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 2018, the S & P 500 index had these early gains in 44 years (in 2011 there was virtually no change). In 36 of these 44 years, the S & P 500 Index increased for the entire year. Assuming that this indicator is a random event with not predictive values, 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 36 years if the true probability if an increase is 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.