Randomly selecting from an infinite number of choices

Suppose I simply tell you "There is some integer, i" and don't give you any more information than that. Aren't you now faced with an infinite number of possible values for i, all equally likely? I mean, it might lead to apparent mathematical contradiction, but it seems like that would definitely put you in a situation were 1) i could be infinitely many integers and 2) you don't have any reason to suspect any given integer over any other integer.

 

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In the OP it's implied that the demon can move to any room by the moment the angel looks in a room. But then if it takes the demon any time at all to move between rooms, the angel has to wait forever before looking (i.e. the angel can't look), and the story contradicts itself. If we assume instead that the demon takes no time to move between rooms, then the demon can be everywhere in the hotel at once, with a 100% probability of being in every room. In that case I think it's contradictory to say the demon can hide in a particular room. The story seems to be self-inconsistent no matter how it's interpreted.

 

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If you're going to get so caught up on the fluff backstory, you can simplify the scenario down to "If I randomly pick an integer, what are the odds that you will be able to guess my integer on the first try?"

 

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Nasor, and again the answer in theory is ZERO but in reality it is greater than that because our ability truly choose a RANDOM number from an infinite set is not possible. As was mentioned, it is the same problem that arises when you consider things like the size of the hotel the Angel is in, the time it takes to visit each door, etc. The mind experiment breaks from theory when brought into "reality".

 

Wait... can the position of an event be specified to infinite precision?

 

I tend to agree with RJBerry. To let you randomly pick any of an infinite number of integers requires that I wait forever, if it takes you any time at all to build the number (like "555" takes longer than "55"), in which case I never get to guess, and the question is self-inconsistent (unworthy of a solution). In reality it will always take some time to build the number.

There may well be an infinite number of galaxies. But if a criminal with a spaceship tries to use that fact to have the perfect hideout, it won't work perfectly because it takes time to move between galaxies, so the search space will always be finite.

 

Just a thought but surely

infinity x the infinitismal = 1

I would say the probability of choosing a door is infinitismal but certainly not zero

 

Surely not.

 

There are a lot of things one assumes to have done in math that cannot actually be "accomplished", in some sense, in the physical world.

"Take the square root of 2 - - - - "

The question of meaningfulness does not rest on the physical accomplishment, or even the theoretical possibility of the physical accomplishment.

Suppose the countably infinite set of doors were mapped into the infinity of points on the unit circle. That is a small thing, right in front of us. Now is it possible to select one of the doors "at random" ?

 

How could it ever fit ?

 

Well, you treat the car as point just like you do with bodies of mass. Choose the very tip of the bumper, for example.

 

Yep.

Infinitely small. Not zero.

 

First of all, integers are a countable infinity so probability will never be zero. Infinitely small.

Secondly, I don't understand how we are unable to choose a random number from an infinite set :shrug:

 

The problem is, you are implicitly assuming a uniform distribution. ie the probability of the angel being behind any given room is the same as any other. Of course since we have an infinite number of rooms (1,2,3,...), the probability of the angel being behind a room will be 1/k for some k. But k must approach infinity. So the probability is zero.

Assuming a different distribution, where the probability of the angel being in room n is given by \(2^{-n+1}.\) The sum of the probabilities from n=1 to n=infinity is 1, and this is a more realistic model in the sense that the angel is more likely to be in a door near the entrance to the hotel than one infinitely far away.
This is just one model, you could come up with a zillion others, the important point being a uniform distribution is the worst of all, as it effectively forces every probability to zero.

 

If everyone’s really going to get so whiney about the feasibility of actually picking a number etc, imagine two people both throw a coin so that it lands randomly on a ruler. What are the odds that the exact center of the coin will land on the same point each time? How would you describe the probability distribution for where the coin will land? It would seem that you have infinitely many possible points along the ruler where the coin might land, and all are equally likely. For every single point along the ruler where the coin might potentially land you can easily prove that the odds of it landing there are zero, yet it definitely lands on one of them...

I realize that this is more like randomly picking and guessing real numbers than integers, but my point is it seems like a “real life” example of a situation where you have infinitely many equally-likely possibilites. So do we just have to accept that something can happen even though it has a probability of zero?

 

It's a smaller infinity, so no problem.

Think of relabeling every one of those hotel doors with the multiplicative reciprocal of its room number. Every one of those door label numbers is in the interval [0 -> 1], and so the angel can make its selection of the room to visit from that interval.

 

Anyway, the answer to any such questions about choosing out of an infinite set is that the chance is approaching zero, it's not actually zero.

 

Infinity is not 'a large number' or 'approaching the very largest number', it's 'endless'.

1 over an ever increasing number approaches zero. 1 over infinity equals zero!

I know that in mathematics, we use infinitesimals and integrate them to get areas and that infitesimal areas can have different area ratios (and still have zero areas: 1x0 = πr2x0), but in these cases the conceptual infinities 'cancel out' when you integrate and you get your ratios back.

 

You can't put this into the real world - no position can ever be specified to infinite precision. The leading edge of the electron cloud of the leading atom of the car's bumper doesn't have a precise position. The position of the coins relative to the ruler is always fuzzy, and always changing.

But...

we do have the interesting mathematical exercise of selecting a random point from a line segment, an area, or any other finite space, which would (in abstract) seem to produce a value which had zero probability of occurring.

There are a couple of points of interest in this exercise for me...

Firstly, note that (as Nasor pointed out) this is selecting from a continuum, an uncountable set. Is there any way that a continuum can be uniformly mapped onto the naturals? I.e. can a random selection from a continuum be used to generate a uniform random selection from the naturals?

Secondly (least importantly) I keep getting a "so what" feeling about the idea...
You can say "choose a random point in the unit circle", but can you actually do it... even in the abstract you can't ever specify or use that point in any meaningful way... can you?


Thirdly (most interestingly) note that there are countable computable numbers. This seems to mean that a random selection from a continuum must be uncomputable.
This in turn means that every possible statement of the form:

"The number selected is [insert expression here]"​

...is false (except the tautological case where [insert expression here] is "the number selected").

So where does that leave us?
We can describe in the abstract an event which apparently had P=0, and yet subsequently occurred, but...
There is no true statement for which P=0.

:bugeye:

 

Okay, so use the coin's center of gravity - that should be an exact point, regardless of any difficulties related to deciding where the edge of the coin begins. Also, it's not clear to me what you mean by "no point can ever be specified to infinite precision." I agree that I probably can't look at the coin and actually measure where its exact center of mass is with infinite precision, but that doesn't change the fact that the point should indeed exist somewhere on the coin, regardless of my ability to measure it.