2. Consider the "Freddy" model discussed in class where Freddy is someone who believes in the Law of Small Numbers. Assume that N = 9 (meaning that Freddy incorrectly thinks that performance is modelled as draws without replacement from an urn with 9 balls when in reality the balls are drawn with replacement). Freddy observes two free throws each by a large number of basketball players. There are four possible types of players. An excellent player is sure to make every free throw; an above-average player has a 2/3 chance of making each free throw; a below-average player has a 1/3 chance of making each free throw, and a terrible player is sure to miss every free throw, and Freddy knows all this. The proportion of excellent, above-average, below-average, and terrible players in the population is q. q. q, and -q, where q> 0. Freddy also knows this, but he does not know q- he is trying to infer the proportion of each type of player from his observations. a) (5 pts) In reality, what is the probability that an above-average player makes both of her free throws? What does Freddy think is the probability? b) (5 pts) Explain in one or two sentences the phenomenon that is reflected by the comparison of these two numbers. c) (10 pts) In reality, as a function of q, what proportion of basketball players will make both free throws? d) (10 pts) What proportion of basketball players does Freddy think should make both free throws if he thinks the proportion of excellent, above-average, below-average, and terrible players in the population is -,, q, and 9, respectively? e) (10 pts) Given your answers to the previous two parts, what does Freddy deduce is the proportion of excellent players in the population, i.e. -, as a function of q? How does Freddy's deduction compare to reality? f) (5 pts) Explain in one or two sentences the phenomenon reflected in the answer to part (e).