KöMaL 2013 B4560

The network of roads in the city of Icosapolis corresponds to the edge graph of an icosahedron. The home of Iorgos is at one vertex of the icosahedron, while his favourite theater is situated at the opposite vertex. On his way home from the theatre after dark, he stops at every vertex he reaches, and after some hesitation decides which way to proceed. Assume that at any vertex the probability of meeting someone who will show him one possible direction to get him home along the least possible number of edges is $p$. Otherwise he will chose an edge at random. (He may as well turn back in the direction he came from.) For what value of $p$ is there a 50% chance that he will get home before getting back to the theatre?