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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 -[tex]\infty[/tex] to [tex]\infty[/tex] is not 1.

So how exactly are the real numbers distributed?