According to the “January theory,” if the stock market is up at the end of January, it will be “up” for the year. If it is “down” at the end of January, it will be “down” for the year. Within the last 34 years, this theory proved to be true for 29 years. A different theory is that the market change at the end of January and the market change at the end of the year are unrelated. Specifically, for any market change in January, the probability that the market is “up” or “down” at the end of the year is equally likely—that is, the probability is 0.5. You will need a statistical software package to help you solve this problem.

a. Based on history, what is the probability that a year will end with an “up” market when January ends with an “up” market?

b. If the January market change and the yearend market change are unrelated, the probability that the market is “up” with an “up” January is 0.5. Using 0.5, what is the probability that the market would be up 29 or more years? What would be the mean number of years that the market is “up”?

c. Based on the result in part (b), what is your conclusion regarding the “January theory”?