In this exercise, we examine what happens to the probabilities in the umbrella world in the limit of long time sequences.

a. Suppose we observe an unending sequence of days on which the umbrella appears. Show that, as the days go by, the probability of rain on the current day increases monotonically toward a fixed point. Calculate this fixed point.

b. Now consider forecasting further and further into the future, given just the first two umbrella observations. First, compute the probability P(r_2+k|u₁, u₂) for k = 1. . . 20 and plot the results. You should see that the probability converges towards a fixed point. Prove that the exact value of this fixed point is 0.5.