Is Probability Meaningless in an Infinite Universe?

[windy miller]

Suppose the universe is infinitely big, then even the most improbable thing will happen somewhere in the universe, in fact it will happen an infinite number of times. So what we consider to be probable things and what we consider to be improbable things are both infinite. So how do we rule out improbable things? for example we rule the Higgs signature as being a fluke because that is so unlikely. But in an infinite universe this statistical fluke and the real single both happen an infinite number of times. So how can we say one thing is more probable than another?
of course one easy way out of this is to say the universe is not infinite and maybe it isn't. But it seems pretty ludicrous to me to demand that universe must be a certain size so that we can do probabilities. Is there another way out of this?

 

[Fightfish]

Well, here's a simple thought experiment: suppose you have infinitely many fair coins that you then toss (each coin has a equal probability of being either heads or tails). Then you randomly select one coin - what's the probability that the coin you picked is in the heads orientation?

 

[rootone]

If every coin has an equal probability of being heads or tails then surely this premise implies that any particular coin you chose to examine has a 50% chance of being a head.

 

[Hornbein]

Suppose the universe is infinitely big, then even the most improbable thing will happen somewhere in the universe, in fact it will happen an infinite number of times. So what we consider to be probable things and what we consider to be improbable things are both infinite. So how do we rule out improbable things? for example we rule the Higgs signature as being a fluke because that is so unlikely. But in an infinite universe this statistical fluke and the real single both happen an infinite number of times. So how can we say one thing is more probable than another?
of course one easy way out of this is to say the universe is not infinite and maybe it isn't. But it seems pretty ludicrous to me to demand that universe must be a certain size so that we can do probabilities. Is there another way out of this?

I see this all the time but it isn't true. The most improbable things have probability zero. But they are still possible. In an infinite universe, anything that happens finitely many times has probability zero. It is even possible for things that happen infinitely often to have probability zero.

I learned all this in statistics graduate school. It's called measure theory. I get the impression that it isn't in the standard physics curriculum.

 

[Chronos]

The trouble with infinity is it yields nonsensical results. Let us say, for example, the probability of some event we will call X occurring is zero. The probability of it occurring after an infinite number of trials is 0 x infinity. Does that mean X never happens, happens once or happens an infinite number of times? If all things are possible, then X must and must not eventually occur at the same time and in the same place.

 

[Chalnoth]

I see this all the time but it isn't true. The most improbable things have probability zero. But they are still possible. In an infinite universe, anything that happens finitely many times has probability zero. It is even possible for things that happen infinitely often to have probability zero.

I learned all this in statistics graduate school. It's called measure theory. I get the impression that it isn't in the standard physics curriculum.

The problem here is that there's no uniform way to apply probabilities to an infinite, expanding universe. There are a number of proposals for various ways of applying measure theory to an infinite expanding universe, but there's no way to select which one is the right one.

This is one big motivation for some recent finite universe models, e.g.:
https://arxiv.org/abs/0906.1047

 

[windy miller]

I see this all the time but it isn't true. The most improbable things have probability zero. But they are still possible. In an infinite universe, anything that happens finitely many times has probability zero. It is even possible for things that happen infinitely often to have probability zero.

I learned all this in statistics graduate school. It's called measure theory. I get the impression that it isn't in the standard physics curriculum.

Can you explain how something that happens infinitely often, can have probability zero?

 

[Chalnoth]

Can you explain how something that happens infinitely often, can have probability zero?

He said anything that happens a finite number of times.

 

[windy miller]

He said anything that happens a finite number of times.

HE said "It is even possible for things that happen infinitely often to have probability zero" This is a very intriguing statement I would very much appreciate it if someone could explain how.

 

[Chalnoth]

HE said "It is even possible for things that happen infinitely often to have probability zero" This is a very intriguing statement I would very much appreciate it if someone could explain how.

Well, yes, it is. The quantity $\infty / \infty$ is indefinite, and can evaluate to any value (including zero) depending upon where those infinities come from.

Basically, infinities are complicated.

 

[Hornbein]

Can you explain how something that happens infinitely often, can have probability zero?

The probability that a real number is an integer is zero, even though there are infinitely many integers.

 

[windy miller]

The probability that a real number is an integer is zero, even though there are infinitely many integers.

Forgive me if I feel you've just restated the case rather than offered a genuine explanation. I don't follow this at all I am afraid.

 

[Flatland]

Check out this interesting video by Numberphile.

 

[windy miller]

An interesting video Flatland but i don't see how it is relevant. Googolplex is a very large number but it didn't talk about calculating probabilities in an infinite space.

 

[PeroK]

The probability that a real number is an integer is zero, even though there are infinitely many integers.

This is not true in any meaningful sense. If you have a process to select a real number then either there is a finite probably of selecting an integer or there is no possibility of selecting an integer. There is no way to select a real number at random without specifying a process to do it.

The statement "select a random real number" is actually meaningless.

 

[jbriggs444]

This is not true in any meaningful sense. If you have a process to select a real number then either there is a finite probably of selecting an integer or there is no possibility of selecting an integer. There is no way to select a real number at random without specifying a process to do it.

The statement "select a random real number" is actually meaningless.

We may need to nail down the meaning of "process" here.

If one throws an ideal dart at an ideal dart board and asks whether the angle from the origin to the dart is a rational number then, with probability 1, the answer is "no", although there are an uncountable infinity of possibilities where the answer is "yes".

Admittedly, no terminating real-world measurement process can tell the difference between the two outcomes.

 

[Hornbein]

This is not true in any meaningful sense. If you have a process to select a real number then either there is a finite probably of selecting an integer or there is no possibility of selecting an integer. There is no way to select a real number at random without specifying a process to do it.

The statement "select a random real number" is actually meaningless.

OK. But you have the same problems with an infinite universe, should one restrict oneself to finite processes.