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Old 30th July 2018, 11:44 AM   #81
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Originally Posted by Myriad View Post
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.
How is that germane to a math problem? Pretend magic exists and I've magically handled an infinite set of marbles. What does the math say? What are my odds of getting the blue marble? How I picked the marble shouldn't have any bearing. 2+2 =4, whether we're talking about apples, magic spells, or marbles.

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Old 30th July 2018, 11:47 AM   #82
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Originally Posted by surreptitious57 View Post
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
I agree except countable infinite sets are equal. So if there's an infinite number of decks, there's an equal number of every card, so doesn't that throw the odds calc off and if it doesn't, why not? Instead of there being less aces in a finite collection of decks then all the other cards (which allows you to calculate the odds), once you stipulate an infinity of decks, there aren't less aces anymore.

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Old 30th July 2018, 11:53 AM   #83
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Originally Posted by SOdhner View Post
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.


That is a really, really, interesting question.

There might be a "proof" in there somewhere, using Fudbucker's premise that working brains can arise spontaneously from quantum fluctuations, that the universe/multiverse cannot actually contain an infinite number of brains. But I haven't got it worked out yet.
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Old 30th July 2018, 12:03 PM   #84
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It does not matter if there are an infinite number of decks as long as every one is identical. The number
is irrelevant. It could be one. It could be infinite. It could be anything. It makes absolutely no difference
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Old 30th July 2018, 12:06 PM   #85
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Originally Posted by Fudbucker View Post
How is that germane to a math problem? Pretend magic exists and I've magically handled an infinite set of marbles. What does the math say? What are my odds of getting the blue marble? How I picked the marble shouldn't have any bearing. 2+2 =4, whether we're talking about apples, magic spells, or marbles.

Math says that certain things, such as the logarithm of zero, are undefined. The probability of getting a blue marble, when choosing equally from an infinite set of marbles of which a certain proportion are blue, is one of those things.
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Old 30th July 2018, 12:08 PM   #86
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Originally Posted by Myriad View Post
Math says that certain things, such as the logarithm of zero, are undefined. The probability of getting a blue marble, when choosing equally from an infinite set of marbles of which a certain proportion are blue, is one of those things.
The probability is undefined??? Then what do we say? Do we just shrug in these kinds of cases?
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Old 30th July 2018, 12:09 PM   #87
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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?
Well that's what I do mostly, and it seems to be working out fine so far.
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Old 30th July 2018, 12:10 PM   #88
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Originally Posted by surreptitious57 View Post
It does not matter if there are an infinite number of decks as long as every one is identical. The number
is irrelevant. It could be one. It could be infinite. It could be anything. It makes absolutely no difference
It makes a difference because in a finite amount of decks, there are less aces than other cards. You can assign a probability based on that. With an infinite amount of decks, that's no longer true.
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Old 30th July 2018, 12:11 PM   #89
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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?
You are swapping your infinite for a finite number of something so of course the same maths doesn't apply.
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Old 30th July 2018, 12:12 PM   #90
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Originally Posted by theprestige View Post
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.
Myriad explains that in post 65.
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Old 30th July 2018, 12:12 PM   #91
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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?
Ahhh... yeah. We can do that in cases were it can literally never happen.

You're asking us to predict what happens at the end of an infinite task.

We don't know because the question is incomplete. I don't know if Thompson's Lamp is on or off or which flag Achilles is holding up when he finishes the race.
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Old 30th July 2018, 12:13 PM   #92
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Originally Posted by SOdhner View Post
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.
If there's only two hotels, you flip a coin.
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Old 30th July 2018, 12:16 PM   #93
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Originally Posted by Darat View Post
You are swapping your infinite for a finite number of something so of course the same maths doesn't apply.
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?
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Old 30th July 2018, 12:24 PM   #94
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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?

The probability is undefined because the concept of a "single item selected randomly with equal distribution from a countably infinite set of discrete elements" is undefined.

