Chapter 5, Problem 42PS

One theory concerning the S&P 500 Index is that if it increases during the first trading days of the year, it is likely to increase during the entire year. From 1950 through 2016, the S&P 500 Index had these early gains in 42 years (in 2011 there was virtually no change). In 37 of these 42 years, the S&P 500 Index increased for the entire year. 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 37 or more years if he true probability of an increase in the S&P 500 Index is

$\begin{array}{l}\text{a}\text{.}\text{0}\text{.50}\\ \text{b}\text{.}\text{0}\text{.70?}\\ \text{c}\text{.}\text{0}\text{.90?}\\ \text{d}\text{. Based on the results of (a) through (c)}\text{. What do you think is the probability that the S\&P 500 Index will increase if there is an early gain in the first five trading days of the year? Explain}\text{.}\end{array}$