Random "Icosahedron" Walk

Discrete Mathematics Level 4

A bug starts on one vertex of an icosahedron. 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 opposite to the original vertex he was on?

Clarification: Every second he chooses randomly between the five edges available to him, including the one he might have just walked along.

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