2. Consider the "Freddy" model of the representativeness heuristic with N = 4. Freddy starts off thinking that there is a positive probability mutual-fund manager Helga is good, a positive probability that she is mediocre, and a positive probability that she is bad. A good mutual- fund manager has a 3/4 chance of beating the market each quarter, a mediocre manager has a 1/2 chance of beating the market each quarter, and a bad manager has a 1/4 chance of beating the market each quarter (Freddy knows all this). In reality the performance of Helga across quarters is independent (but Freddy of course does not understand this). Freddy observes three quarters of performance of mutual-fund manager Helga.

(a) Suppose Helga beats the market in the first quarter. What does Freddy think is the chance that a good manager will beat the market in the next quarter? What does Freddy think is the chance that a mediocre manager will beat the market in the next quarter? What does Freddy think is the chance that a bad manager will beat the market in the next quarter? Explain how your answers are related to the gambler's fallacy.

(b) Suppose Helga beats the market in all three quarters. What does Freddy conclude? Explain intuitively.

(c) Now suppose that Freddy does not know the proportion of good, mediocre, and bad managers in the population, and is trying to infer this from observing three quarters of performance by a large number of managers. Assume that in reality all mutual- fund managers are mediocre. This question asks you to solve for what Freddy concludes from observing these completely homogenous managers.

i. In reality, what proportion of managers will have three good performances? Two/one/zero good performances? These are the proportions Freddy observes.

ii. Suppose Freddy thought that the proportion of good, mediocre, and bad managers in the population was , 1-2 and , respectively. What does Freddy expect should be the proportion of managers who show three/two/one/zero good performances?

iii. Given your answers to the previous two parts, what does Freddy deduce is the proportion of good managers in the population? Give an intuition for your answer.

(d) Explain intuitively how part (c) relates to the difficulty of explaining to a basketball fan that there is no such thing as a hot hand.