Randomly selecting from an infinite number of choices

A while ago a math teacher of mine told a little story about a demon that's being chased by an angel. The demon goes into a hotel with an infinite number of rooms, and randomly picks a room to hide in. The point of the story was that if the angel only gets to look behind one door, there is no possible way that he could ever find the demon. Since the odds of the demon being in the room that the angel randomly selected are 1 over infinity, the chances of the angel finding the demon on the first try are exactly zero.

I thought about this for a while, and it seems like there's a problem with the concept of the scenario. Specifically, it seems like this argument could also be used to prove that the angel couldn't look behind any door, since before the angel selects a door you could prove that the odds of any given door being selected are zero - and how could the angel ever pick a door if every potential door that might be picked has zero chance of being selected? But on the other hand, there are an infinite number of doors to choose from, so it seems like the angel should be able to pick one.

So is the concept of randomly picking one choice from an infinite set of choices even mathematically meaningful? Does the very statement of the scenario violate the rules of math by assuming that the angel can randomly pick one room to look in from the infinitely many rooms available?

 

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What if the demon goes into the first door and the angel does the same?

The thing is, it is meaningless. A hotel with infinitely many rooms is impossible anyway.

 

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I think you're confusing practicality with mathematical concepts. If we assume that the demon can pick from an infinite number of doors, then in a practical world these doors would each take up a finite space. To allow him equal probability of choosing any door at random, he needs to be given infinite time to choose his door. However, this isn't very practical due to the fact that the angle is chasing him and could possibility use some logic to deduce where he is most likely to be.

 

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Is it meaningful mathematically?
Is there any meaning in a random variable with a flat probability density function over all the naturals?

Isn't the probability that such a variable is less than any finite number exactly zero?

 

I don't know. Perhaps you could enlighten me?

 

Nasor: your paradox arises because the story applies physicality to the abstract math concept of infinity. Infinity does not exist in reality, it is simply a useful tool in mathematics. Think of it like this: what are the odds of you and I picking the same truly random rational number between 0 and 1? The mathematical answer is ZERO, however in reality the answer is non-zero because any method that we employ (using a computer, picking one out of thin air, etc) necessarily reduces the target set to a finite number.

 

No, I can't. That's why I'm asking. Specifically, I'm reiterating part of your original question in response to Absane.

 

Sorry, I thought it was rhetorical.

 

In probability theory, there is no way to have a countably infinite set of possibilities where the probabilities are all the same. As it has been noted, that to be all the same they have to be all 0, which when added up contradicts the requirement that the total probability be 1.

 

Could you elaborate a bit more, please? What is "it?"

And if you're dealing with a probability density function, then the random variable must be an interval in the reals. So, you cannot talk about just the naturals.



The third postulate of probabilities is:

If \(A_1, A_2, A_3, ..., \)is a finite or infinite sequence of mutually exclusive events of sample space S, then
\(P(A_1 \cup A_2 \cup A_3 \cup ...) = P(A_1) + P(A_2) + P(A_3) + ...\).

If we consider the finite sequence \(A_1 = 1. A_2 = 2, ..., A_n = n\), then \(P(A_k) = \frac{1}{k}\) for \(1 \leq k \leq n\).

As \(n \rightarrow \infty\), \(P(A_k) \rightarrow 0\).

So in dealing with a sample with and infinite number of members the calculated probability is, indeed, zero.

This does not imply that event \(A_k\) cannot happen. I honestly do not know how to explain this. Some handwaving could suffice but I won't do that.

 

As others have noted, the idea of random selection amongst a (countably) infinite number of choices is ill-posed, at least in terms of classical probability. This same issue underlies the two envelopes problem, btw.

But just saying that classical probability doesn't handle this case is not very satisfying. We'd like to know why not, since the idea of randomly selecting from an infinite number of choices seems intuitively okay, at least at first glance. But perhaps it is not so simple. How would you go about choosing a door at random?

Nevertheless, improper distributions like this are used in statistics, specifically in forming uninformative prior in the context of Bayesian estimation.

 

Didn't you already post a thread on this?

 

Hi Absane,

"It" was supposed to be clarified in the second sentence, but I screwed up the terminology.

What I meant to ask was whether the idea of a uniform distribution over the integers is meaningful... I notice that in the other thread, [post=2118139]DH calls this a "nonsense concept"[/post].

Right with you.

Surely you jest?
P=0 doesn't just imply "cannot happen"... isn't that precisely what P=0 means?


I suggest that if some event happens, then its probability was not zero. But, perhaps I'm wrong... can you think of a counterexample?

 

Now that I fully understand what you meant, DH is correct.

While I won't construct the actual equations (because I am comfortable in my underwear right now) I will provide a counterexample. On a stretch of road 1 meter long you know there was a car accident. You discover that it crashed at exactly 91.1212... centimeters. What were the odds that it could have crashed elsewhere on the meter stick? Calculated probability is 1 (or 100%) according to integration over the density function from \([0, 91.1212...) \cup (91.1212..., 100]\). Surely, this is in contradiction.

 

i don't care how anyone looks at it. It is still statistically speaking 1 in an infinite amount.

 

Looks like infinity and probability don't go very well together

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I think the best way of explaining this one is tha there are 2 types of infinity. Countable and uncountable (http://www.sciforums.com/encyclopedia/Countable_infinity

As we are dealing with a countable infinity (i.e. number of doors) the probability would not be zero but infinitely small.

 

Does it matter?

Consider the set of integers, which is a countable infinity. Pick on. What is the probability that you picked Avagadro's number?

 

No, although some of the things that were said in that thread prompted me to post this one. In that thread I was asking about how to mathematically describe the odds of selecting a certain element from an infinite set given an infinite number of guesses, and whether the odds of a specific element being picked trended toward zero or 1 as the number of guesses went to infinity (as well as some other things). Someone in that thread said that the very concept of being able to randomly pick from an infinite set was flawed, and it reminded me of the whole angel/demon scenario. Since it wasn't directly related to the topic of that thread I decided to start a new one, rather than resurrecting a thread that was over a month old.

 

That would be me. Specifically, the concept of uniform distribution over the integers is a flawed concept. OTOH, suppose the demon runs by each door in sequence, each time flipping a coin to determine whether this is the room he enters. This has a well-defined mass distribution function. The angel should either choose the first door (mode) or the second (mean).

That's mighty decent of you. Some of the members here have no qualms in necromancing a five year old thread, let alone a one month old thread.