# Is selecting a random person from an infinite population of people an invalid premise to begin with? [closed]

This was initially sparked by a hypothetical question:

There are two scenarios. In the first, an infinite number of people are living in a completely blissful paradise, but every day a person is selected randomly and sent to eternal torturous hell with no chance of ever returning. In the second, an infinite number of people are living in hell to begin with, but every day a person is selected randomly and sent to eternal paradise with no chance of ever returning.

My friend is arguing that they would pick heaven since the chance of ever getting selected (or anyone ever getting selected, for that matter) is zero, therefore they will remain in heaven forever. They argue that randomly choosing one thing from an infinite set is a mathematical impossibility.

However, this contradicts the premise of the question. Surely someone must be chosen at the end of each day, right? Is the premise of the question invalid?

If the question is indeed valid, is there any way to prove that any given person among the infinite population in paradise has a finite amount of time before they are sent over to hell?

• – pie
Commented Jul 13 at 2:20
• Are you basically asking "Can one choose (take, select, touch, point at) an object in an infinite set of objects" ... ? Is that what you're asking? Commented Jul 13 at 15:27
• Interesting question, but more realisticly? I'm choosing Hell first because I'm betting that there will only ever be a finite number of human beings in the whole history of the universe, and so whichever place I choose first, I'll only spend a finite number of days there before I spend eternity in the other place. Commented Jul 13 at 16:38
• Randomly choosing one thing from a countable set with uniform probability is mathematically impossible. A non-uniform probability that is non-zero for everyone in the population is perfectly possible. Do you want to consider what happens if we suppose a non-uniform distribution? Commented Jul 13 at 17:38
• Commented Jul 14 at 0:23

This is a result that is fundamental enough to deserve its own name but I don't know that anyone has bothered to give it one: there is no uniform probability distribution on a countably infinite set.

That is, we're used to this fundamental fact about finite sets that there is a unique probability distribution which is as "fair" or "unbiased" as possible: the uniform distribution, where for a set, say $$\{ 1, 2, \dots n \}$$ of size $$n$$, we pick each element with equal probabilities $$p_i = \frac{1}{n}$$. There is no way to do this on a countably infinite set at all! And it's easy to see why: you have to somehow assign the same probability $$p$$ to every member of the countably infinite set, say $$\{ 1, 2, \dots \}$$, and it can't be zero (because then the probabilities would sum to $$0$$) and it also can't be positive (because then the probabilities would sum to $$\infty$$).

So, there's no such thing as choosing a random element of a countably infinite set uniformly at random. For example there's no such thing as choosing a random integer, or random natural number, etc. This means that every probability distribution on a countably infinite set is necessarily "unfair" or "biased": some elements must be more probable than others, in order for the probabilities to all sum to $$1$$.

A simple example of such a probability distribution on the natural numbers $$\{ 1, 2, \dots \}$$ is the geometric distribution $$p_i = \frac{1}{2^i}$$; in your hypothetical you could imagine countably infinitely many people in heaven ranked by how sinful they are, and every day the $$i^{th}$$ most sinful person is sent to hell with probability $$\frac{1}{2^i}$$. In this case, yes, you can prove that with probability $$1$$ every individual person ends up in hell after finitely many days. You could replace the $$p_i$$ with any positive real numbers such that $$\sum p_i = 1$$ and that would still be true; the important thing is that with this setup the probability that you go to hell each day is 1) nonzero and 2) doesn't decrease.

This setup makes the comparison between the heaven-to-hell and hell-to-heaven scenarios interesting because which one seems better than the other appears to differ dramatically depending on whether you look at it from the perspective of any individual person or from the perspective of the whole group. In the heaven-to-hell scenario, any individual person only spends a finite number of days in heaven but then spends the rest of eternity in hell. But 100% of the whole group stays in heaven the whole time and only 0% of the group is ever in hell on any given day! And the hell-to-heaven scenario is reversed. It's quite dramatic. Sounds like a Borges short story.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Mathematics Meta, or in Mathematics Chat. Comments continuing discussion may be removed. Commented Jul 13 at 18:29
• Having chosen a probability measure on the group of people, it's a bit strange to use a different meaning of 100%. And if you use the same probability measure, I'm pretty sure that the group staying in heaven converges to 0% with probability $1$. Commented Jul 14 at 2:48
• @ronno: I am not fixing a probability measure on the set of people. People get removed every day so the set of people isn't fixed over time, and the probability assignments aren't fixed over time either. That "100%" is meant in the sense of natural density (en.wikipedia.org/wiki/Natural_density) but if you're unhappy with that replace it with "all but finitely members of the group are in heaven on any given day." Commented Jul 14 at 2:51
• Why put the last paragraph in a "spoiler" box? I think it's a very interesting fact about this setup. Commented Jul 14 at 2:56
• @David: I've started using spoiler text for comments which I think of as "asides." I don't mean I don't want people to read them - there's a reason they're so easy to click, after all - just that I don't consider them as "primary" as the rest of the text in the answer. Commented Jul 14 at 7:11

