# Living life on the edge!

A bug starts on one vertex of a dodecahedron. Call it A. Define a second vertex adjacent to the one he starts on, and call it B.

Every second he randomly walks along one edge to another vertex. What is the expected value of the number of seconds it will take for him to reach the vertex B?


Clarification: Every second he chooses randomly among the three edges available to him, including the one he might have just walked along. On his first move, he has a $$\frac13$$ probability of reaching B.

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