Are real numbers generated uniformly at random guaranteed to be unique? Suppose I can generate numbers uniformly at random from an infinite set, such as:
$$r \in \mathbb{R} :  0 < r < 1$$
Each number has an infinitely small probability of being generated. Does that mean any two randomly generated $r$ are guaranteed to be unique?

I think this is a weaker version of the same question: Suppose an algorithm generates two bits uniformly at random. If the two bits are different, it starts over. Otherwise, it halts.
Is the algorithm guaranteed to halt?
(Assuming the universe can provide infinite entropy/has an infinite lifespan.)

If each generation is an independent event (and in this case I think we can assume it is), then no.
The outcome of one generation does not affect the outcome of any other generation (independence), so the outcome of generating a particular number $r$ has the same non-zero probability each time you generate a random number.
