In a good weather year, the number of storms is Poisson distributed with mean 1;

in a bad year it is Poisson distributed with mean 3. Suppose that any year’s weather condition depends on past years only through the previous year’s condition.

Suppose that a good year is equally likely to be followed by either a good or a bad year, and that a bad year is twice as likely to be followed by a bad year as by a good year. Suppose that last year—call it year 0—was a good year.

(a) Find the expected total number of storms in the next two years (that is, in years 1 and 2).

(b) Find the probability there are no storms in year 3.

(c) Find the long-run average number of storms per year.

(d) Find the proportion of years that have no storms.