It's like asking the probability of the last base-ten digit of pi being 4. It might seem reasonable that such a probability should exist, and even seem reasonable that that probability should have a particular value (0.1). But there is no last digit of pi, so there's no such probability.

A procedure to compute the last digit of pi will never terminate. Nor will the procedure to select a random integer with equal probability from the set of positive integers (even though my ingenious 1/n swap procedure is a really good start) never terminates either. Neither the "last digit of pi" nor the "ultimately selected random integer" is any more valid than the other, so there's no sense in assigning probabilities to it. What is the probability that a snark is a boojum?
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Old 30th July 2018, 12:26 PM   #95
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Originally Posted by Myriad View Post
The probability is undefined because the concept of a "single item selected randomly with equal distribution from a countably infinite set of discrete elements" is undefined.

It's like asking the probability of the last base-ten digit of pi being 4. It might seem reasonable that such a probability should exist, and even seem reasonable that that probability should have a particular value (0.1). But there is no last digit of pi, so there's no such probability.

A procedure to compute the last digit of pi will never terminate. Nor will the procedure to select a random integer with equal probability from the set of positive integers (even though my ingenious 1/n swap procedure is a really good start) never terminates either. Neither the "last digit of pi" nor the "ultimately selected random integer" is any more valid than the other, so there's no sense in assigning probabilities to it. What is the probability that a snark is a boojum?
Is the problem here that we're unable to define the probability space because we're dealing with infinities? Is that why you wanted to know how I was picking the marbles?
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Old 30th July 2018, 12:28 PM   #96
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Originally Posted by Fudbucker
It makes a difference because in a finite amount of decks there are less aces than other cards
You can assign a probability based on that. With an infinite amount of decks that is no longer true
With an infinite amount of decks the ratio between aces and all other cards remains the same however. In a standard deck of 52 cards there are
4 aces so the ratio is 4 / 52 which is the same for an infinite number of decks. Why must the ratio change simply because the decks are infinite?
Infinite sets exist just the same as finite ones do. The principle is exactly the same
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Old 30th July 2018, 12:31 PM   #97
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Originally Posted by surreptitious57 View Post
With an infinite amount of decks the ratio between aces and all other cards remains the same however. In a standard deck of 52 cards there are
4 aces so the ratio is 4 / 52 which is the same for an infinite number of decks. Why must the ratio change simply because the decks are infinite?
Infinite sets exist just the same as finite ones do. The principle is exactly the same
Because infinite countable sets are equal. The set of cards that are aces is infinite and therefore equal to the infinite set of cards that aren't aces. The ratio gets all thrown out of whack.
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Old 30th July 2018, 12:37 PM   #98
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Originally Posted by Fudbucker View Post
Because infinite countable sets are equal. The set of cards that are aces is infinite and therefore equal to the infinite set of cards that aren't aces. The ratio gets all thrown out of whack.
Why does it? It's the number of decks that's infinite, not the cards within one deck of an infinite series of them. The ratio remains the same.
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Old 30th July 2018, 12:38 PM   #99
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Originally Posted by Fudbucker View Post
If there's only two hotels, you flip a coin.
And if there were infinite hotels you would just declare that you're arbitrarily ignoring those and so you must not be in any of them.
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Old 30th July 2018, 12:40 PM   #100
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Originally Posted by SOdhner View Post
And if there were infinite hotels you would just declare that you're arbitrarily ignoring those and so you must not be in any of them.
I was unclear. I meant if there were only two types of hotels, and there were an infinite amount of each type, you would flip a coin.
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Old 30th July 2018, 12:44 PM   #101
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Originally Posted by Fudbucker View Post
Is the problem here that we're unable to define the probability space because we're dealing with infinities? Is that why you wanted to know how I was picking the marbles?

A mathematician on askamathematician.com stated the problem this way:

Originally Posted by A Mathematician
A probability mass function (which is the kind of probability distribution we need in this case) is defined to be a positive function that has a sum of values equal to 1. But any positive function that assigns an equal value to each integer must have probabilities that sum to either infinity or zero, so the desired distribution is impossible to construct.

