4. Mutual fund managers can be average, good or bad. The chance that an average manager beats the market in a given year is 50%, but good managers beat the market 90% of the time and bad managers beat the market 10% of the time. 50% of all managers are average, 30% are bad and 20% are good. Tyler is a new mutual fund manager.

(a) How many years in a row would Tyler have to beat the market to be at least 90% sure that he is a good manager? (2 points)

(b) How many years in a row would Tyler have to underperform the market (i.e. NOT beat the market) to be at least 90% sure that he is a bad manager? (2 points)

(c) Now suppose Tyler believes there is a 75% chance he is a good manager and a 25% chance he is an average manager (and no chance that he's a bad manager!). How many years in a row would Tyler have to beat the market for him to be 90% sure that he's a good manager? (2 points)

(d) Tyler's belief in (c) is an example of what kind of cognitive bias? (1 points)