Apologies if this is a duplicate. I searched and didn't find anything quite like it.
Suppose I have a drawer with an equal number of N black socks and N white socks. They're all mixed up. So, my chances of picking a matching pair in the first two selections is (N-1)/(2N-1), right? Well, what if, before I pick the first sock, I randomly (so I don't know the colors of the socks I'm moving) partition the drawer so that there are N socks on each side, and I draw one sock from each side. Do the chances of drawing a matching pair change?
On the one hand, we can see that selection from one side doesn't change the composition of socks on the other side of the partition. However, whichever color I choose from the "first" side, it's likely that there are more of that color on that side. On other words, if I draw a black sock from one side, it's more likely that that side had N-1 blacks and 1 white than it is that that side had 1 black and N-1 whites.
My suspicion is that I need to do some kind of hypothesis testing, where I consider the chances of every possible partitioning, but that's way above my skill level.