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Probability in an Infinite Set


EquisDeXD

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So I know there's a specific way to derive the probability of something in an infinite set with some kind of arbitrary label of 0. Normally it would seem like the chances of picking any element in an infinite set would be 1/infinity, but you can't divide something by infinity, so how does probability and permutation get around that? I mean there's infinite area of matter to occupy, but if it had 0 chance of existing in a particular location, how could all this be here? Or like with the domain of a function that extends indefinitely but that models probability as y values being more likely at certain x values, like a probability bell curve, how is this modeled? What else could you say besides "1/infinity"? How isn't it 1/infinity? Is it 3/infinity in places of higher probability? That doesn't seem to make sense.

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  • 2 weeks later...

It only makes sense to talk about ranges in an infinite set.

 

I'll use the same example from the other thread.

 

Consider a uniform distribution from 0 to 1.

 

That means [math] f(x) = 1 [/math] for 0 < x < 1. And we'll agree that there are infinite number of numbers between 0 and 1.

 

The definition of the probability of a single sample X located between a and b is

 

[math]P(A < x < B) = \int_A^B f(x) \, \mathrm{d} x[/math]

 

And, you can see if A = B... that integral is 0.

 

So, the probability of choosing any individual element in the infinite set is 0. You only get non-zero answers when talking about ranges withing the infinite set.

 

Now, the above definition is true for any f(x). And there are functions that have more weight on places that in others, such as the above mentioned bell curve a.k.a. the normal or Gaussian distribution. Again, you have to talk about ranges. If you take a range with a fixed width about the mean of the normal distribution, the above integral will evaluate to a higher number than if take that same fixed width on any range on the curve that doesn't include the mean. Again, any single number that is sampled will have probability zero because that integral evaluates to zero. But the probability density curve, f(x), can take on most any form, within a few restrictions. Namely, not-negative, and the integral over the entire range should evaluate to exactly 1.

Edited by Bignose
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It only makes sense to talk about ranges in an infinite set.

 

I'll use the same example from the other thread.

 

Consider a uniform distribution from 0 to 1.

 

That means [math] f(x) = 1 [/math] for 0 < x < 1. And we'll agree that there are infinite number of numbers between 0 and 1.

 

The definition of the probability of a single sample X located between a and b is

 

[math]P(A < x < B) = \int_A^B f(x) \, \mathrm{d} x[/math]

 

And, you can see if A = B... that integral is 0.

 

So, the probability of choosing any individual element in the infinite set is 0. You only get non-zero answers when talking about ranges withing the infinite set.

 

Now, the above definition is true for any f(x). And there are functions that have more weight on places that in others, such as the above mentioned bell curve a.k.a. the normal or Gaussian distribution. Again, you have to talk about ranges. If you take a range with a fixed width about the mean of the normal distribution, the above integral will evaluate to a higher number than if take that same fixed width on any range on the curve that doesn't include the mean. Again, any single number that is sampled will have probability zero because that integral evaluates to zero. But the probability density curve, f(x), can take on most any form, within a few restrictions. Namely, not-negative, and the integral over the entire range should evaluate to exactly 1.

 

I feel like your talking about calculus more than measure theory, because in measure theory you normally have some kind of cardinal infinity as the magnitude of a of an infinite set.

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you do realize that measure theory and integrals are intimately related, yes? Most people are far more familiar with integrals, so I think it is generally more useful to try to explain these concepts using that more familiar framework.

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  • 4 weeks later...

The previous answer provides a good basic explanation of how probability works over infinite sets. Indeed, the purpose of measure theory is primarily to properly understand integrals and probability.

 

In basic measure theory, the relations between infinity and other numbers are specifically defined. infinity*0=0. So an integral over a set of measure zero is always zero, even if you are integrating a function which is infinite on that set, and an integral of a zero function is always zero, even if you integrate it over a set of measure infinity.

 

This definition ONLY applies to measure theory (and specifically to the definition of the integral)! You cannot use this identity in any other context (for example, you cannot say that lim_{n->infinity}(n/n)=0 just because lim{n->infinity}(n)=infinity and lim{n->infinity}(1/n)=0).

 

Note that measure theory kind of gets rid of ratios in probability where they don't make sense. Everything is expressed in terms of products, sums, and limits. So you don't really have to worry about dividing by infinity. In general, it's probably always a safe bet to say that 1/infinity is zero though. infinity/infinity or 0/0 is another matter. 0/0 can appear when conditioning on an event of probability zero; the way this is handled in infinite sets is with conditional probability density functions.

Edited by Zorgoth
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