In other words, what is the probability that our randomly selected integer will be, say, 29? If that probability is exactly zero, then the sum of the probabilities for all the integers will still be zero. But if that probability is greater than zero by even the tiniest amount, the sum of the probabilities for all the integers, all also equal to that amount, must sum to greater than 1 (in fact, to infinity). Either way, the function cannot be valid.
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Old 30th July 2018, 12:47 PM   #102
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Originally Posted by Fudbucker
infinite countable sets are equal. The set of cards that are aces is infinite and therefore
equal to the infinite set of cards that are not aces. The ratio gets all thrown out of whack
But not every infinite set is equal. The infinite set of primes for example is a smaller set than the
infinite set of integers because primes occur less frequently in the number line than integers do

And so it is with aces and all other cards. The infinite set of aces is smaller than the infinite
set of all other cards because aces occur less frequently in the deck than all other cards do
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Old 30th July 2018, 12:47 PM   #103
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Originally Posted by GlennB View Post
Why does it? It's the number of decks that's infinite, not the cards within one deck of an infinite series of them. The ratio remains the same.
Instead of being handed a card, just pretend you have an infinite amount of time to walk along a row of infinite decks of facedown cards and you finally pick one. What are the odds it's the ace of spades? Well, there's an infinite number of them, and an infinite number of non-ace-of-spades cards and those two infinities are equal, so... 50/50? But intuitively, that doesn't seem to make any sense.
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Old 30th July 2018, 12:48 PM   #104
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Originally Posted by surreptitious57 View Post
But not every infinite set is equal. The infinite set of primes for example is a smaller set than the
infinite set of integers because primes occur less frequently in the number line than integers do

And so it is with aces and all other cards. The infinite set of aces is smaller than the infinite
set of all other cards because aces occur less frequently in the deck than all other cards do
I don't think that's right. If the set is countable and infinite, it's equal to any other set that's countable and infinite.
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Old 30th July 2018, 12:51 PM   #105
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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?

The odds of you holding the ace or of holding any card are zero, because you'd still be waiting (forever) for the someone to actually hand you the card.

There's an argument between two invalid answers going on here. One says the probability is 1 in 52 because "the proportion stays the same" in the infinite set. The other is the probability is 1 in 2 because "countably infinite sets are equal."

Neither answer is valid, because the premise is invalid. There is no such thing as a card picked fairy (that is, with an equal distribution) from an infinite number of decks. It's not only impractical to do in reality, it's undefined mathematically.
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Old 30th July 2018, 12:52 PM   #106
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Originally Posted by surreptitious57 View Post
The infinite set of primes for example is a smaller set than the infinite set of integers because primes occur less frequently in the number line than integers do
That's not what all the mathematicians say. I'm going to have to trust them on this, despite it being counter-intuitive. Any countable infinity is the same "size" as any other countable infinity.

The probability question is still hard to answer in any kind of useful way, but the part about there being just as many aces as all the other cards combined is mathematically correct. Because infinity is strange.
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Old 30th July 2018, 12:53 PM   #107
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Originally Posted by surreptitious57 View Post
But not every infinite set is equal. The infinite set of primes for example is a smaller set than the
infinite set of integers because primes occur less frequently in the number line than integers do
That's actually less clear than you'd think. You could argue it either way.

1. Obviously there is a smaller number of prime numbers than whole integers because while all prime numbers are whole integers, not all whole integers are prime numbers.

2. However since an infinite set is... infinite that doesn't matter. If Bill starts listing all the prime numbers in sequence (2, 3, 5, 7, 11, 13, 17, 19, 23...) saying a prime number every second (we'll just ignore the fact that after a while the numbers would take far longer than a second to say...) and Ted starts listing all whole integers in order (1, 2, 3, 4, 5, 6, 7....) also at the rate of 1 a second at no point will Ted and Bill ever... not being saying numbers or "take a lead" over the other one in any way that can be measured because they are both going to have the same amount of numbers left to sound off, infinite. A prime number being spoken will always be said at the same time as a whole integer being spoken with no potential stopping point. 1:1 ratio confirmed.
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Old 30th July 2018, 12:54 PM   #108
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Originally Posted by Myriad View Post
The odds of you holding the ace or of holding any card are zero, because you'd still be waiting (forever) for the someone to actually hand you the card.
Right, this is why I asked the AI massage bed version of the hotel question. We're getting hung up on the "choosing" part, but for what Fudbucker wants to apply it to there isn't really a choice being made.

