Question

The Chicago Art Museum's Renaissance display consists of four hallways bordered around a square courtyard. A single guard is assigned to patrol the four hallways. When the guard starts working, he begins in one of the corners and walks clockwise. When he arrives at a subsequent corner he flips two coins. If both coins are heads, he changes the direction he is walking. Otherwise, he continues in the same direction. Let E be the expected number of lengths of hallway that he walks before he first returns to his starting corner. Let p be the probability that he walks strictly more than E lengths of hallway before returning to his starting corner. p can be expressed as ab where a and b are coprime positive integers. What is the value of a+b?

The guard may walk in the same hallway more than one time. Each time he walks in it counts as one length of hallway.