Q:

Probability distribution of choosing a real number at random

When one chooses a real number at random, what is the probability that the chosen number is between 0 and 1? Intuitively, I would guess that the probability is 0, but since there is no uniform distribution over the real line, I don't know how to express it mathematically on a more rigorous footing. Can someone help? (I'm not so strong in measure theory, so please use as elementary language as possible, please. Thank you.)

A:

You've pretty much answered your own question: the answer depends upon the specific probability distribution you are using to choose a real number at random.
The natural measure on the real numbers, namely Lebesgue measure/Haar measure, assigns infinite measure to the real numbers, and thus cannot be normalized to be a probability distribution. There is no translation-invariant measure on the real numbers that assigns a finite nonzero measure to the whole set.
So, without a natural probability measure on the real numbers, the question cannot be answered as stated.
One might say that the most natural thing to do is to consider the question on the interval $[-T,T]$ with the uniform distribution (for which the probability that the chosen number is in $[0,1]$ is $\frac1{2T}$) and then let $T$ tend to infinity; the limit of the probabilities that the chosen number is in $[0,1]$ equals $0$. But that does not correspond to a specific probability distribution on the real numbers, only a limit that people might find natural enough. Maybe that's the best we can do....

