(solution) Suppose an island lies in the monsoon zone, and has just two

(solution) Suppose an island lies in the monsoon zone, and has just two

Suppose an island lies in the monsoon zone, and has just two seasons: Wet and Dry. The wet season lasts for 1/3 of the year, and the dry season for 2/3 of the year. During the Wet season, the probability that it is raining is 4/5; during the Dry season, the probability that it is raining is 1/4. Suppose I visit this island on a random day, and it is raining when I arrive. What is the probability it will be raining when I return to this island in a year’s time? (You may assume that in a year’s time the season will be the same as today but, given the season, whether or not it is raining is independent of today’s weather.)