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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?

b$)$ Now suppose that the manager is a believer in the law of small numbers. That is$,$ she believes $($erroneously$)$ that across $1$ consecutive quarter, 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?

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.

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