Much is made over the fact that some 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. Suppose there are 5000 mutual funds in which you can invest. Suppose each has a probability of 0.5 of outperforming the market in any given year and that performance in each year is independent of performance in any other year.
a. What is the probability that any particular fund outperforms the market in 5 consecutive years? Ten consecutive years? Twenty consecutive years?
b. What is the probability that at least one of the 5000 funds outperforms the market in 5 consecutive years, 10 consecutive years, 20 consecutive years.
c. Find the probability that at least 10 funds outperform the market in 5 consecutive years, 10 consecutive years, 20 consecutive years.