The following is an interview question.
There is an ante sitting on a vertex of a cube, at each time step, there is $\frac{1}{3}$ probability for the ante to move an each adjacent vertex. What is the expected time for the ante to visit all the 8 vertices?
Let $X$ be the time for the ante to visit all the $8$ vertices. Then $$E(X) = \sum_{x=1}^\infty x P(X=x).$$ However, I have difficulty calculating $P(X=x).$