According to the "January theory," if the stock market is up in January, it will be up for the whole year (and vice versa). Suppose that there is not truth whatever in this theory, and that the likelihood of the stock market moving up or down in any given year is completely independent of the direction of movement in January. Suppose furthermore that the probability of the stock market being up in January is 0.60, and that the probability that the stock market is up for the year is 0.90.

a) what is the probability that the stock market movement in January will agree with the stock market movement for the entire year?

b) over a twenty year period, what is the probability that the January movement and the annual movement of the stock market will agree for all twenty years?

c) what is the probability of agreement between the January and annual movements in the stock market in at least 15 of the 20 years?

d)what is the probability of agreement between the January and annual movements in the stock market in at least 17 of the 20 years?

e) given your answers to the previous parts, how might you test whether the January theory is true or false?