Much is made of the fact that certain mutual funds outperform the market year after year (that is, the return from holding shares in the mutual fund is higher than the return from holding a portfolio such as the S&P 500). For concreteness, consider a 10-year period and let the population be the 4,170 mutual funds reported in The Wall Street Journal on January 1, 1995. By saying that performance relative to the market is random, we mean that each fund has a 50–50 chance of outperforming the market in any year and that performance is independent from year to year.

(i) If performance relative to the market is truly random, what is the probability that any particular fund outperforms the market in all 10 years?

(ii) Of the 4,170 mutual funds, what is the expected number of funds that will outperform the market in all 10 years?

(iii) Find the probability that at least one fund out of 4,170 funds outperforms the market in all 10 years. What do you make of your answer?

(iv) If you have a statistical package that computes binomial probabilities, find the probability that at least five funds outperform the market in all 10 years.