[Request] Spoon and fork probability problem

I encountered a post online about the best strategy to always pick up exactly 1 spoon and 1 fork from a container with mixed pieces.

If there are equal numbers of spoons and forks, what is the probability of getting exactly 1 of each when you pick up 2?

Intuitively, you'd think the chance is 50%. If you lost the combinations, you can only get the following (SS, FS, FF, SF) where 2/4 is correct. However, the actual chance is always higher than 50% because it doesn't matter what the first one you pick up is. It will determine what you need to pick up on the 2nd try. 1 pair = 100%, 2 pairs = 2/3, 3 pairs = 3/5, 4 pairs = 4/7.

So if you wanted to prove this, how would you go about it?