Forever, there are some persons in the heaven,

though each specific person cannot remain there forever.

The reason is that persons are sent out daily.

If the time pattern changes, the following can be possible:

At a certain time, no one remains in the heaven.

For example, if persons are sent out at time instants $$t_0\left ( 1-\left ( \frac12 \right )^i \right ), i\in \mathbb N$$, then there will be no one in the heaven at time $$t_0$$.

Surely someone must be chosen at the end of each day, right?

And surely someone must win the lottery, right? But the chance that it will be you (or any particular person we select) is very small.

As others have said, we can't really have a uniform probability distribution over an infinite set. When reasoning about infinite numbers, a common solution is to use limits.

First, imagine that there are a thousand people in Paradise (or Hell). Your chance of being selected on day 1 is 1/1,000, 1/999 on day 2, and so on. You'll last at most 1,000 days.

Then, suppose we increase it to a million people. Now your chance of being selected on day 1 is much lower, 1/1,000,000, and you'll last at most a million days.

As we keep increasing the number of people, your chance of being selected on any particular day keeps getting smaller, and the worst-case length of time you expect to last increases. As the number of people approaches infinity, the chance of being selected approaches 0, and the expected length of stay approaches an infinite time.

• we can't really have a probability distribution over an infinite set > Yes, we can. The Lebesgue measure on $[0,1]$ is a probability density over $[0,1]$ which is infinite. Similarly, $\mathbb{P}(n) = \frac{1}{2^{n}}$ defines a probability distribution over $\Bbb N = \{1,2,\ldots\}$. Paraphrasing @DanielR.Collins : Example of randomly picking an item from the infinite set of natural numbers: Flip a coin and if heads take 1. Else flip a coin and if heads take 2. Etc. Every natural number $n$ has positive probability $1/2^n$ of being picked. Commented Jul 14 at 12:37
• I changed it to say "uniform probability distribution". And in the question's hypothetical situation, your suggested method can only be applied if we assign all the denizens an index. Commented Jul 14 at 13:57
• Ok, then I agree with the non existence of a uniform probability distribution, provided the infinite set is countable. The Lebesgue measure on $[0,1]$ is considered uniform, for an interval of length $\ell \in [0,1]$ has probability $\ell$. Commented Jul 14 at 14:00

If the question is indeed valid

I'm pretty sure the question is simply not valid.

There is physically no such thing as an infinite number of discrete objects all in existence at the same moment.

An "infinite set" is purely a philosophical construction.

It is writing in a book somewhere, the book being titled "On infinite sets ..."

The notion

choose (take, select, touch, point at) an object in an "infinite set" of objects

simply depends on the epistemology, stated in the book, by whoever wrote the book in question on infinite sets, or, whichever book you choose to discuss on the topic if there is more than one.

The notion

choose (take, select, touch, point at) an object in an "infinite set" of objects

is completely abstract, and the concept depends only on the totally abstract definition of "choosing" within the totally abstract definition of "infinite set" of whichever definition you're using.

As user @didier points out, if you are asking a pure mathematical question (using an analogy about people, hotels, hell, etc) then you are very simply asking is there a uniform probability distribution on the set of integers to which the answer is well-known in mathematics, given the common and well-established concepts in math and math language.

• This is a mathematical question. It doesn't matter if things are unphysical and made up. Commented Jul 14 at 10:38
• @paulina But this question is not even mathematically valid. We cannot choose an object out of an infinite set such that every object has the same probability to be chosen. Commented Jul 23 at 22:02