Suppose Freddy observes two free throws each by a large number of basketball players. There are four possible types of players Freddy imagines can exist. An excellent player is sure to make every free throw; an above-average player has a two-thirds chance of making each free throw; a below-average player has a one-third chance of making each free throw, and a terrible player is sure to miss every free throw. In reality, exactly half the players are above-average and half are below-average, and none are excellent or terrible. Freddy correctly knows that the proportion of above-average players equals the proportion of below-average players, and that the proportion of excellent players equals the proportion of terrible players. But he does not know how many excellent and terrible players there are, and is trying to deduce this from his observations.

In reality, a basketball player's two free throws are independent from each other. Hence, a player's two free throws can be modelled correctly as draws with replacement from an urn with 3 balls, where the appropriate number of balls (3, 2, 1, and 0 for excellent, above-average, below-average, and terrible players, respectively) correspond to successful free throws, and the rest correspond to missed free throws. Freddy, however, incorrectly thinks that the balls are being drawn without replacement. (Note that using the notation from class that means N = 3.)

(a) (6 points) Briefly but carefully describe the heuristic that this model is trying to capture and how the model capture it.

(b) (6 points) In reality, what is the probability that an above-average player makes both of her free throws? What does Freddy think is the probability? Explain in one or two sentences the phenomenon that the comparison of these two numbers reflects.

(c) (4 points) In reality, what proportion of basketball players will make both free throws?

(d) (7 points) What proportion of basketball players does Freddy think should make both free throws if the proportion of excellent, above average, below-average, and terrible players in the population is 1/2 q, q, q, and 1/2 q, respectively? Your final answer should be an expression of the form ax + b for some real numbers a and b. (e) (8 points) Based on your answers to the previous two parts, what does Freddy deduce is the proportion of excellent players? How is this related to the truth? Explain the intuition

(f) (6 points) Explain briefly how this question is related to the hot hand fallacy in basketball.