
Do the odds change when we're dealing with an infinite amount of decks?
Right, I agree that's the question. I was just explaining why they were caring about the method of selection - if you're selecting from a finite area we all know how it works already so it doesn't really answer your question. I tried to help your cause out by giving an alternate hotel example where no "selection" is happening, but nobody has responded to it.
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Old 30th July 2018, 11:30 AM   #76
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Originally Posted by Fudbucker View Post
The point I was making is that my inability to reach a marble or card shouldn't have a bearing on the odds of what the marble or card are. If you take 52 cards and hand me just one at random, there's a 1 in 52 chance it's the ace of spades. If you picked a card at random from ten decks and handed it to me, the odds it's the ace are 1 in 52. Same with a hundred decks, a thousand, and so on. Do the odds change when we're dealing with an infinite amount of decks?
No, because that would be insane
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Old 30th July 2018, 11:36 AM   #77
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Originally Posted by Robin View Post
No there aren't.
There are certain limitations of the English language that make it impossible to declare that your answer is either "right" or "wrong". However, if you happen to be taking a math class where the answer is relevant, you will get points on the test if you go with SOdhner's answer, not yours.


If you are in a philosophy class, how you score points might be different.
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Old 30th July 2018, 11:40 AM   #78
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Originally Posted by Fudbucker View Post
Just posit a machine that handles all the marbles in its little claws and eventually picks one at random (and nevermind the fact that it would take an infinite amount of time to do it, that's not relevant). What are the odds the machine picked a blue marble?

There is no plausible way the machine could perform that function. It's like saying "just posit a magic spell that picks a marble at random (and never mind the fact that magic spells don't work)."

If it takes an infinite amount of time, then it literally never picks a marble. That is to say, there is no time you can specify, however far in the future, at which the machine has handled all the marbles and selected one. If it takes a finite amount of time, while moving at finite speeds, then it can only have selected from a finite subset of the marbles, and no matter how large that subset is, it is infinitesimally small compared with the whole set of marbles.

Even for a hypothetical impossible machine that, for instance, handles marbles at ever-increasing speeds and so handles all the marbles within some finite time interval T, there's still no way to explain how its actual "select this marble now" function would make that choice. [ETA: Actually there is, for that part; see below] We can predict that at any time t<T, the machine will not have picked a marble yet and will have so far only scanned a finite (infinitesimal, relatively) subset of the marbles. After time T it will have scanned all the marbles and supposedly picked one. But at time T is a singularity; there is no way to predict or even describe the machine's state at that time, so the assumption that its state will make sense after time T, or will follow causally from what it did during the singularity, cannot be made. (See discussions of the Thompson's Lamp paradox for more about that scenario. Forum thread here.)

There might actually be a clever way to select the marble with equal probability given infinite time or an ever-accelerating machine, involving provisionally selecting each marble scanned by swapping it for any marble previously selected. But you still have the singularity problem as with the Thompson's Lamp and other so-called supertasks.

ETA: Yeah, actually it appears to be simpler than I thought. Take the first marble. At the second marble, with probability 1/2, pick it up and drop the first marble. At the third marble, perform the swap with probability 1/3, and so forth. At the end of n marbles there's an equal probability 1/n of each of the n marbles being the currently selected one. But n can only ever be a finite subset of all the marbles, until the singularity at time T, and after the singularity there's no telling what state the machine is in. For all we can predict, it might have selected a tiger.
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Old 30th July 2018, 11:40 AM   #79
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The odds remain the same regardless of number of decks because each deck automatically repeats itself after 52 cards and the cards are always
the same in every one. And as long this consistency is maintained there is no increased or decreased probability of a particular card being picked
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Old 30th July 2018, 11:44 AM   #80
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Originally Posted by Myriad View Post
(See discussions of the Thompson's Lamp paradox for more about that scenario. Forum thread here.).
Yeah this is just the Supertask / Paradox of the Grand Hotel problem. It's not true, false, or unanswerable, it's incomplete.
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