Provide a uniformly random number chosen from $[0,1]$

Question

This is a soft-question and a riddle.

I am just wondering, that the phrase "pick a random number uniformly from [0,1]" is used very often in education, but in reality it can not be achieved. I mean, you can provide any number, but justifying that it was sampled uniformly seems impossible to me. Is it?

Provide (preferably explicitly) a number $x$ uniformly chosen from $[0,1]$ and justify how it was sampled uniformly.

...or justify this can not be done.

Answer 1

The phrase

pick a random number uniformly from $[0,1]$

is just a way to describe the standard uniform continuous random variable. That is, one which has a probability distribution which is known as continuous uniform distribution. It is precisely defined.

Actually producing a random number is a completely different task. As someone wrote

Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.

Not only that, but using physical sources of randomness using nondeterministic means is doomed by the requirement to "prove" that it produces a uniform random real number because, it would require infinite precision which is not attainable by measurements.

Long story short. Picking a random real number uniformly is just a manner of speaking. No actual picking is done.