Of course it doesn't matter because he's wrong for other reasons, but I guess I shouldn't derail any more in this thread.
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Old 30th July 2018, 01:00 PM   #109
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Originally Posted by Myriad View Post
The odds of you holding the ace or of holding any card are zero, because you'd still be waiting (forever) for the someone to actually hand you the card.
Is it possible to evaluate the problem in terms of pure math? These conversations about physically unphysical scenarios seem like a red herring.
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Old 30th July 2018, 01:02 PM   #110
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Originally Posted by theprestige View Post
Is it possible to evaluate the problem in terms of pure math? These conversations about physically unphysical scenarios seem like a red herring.
They are. Again that's the entire mathmatic concept of a Supertask. They are interesting to think about but by definition have no possible practical answer because they can't actually happen, even mathematically one could argue.
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Old 30th July 2018, 01:04 PM   #111
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Originally Posted by theprestige View Post
Is it possible to evaluate the problem in terms of pure math? These conversations about physically unphysical scenarios seem like a red herring.
Yes, but there's actually like three or four things being discussed by now.

1. Selection of a random item from a countably infinite set
2. Whether or not countably infinite sets are all the same size
3. Whether or not it's meaningful to discuss a ratio of things within a countably infinite set
4. Physical limitations to actually doing stuff with hypothetical infinite things

So we know you don't want to talk about #4, which is fair. What's your question?
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Old 30th July 2018, 01:05 PM   #112
Myriad
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Originally Posted by Fudbucker View Post
Instead of being handed a card, just pretend you have an infinite amount of time to walk along a row of infinite decks of facedown cards and you finally pick one. What are the odds it's the ace of spades? Well, there's an infinite number of them, and an infinite number of non-ace-of-spades cards and those two infinities are equal, so... 50/50? But intuitively, that doesn't seem to make any sense.

How many different ways can you find to ask the same question?

If you use a finite amount of time (that is to say, you do "finally" pick one at some future time) then you're picking from the subset of cards you've been able to reach by that time, which has a certain proportion of aces of spades. You're not actually picking from infinite sets of cards, so the answer is 1/52 (or 1/54 with two jokers in the deck).

If you use an infinite amount of time, then there is no "finally" so there is no picked card and nothing we can say about the nonexistent picked card. (Compare: "You have an infinite amount of time, so you use it to count the integers, and when you finally get to the highest one, is it odd or even?" See, that doesn't work.)
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Old 30th July 2018, 01:08 PM   #113
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Originally Posted by Fudbucker View Post
I don't think that's right. If the set is countable and infinite, it's equal to any other set that's countable and infinite.
This is partly true. If a hotel has an infinite number of rooms then it has an infinite number of opulent rooms and an infinite number of dingy rooms. It would make no sense to talk about the probability of getting assigned an opulent room because you can't divide infinity by infinity (every possible answer is equally valid).

BUT

Any room number you are assigned to will be a finite number (you can't assign room number "infinity"). Once you have a finite number, you can compute odds to your heart's content.

Mathematically we would say that a hotel has a certain number of rooms and that number tends towards infinity.
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Old 30th July 2018, 01:09 PM   #114
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I knew I had read about this before:

Countably Infinite Sets
Our baseline level of infinity will come from our most basic infinite set: the previously mentioned natural numbers. A set that is the same size as the natural numbers ó that can be put into a one to one correspondence with the natural numbers ó is called a countably infinite set.


