incorrect answer deleted.
sim shows 8/15 is correct.
Ok, it's not that big a deal.
You choose the first sock with probability 1 and the second unmatched sock with probability 4/5
For clarity let's say you chose (r,w) as the first pair. We are left with (r,w,b,b)
For the third sock you can choose, either r or w, and be forced to choose b as the 4th, or
You can choose b, and then be forced to choose r or w as the 4th. This is
2/4 x 2/3 + 2/4 x 2/3 = 2/3
I.e. there is a 2/3 probability you will choose a non-matching pair for the 3rd and 4th socks.
P[non-matching pair] = 4/5 x 2/3 = 8/15