I'll give the general solution for a nxnxn cube.
When painted 6n^2 unit squares will be painted. There are 6n^3 total sides. The chosing and rolling assures that each of these total sides has the same probability of being chosen.
(6n^2)/(6n^3)=1/n.
So the probability when n=7 is 1/7

Another very slick way of looking at this.
Consider the top face of the big cube. Each little cube has paint on this side, but there are n1 cubes below it that don't. The same holds in each direction so the answer is 1/n.