A surprising number of infinite sets are actually countable. At first glance, the set of integers, made up of the natural numbers, their negative number counterparts, and zero, looks like it should be bigger than the naturals. After all, for each of our natural numbers, like 2 or 10, we just added a negative number, -2 or -10. But the integers are countable ó we can find a way to assign exactly one integer to each natural number by bouncing back and forth between positive and negative numbers:

If we continue the pattern suggested above, we end up assigning exactly one integer to each natural number, with each integer assigned to a natural number, giving us the kind of one to one pairing that means the two sets are the same size.

This is a little freaky, since the natural numbers are a subset of the integers ó each natural number is also an integer. But even though the natural numbers are fully contained in the integers, the two sets actually do have the same size.

https://www.businessinsider.com/the-...finity-2013-11
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Old 30th July 2018, 01:09 PM   #115
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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?
If the card isn't selected randomly then it is unknowable, and there is no way to select it randomly.
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Old 30th July 2018, 01:10 PM   #116
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Originally Posted by JoeMorgue

A prime number being spoken will always be said at the same time as a whole integer being spoken with no potential stopping point 1:1 ratio confirmed
The infinite set of primes still has less members than the the infinite set of integers
Not all infinite sets are therefore the same size even though they are all still infinite
It is wrong from a logical perspective to treat them as if they were all the same size
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Old 30th July 2018, 01:13 PM   #117
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Originally Posted by SOdhner View Post
Right, this is why I asked the AI massage bed version of the hotel question. We're getting hung up on the "choosing" part, but for what Fudbucker wants to apply it to there isn't really a choice being made.

Yes, and that's why I said that version is extremely interesting, to the point where I haven't been able to mentally come to grips with it and offer an answer. There's a chance you and Fudbucker have together come up with a truly original paradox there.

Quote:
Of course it doesn't matter because he's wrong for other reasons, but I guess I shouldn't derail any more in this thread.

I'm still hoping Fudbucker will take notice of your version and run with it (that is, to start with, re-state his case in terms of it).
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Old 30th July 2018, 01:16 PM   #118
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Originally Posted by surreptitious57 View Post
The infinite set of primes still has less members than the the infinite set of integers
Not all infinite sets are therefore the same size even though they are all still infinite
It is wrong from a logical perspective to treat them as if they were all the same size
No but two countable infinite sets would be, under a certain point of view.

A countable set versus an uncountable set is comparable, two countable sets are not.

Again this is all... meaningless. Infinity isn't a number you plug into an equation. Both arguments are correct because... they aren't arguments.

Is Infinity minus Infinity 0 or Infinity? Yes.

If there's a lighthouse an infinite number of miles away from you and you walk an infinite number of miles towards it are you are at the lighthouse or an infinite number of miles away from it? Yes.

If I have an infinite number of rooms in my hotel and an infinite number of guest, do I have rooms in my hotel left to rent out for the night? Syntax Error.

We could do this all day.
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Old 30th July 2018, 01:16 PM   #119
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Originally Posted by Myriad View Post
How many different ways can you find to ask the same question?

If you use a finite amount of time (that is to say, you do "finally" pick one at some future time) then you're picking from the subset of cards you've been able to reach by that time, which has a certain proportion of aces of spades. You're not actually picking from infinite sets of cards, so the answer is 1/52 (or 1/54 with two jokers in the deck).

If you use an infinite amount of time, then there is no "finally" so there is no picked card and nothing we can say about the nonexistent picked card. (Compare: "You have an infinite amount of time, so you use it to count the integers, and when you finally get to the highest one, is it odd or even?" See, that doesn't work.)
I agree with ThePrestige. All this seems like a red herring. There should just be a mathematical solution aside from all this card selection and whatnot.
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Old 30th July 2018, 01:18 PM   #120
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Originally Posted by JoeMorgue View Post
They are. Again that's the entire mathmatic concept of a Supertask. They are interesting to think about but by definition have no possible practical answer because they can't actually happen, even mathematically one could argue.
Ah, there's a term for all this. Fascinating.
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