I have wondered this for quite some time, particularly 4-A below.   
Maybe  Google Answers can solve it.

1.  Is it possible to select a truly random real number within a given
interval?  There must be exactly equal chance of selecting any real
number within the given range.  If possible, provide a method.  If
impossible, prove so, or prove it is unprovable.  You can assume the
endpoints are A) real numbers, B) rational numbers, or C) integers,
whichever you prefer.

Alternate formulations:

2)Is is possible to select a truly random rational number within a
given interval, with endpoints A)real numbers, B)rational numbers, or
C)integers?

3)Is it possible to select a truly random prime number?

4)More generally, is it possible to select a truly random member of an
A)Uncountably infinite set or B)Countably infinite set?

Thanks,

Josh Perlin

Regarding question 4: "Randomness" involves probability, which
requires a probability distribution. These exist for countable and
uncontable sets. In fact there are lots and lots of them. The question
is which one are we talking about?

Naively if we think of picking one random member from a countable set,
each has the same chance of being selected. The sum of these chances
is 1. Therefore each must have probability 0 of being selected. But
then the sum is 0, not 1. What this means is that there is no
underlying probability distribution that makes sense of the question.

For an uncountable set, like the interval [a,b], one can associate a
natural distribution with the question: f(x)=1/(b-a). So now we're at
second base. But still the probability of any number being selected is
0. So one can't really randomly select a number. But intervals can be
replaced in a computer by very large finite sets of computer
representations of the numbers in the interval. Then one just has to
draw a number randomly from a finite set.

Even that is tricky, and procedures for doing this are called
"pseudo-random" since any procedure by definition cannot be random.