1. 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.

2. Now consider

*forecasting*further and further into the future, given just the first two umbrella observations. First, compute the probability $P(r_{2+k}|u_1,u_2)$ for $k=1 \ldots 20$ 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.

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

1. 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.

2. Now consider *forecasting* further and further into the
future, given just the first two umbrella observations. First,
compute the probability $P(r_{2+k}|u_1,u_2)$ for
$k=1 \ldots 20$ 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.