Show transcribed image text In a particular region of the Pokémon world, there are just three types of Pokémon: Dark, Psychic, and Fighting. Kecleon, the Color Change Pokémon, has the ability to change its type to blend into its surroundings. Its goal is to become the best type. It seeks to accomplish this goal by choosing a type of Pokémon to battle uniformly at random; it may choose a Pokémon of 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 Pokémon that defeated it. It continues this process indefinitely For all questions assume that Dark Pokémon always defeat Psychic Pokémon, Psy chic Pokémon always defeat Fighting Pokémon, and Fighting Pokémon always defeat Dark Pokémon. If two Pokémon 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? b) After many (approaching infinity) battles, what are the steady-state fractions of battles that Kecleon spends as each of the three types? A nearby graveyard is haunted! Ghost-type Pokemon invade the population. Assume that Ghost Pokémon always lose to Dark Pokémon, and always win against Psychic Pokémorn and Fighting Pokémon. c) Suppose that Kecleon starts as the Fighting type when the Ghost Pokémon 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?
In a particular region of the Pokémon world, there are just three types of Pokémon: Dark, Psychic, and Fighting. Kecleon, the Color Change Pokémon, has the ability to change its type to blend into its surroundings. Its goal is to become the best type. It seeks to accomplish this goal by choosing a type of Pokémon to battle uniformly at random; it may choose a Pokémon of 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 Pokémon that defeated it. It continues this process indefinitely For all questions assume that Dark Pokémon always defeat Psychic Pokémon, Psy chic Pokémon always defeat Fighting Pokémon, and Fighting Pokémon always defeat Dark Pokémon. If two Pokémon 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? b) After many (approaching infinity) battles, what are the steady-state fractions of battles that Kecleon spends as each of the three types? A nearby graveyard is haunted! Ghost-type Pokemon invade the population. Assume that Ghost Pokémon always lose to Dark Pokémon, and always win against Psychic Pokémorn and Fighting Pokémon. c) Suppose that Kecleon starts as the Fighting type when the Ghost Pokémon 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?
