I have produced a new perfect playpen for my pig Percy.

The playpen is a 3D cube. There are tunnels along each edge and a different entertaining pig toy/mud bath/swill trough at each vertex. Percy's pretty smart, so he can get up and down the vertical edges no problem! So starting at any vertex, he can get to any other vertex by going along edges.

Percy is so happy with his new pigpen he starts running around it randomly. After arriving at a vertex, he then moves along one of the three adjacent edges at random (so he might end up visiting somewhere he's been before: he doesn't mind).

Find the probability that after he has moved along seven edges, he has visited every vertex at least once and is at the vertex diagonally opposite the one he started at. (That is, diagonally opposite across a diagonal of the cube, not a face!) 

What about the probability that after he has moved along seven edges, he has visited every vertex at least once (but now I don't mind where he finishes)?

 

Extension

What is the probability that after he has moved along seven edges, he finishes in the corner diagonally opposite his starting corner?