Much is made of the fact that certain mutual funds outperform the market year after year (that is,..

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 5050 chance of outperforming the
market in any year and that performance is independent from year to year.

(i) If performance

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 5050 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 10years. 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.