KöMaL 2011 B4374

Pacworm the cheese mite is sitting in the middle of a cheese cube of 5 cm edges. He eats a tunnel through the cheese. He always proceeds 1 cm parallel to an edge of the cube and then changes direction. While turning, he makes sure that he makes a 90-degree turn and that facing in the new direction he has more than 1 cm of cheese in front. He always chooses a new direction among the possibilities with equal probability. What is the probability that after travelling 5 cm he ends up in a position where there is exactly one edge at a distance of at most 0.8 cm?