How are the Real Numbers distributed?

[HyperbolicMan]

Question: What is the probability that a random variable X with domain all real numbers will take a value in the closed interval [a,b]?

It seems to me that in order to answer this question you have to know how the real numbers are distributed. Given the appropriate distribution function, you can integrate it from a to b to find the probability.

Common sense says that the real numbers should have a constant distribution (e.g. P(x)=c for all x). However, the integral of any constant function from -\infty to \infty is not 1.

So how exactly are the real numbers distributed?

 

[transphenomen]

I am pretty sure that for any finite numbers a and b, the chance that x is in [a,b] is zero.

For any finite number a, the chance that x is in [a,infinity) is 50%. The same with (-infinity,a]

 

[rasmhop]

So how exactly are the real numbers distributed?

They are distributed how you want them to be. A set does not have an intrinsic distribution. For instance you may define a probability density function p(t) by p(t) = 1/2 for t in [1,3] and p(t)=0 otherwise.

If what you want is a uniform distribution, then no such thing can exist on the real numbers for the reason you mentioned (that if p(x) =c for all x, then \int_{-\infty}^\infty p(x) dx is 0 or \infty depending on whether c =0 or c>0, but never 1).