Let's say I have a set composed of N elements, and a non-empty subset of that set composed of n elements. I aim to compare two scenarios: in one, I'm picking a random element of the set twice, and in the other I'm picking a random element of the set once, and then picking a random element of the subset once. I want to compare the probability of picking the same element twice in both scenarios.
Now, if I remember my elementary school math correctly, the odds of getting the same element twice in the full set can be described by $$\frac{N}{N}*\frac{1}{N}=\frac{1}{N}$$ or, textually: you're picking a random element from the universe of possible elements on the first pick, and picking the very same element on the second pick. Easy enough.
The second scenario is a bit trickier, though. You'll have to pick a random element already extant in the subset on the first pick, and then pick the very same element, now in the subset, on the second pick. To calculate those odds, then, you can go with $$\frac{n}{N}*\frac{1}{n}=\frac{1}{N}$$ where we're limiting the possibilities of the first pick, and following the same logic as the previous scenario regarding the second pick, though now applied to the subset.
As anyone can see, though, the probability of both scenarios is identical. Is that correct? Am I misunderstanding something here? This feels counterintuitive as heck!