Question: One theory concerning the S&P 500 index is that if it increases during the first five trading days of the year, it is likely to increase during the entire year. From 1950 through 2005, the S&P 500 index had these early gains in 35 years. In 30 of these 35 years, the S&P 500 index increased. Assuming that this indicator is a random event with no predictive value, you would expect that the indicator would be correct 50% of the time. What is the probability of the S&P 500 index increasing in 30 or more years if the true probability of an increase in the S&P 500 index is

a. 0.50?

b. 0.70?

c. 0.90?

d. Based on the results of (a) through (c), what do you think is the probability that the S&P 500 index will increase if there is an early gain in the first five trading days of the year? Explain.