1. [3] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started?
$(0,0,0)$. If after 3 moves he is at $(1,1,1)$ he cannot get back to $(0,0,0)$. From any other vertex he has a $\frac{1}{3}$ chance of getting back on the final move. There is a $\frac{2}{9}$ chance he ends up at $(1,1,1)$, and thus a $\frac{7}{9}$ chance he does not end up there, and thus a $\frac{7}{27}$ chance he ends up at $(0,0,0)$.