In a particular region of the Pokemon world, there are just three types of Pokemon: Dark, Psychic, and Fighting. Kecleon, the Color Change Pokemon, has the ability to change its type to blend into its surroundings. Its goal is to become stronger, and it seeks to accomplish this goal by choosing a type of Pokemon to battle uniformly at random (including its own type). If it wins the battle, it will keep its old type. If it loses, it will take on the type of the Pokemon that defeated it. It continues this process indefinitely. For all questions below, assume that Dark Pokemon always win against Psychic Pokemon, Psychic Pokemon always win against Fighting Pokemon, and Fighting Pokemon always win against Dark Pokemon. If two Pokemon of the same type battle, each has an equal chance of winning.

(a) Suppose that Kecleon starts as the Fighting type. What is the probability that it will be each of the three types after two battles? I.e., what is the probability that Kecleon will be a Fighting type after two battles, a Psychic type after two battles, and a Dark type after two battles?

(b) After many (approaching infinity) battles, what are the steady-state fractions of battles that Kecleon spends as each of the three types?

(c) A nearby graveyard is haunted! Ghost-type Pokemon invade the population. Assume that Ghost Pokemon always lose to Dark Pokemon, and always win against Psychic Pokemon and Fighting Pokemon. Suppose that Kecleon starts as the Fighting type when the Ghost Pokemon arrive. What is the probability that it will be each of the four types after two battles?

(d) After many (approaching infinity) battles, what are the steady-state fractions of battles that Kecleon spends as each of the four types?