Jubal

The greatest number of socks to get a matching pair is 5. If your first pull is white, your second blue, your third grey, and your fourth black, then the fifth sock you pull has to be one of those same colors, making a pair.

There are choose(10,2) = 45 ways to choose two things from ten if order doesn't matter. There are choose(7,2) = 21 ways to choose two blue socks. So 21/45 = 7/15 (This is the same answer as Bob's... do you see why?)