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 80% of the time and bad managers beat the market 10% of the time. 50% of all managers are average, 15% are bad and 35% are good.
T is a new mutual fund manager. (a) How many years in a row would T have to beat the market to be at least 80% sure that he is a good manager? (b) How many years in a row would T have to underperform the market (i.e. NOT beat the market) to be at least 80% sure that he is a bad manager? (c) Now suppose T believes there is a 60% chance he is a good manager and a 40% chance he is an average manager (and no chance that he's a bad manager!). How many years in a row would T have to beat the market for him to be 80% sure that he's a good manager? (d) T's belief in (c) is an example of what kind of cognitive bias