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30th July 2018, 11:44 AM  #81 
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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.

30th July 2018, 11:47 AM  #82 
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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.

30th July 2018, 11:53 AM  #83 
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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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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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30th July 2018, 12:06 PM  #85 
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30th July 2018, 12:08 PM  #86 
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30th July 2018, 12:09 PM  #87 
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30th July 2018, 12:10 PM  #88 
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30th July 2018, 12:11 PM  #89 
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30th July 2018, 12:12 PM  #90 
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30th July 2018, 12:12 PM  #91 
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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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30th July 2018, 12:13 PM  #92 
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30th July 2018, 12:16 PM  #93 
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30th July 2018, 12:24 PM  #94 
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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 baseten 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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30th July 2018, 12:26 PM  #95 
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30th July 2018, 12:28 PM  #96 
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Originally Posted by Fudbucker
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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30th July 2018, 12:31 PM  #97 
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30th July 2018, 12:37 PM  #98 
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30th July 2018, 12:38 PM  #99 
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30th July 2018, 12:40 PM  #100 
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30th July 2018, 12:44 PM  #101 
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A mathematician on askamathematician.com stated the problem this way:
Originally Posted by A Mathematician
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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30th July 2018, 12:47 PM  #102 
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Originally Posted by Fudbucker
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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30th July 2018, 12:47 PM  #103 
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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 nonaceofspades cards and those two infinities are equal, so... 50/50? But intuitively, that doesn't seem to make any sense.

30th July 2018, 12:48 PM  #104 
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30th July 2018, 12:51 PM  #105 
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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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30th July 2018, 12:52 PM  #106 
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That's not what all the mathematicians say. I'm going to have to trust them on this, despite it being counterintuitive. 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. 
30th July 2018, 12:53 PM  #107 
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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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30th July 2018, 12:54 PM  #108 
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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. 
30th July 2018, 01:00 PM  #109 
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30th July 2018, 01:02 PM  #110 
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30th July 2018, 01:04 PM  #111 
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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? 
30th July 2018, 01:05 PM  #112 
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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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30th July 2018, 01:08 PM  #113 
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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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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 — that can be put into a one to one correspondence with the natural numbers — 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 — 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 — 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...finity201311 
30th July 2018, 01:09 PM  #115 
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30th July 2018, 01:10 PM  #116 
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Originally Posted by JoeMorgue
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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30th July 2018, 01:13 PM  #117 
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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:
I'm still hoping Fudbucker will take notice of your version and run with it (that is, to start with, restate his case in terms of it). 
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30th July 2018, 01:16 PM  #118 
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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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30th July 2018, 01:16 PM  #119 
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30th July 2018, 01:18 PM  #120 
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