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Please listen to this song for inspiration during this homework.

Exercise 1

Suppose you are a Pokémon Trainer in the Kanto region. You are currently walking through the tall grass of Viridian Forest on your way to Viridian City. When walking through talk grass, you will encounter wild Pokémon. Pokémon encounters happen one-at-a-time and you may assume they are independent.

In Viridian Forest, there are five potential Pokémon that you may encounter. The following shows the probability that any one encounter is a particular Pokémon, as well as the Pokémon’s type.

Caterpie Metapod Weedle Kakuna Pikachu
Pokémon Caterpie Metapod Weedle Kakuna Pikachu
Probability 0.05 0.05 0.50 0.35 0.05
Type Bug Bug Bug, Poison Bug, Poison Electric

(a) Suppose you encounter 10 Pokémon. What is the probability that at least 1 is a Pikachu?

(b) Suppose you encounter 10 Pokémon. What is the probability that more than 7 are type poison?

(c) What is the probability that you see your first Pikachu after your tenth encounter?

(d) What is the probability that you see your third Pikachu on your twelfth encounter?

(e) Suppose you encounter 8 Pokémon. What is the probability that it is 2 Pikachus, 4 Weedles and 2 Kakunas?

Exercise 2

The Pokémon probabilities in Exercise 1 are from the 1998 video game Pokémon Red. Also released at the same time was Pokémon Blue. The games are extremely similar. One difference, is the proportion of certain Pokémon found in certain locations. For example, in Viridian Forest, the proportion of Metapod is:

Suppose it is 1998 and you are 12 years old. Assume Pokémon Red and Blue were both played equally. Your friend at school tells you they encountered 15 Pokémon in Viridian Forest, 3 of which were Metapod. What is the probability that your friend was playing Pokémon Red?

Exercise 3

As we have already noted, walking in tall grass in the Pokémon world can lead to encountering Pokémon. The mechanics of the encounter rate are actually somewhat complicated. For our sake, consider that you encounter Pokémon according to a Poisson process with an average rate of 1.5 per 10 steps in tall grass. (Assume you could have multiple encounters in one step.)

(a) Suppose you are walking in the tall grass. What is the probability that you encounter more than one Pokémon in ten steps?

(b) Suppose you are walking in the tall grass. What is the probability that you encounter any Pokémon in a five steps?

Exercise 4

You have been a Pokémon Trainer for some time now, and your Pokédex is filling up. Suppose you have Pokémon of the following types:

Technically Pokémon can have multiple types, but for this exercise, assume each Pokémon can be only one type.

For battling other Pokémon Trainers, a lineup consists of six Pokémon that you carry with you in Poké Balls.

(a) Suppose you select your lineup at random. What is the probability that it will contain 2 Water Pokémon?

(b) Suppose you select your lineup at random. What is the probability that it will contain at most 2 Psychic Pokémon?

Hofstadter’s Law: “It always takes longer than you expect, even when you take into account Hofstadter’s Law.”

— Douglas Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid