Consider the Freddy model from class, and let N = 8. Suppose Freddy observes quarters of performance by mutual-fund manager Helga. Helga may be skilled, mediocre, or unskilled. A skilled 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 an unskilled manager has a 1/4 chance of beating the market each quarter (and Freddy knows all this). In reality, the performance of a manager is independent from quarter to quarter.

(a) Suppose first that Freddy thinks Helga is mediocre. What does Freddy think is the probability that Helga beats the market in the first quarter? Suppose that she does actually beat the market in the first quarter. What does Freddy think is the probability she does it again? Suppose that she beats the market again. What does Freddy think is the probability that she will do so a third time?

(b) Assuming Freddy is correct that Helga is mediocre. How do the three probabilities in part (a) compare to their true probabilities? What phenomenon does this reflect?

(c) Now suppose that Freddy does not know whether Helga is skilled, mediocre, or unskilled. He has just observed three consecutive quarters of below-market performance by Helga. Can he conclude which type of manager Helga is? Can he rule out any type? Explain the intuition.

(d) How many more quarters of below-market performance does Freddy need to observe to be sure of Helgas type?

(e) This part asks you to derive what Freddy concludes about the proportion of skilled, mediocre, and unskilled managers in the population when he observes the performance of a large sample of mutual-fund managers over two quarters. Suppose that in reality, all managers are mediocre.

i. What proportion of managers will have two above-market performances? Two below-market performances? Mixed performances? This is what Freddy observes.

ii. Suppose Freddy thought that the proportion of skilled, mediocre, and unskilled managers in the population was , 12, and , respectively. What does Freddy expect should be the proportion of managers who show two above-market performances in a row?

iii. Given your answers to the previous two parts, what does Freddy deduce is the proportion of skilled managers in the population? What does someone who does not believe in the Law of Small Numbers deduce is the proportion of skilled managers in the population? Give an intuition for your answer.

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