4. (Lehr Exercise 17.9) Promoting Beliefs with Law of Small Numbers. A manager has hired a new investment analyst. The manager accurately knows that 20% of analysts are high ability and 80% are low ability. High-ability analysts make investment decisions that yield a positive quarterly return 60% of the time, whereas low-ability analysts make investment decisions that yield a positive quarterly return 40% of the time. At the end of the analyst's first year, the manager reviews the analyst's performance over the previous 4 quarters and promotes the analyst if she believes that there is at least a 75% chance that the analyst is high ability. Suppose the analyst has a successful first year, with positive quarterly returns in each of the previous 4 quarters. a. Given the above information, if the manager has Bayesian beliefs, what is her belief that the analyst is high ability at the end of the first year? Will she promote the analyst? Box the probability and write yes or no. b. Now suppose that the manager is a believer in the law of small numbers. That is, she believes (erroneously) that across 1 consecutive quarters, a high (low) ability analyst will earn positive quarterly returns exactly 6 (4) times. What is her belief in this case that the analyst is high ability at the end of the first year? Will the manager promote the analyst? Box the probability and write yes or no. c. Provide intuition for your answers in parts (a) and (b). In addition, explain whether the manager's thinking in part (b) illustrates the gambler's fallacy or extrapolation. Box your selection.