Suppose the probability of rain today is .6 if rain fell yesterday, but only .2 if it did not.

What is the average duration (number of days) of a rainy period?

I have found the transition matrix as

P = [.8 .2

.4 .6]

And I have solved the other two parts of the problem which are:

X = 1, as a rainy day, X = 0 as a non rain day

Give rain fell today, what is the probability of ran on the day after tomorrow?

P(X_2=1 | X_0 = 1) = 0.44

Found this by squaring the matrix and taking the second entry on the column and row.

The other question answered was the fraction of days in which rain falls?

I found the fractions as non-rainy days -> 2/3 and rainy days -> 1/3, by solving the steady state equations.

All that as background. I'm just not sure how to find the average duration of a rainy period.