Skill vs. Luck: Analyzing Probability in Stock Picking and Portfolio Management

[uni412]

Hi, I am reading the book "Fooled By Randomness" by Nassim Nicholas Taleb and ran into what I think is a statistics/probability question. In the book Taleb is talking about how even if the process of picking stocks was completely based on luck (ie. in a given year there is a 50% probability of earning money) some portfolio managers would accumulate very impressive track records just by luck. So if there are 10000 portfolio managers 312.5 would earn money 5 years in a row (10000*.5^5). He then says that if there was an initial population of 10 managers and 1 earned money 5 years in a row he would be much more likely to give money to/believe in the skill of that manager than if there was an initial population of 10000 managers and 1 came to him boasting 5 years of positive returns. Why would the size of the pool of portfolio managers change the probability that an individual managers performance was caused by luck rather than skill? How does Taleb's claim make since mathematically?

Thanks a lot for your help

 

[Focus]

Because if you use a bigger sample size, there will be more extreme values. Say that you and your friend were tossing coins and guessing the outcome (assume that it is a symmetric game), if your friend guesses right 10 times in a row, you may find it more impressive than if there was 100000 of you playing the game and someone did that.

It is to do with expected number of people that will get "lucky", in a small pool the expected number of "lucky" people is much lower than at a large pool. Say you have 0.001 chance of getting "lucky" then, you would expect in a group of 10, 0.01 to get lucky, but a group of 10 000 you would expect 10 to get lucky.