We can further divide case B into B2 and B3, where B2 is the door opened by host, and B3 is the third one.
P_B2 = 0 (host will only open door with Goat), Therefore,
P_B3 = P_B - P_B2 = 2/3 - 0 = 2 / 3
P_B2 = 0 (host will only open door with Goat), Therefore,
P_B3 = P_B - P_B2 = 2/3 - 0 = 2 / 3