Choosing a number from infinite numbers

Question

Imagine infinite set of, let's say, natural numbers. I choose one of the infinite numbers randomly. Let's call that number n. If I choose another number too, can it be the same number (n), theoretically?

Answer 1

As mentioned in the comments, there is no way that each natural can have the same probability. See this answer to "Can you pick a random natural number?..., for instance.

However, we can still pick a number randomly with unequal probabilities. For instance, suppose I choose a number by flipping a coin, and the number is the number of heads before the first tail. (If you want to include a case of flipping heads forever, let's say that also counts as the number $0$.)

I just got the following sequences of flips: $HHT,T,HT,T$. Which means that the number $0$ was randomly selected twice.

Answer 2

I'd loved to put this in a comment, but I can't because I am new here.

I just wanted to say that since you picked that particular number the first time, it has a non-zero probability of being picked. Thus it definitely can be picked again.