Question

In a certain city during the month of May, the probability that a rainy day will befollowed by another rainy day is 0.80 and the probability that a sunny day will be followed by a rainy day is 0.60. Assuming that each day is classified as being either rainy or sunny and that the weather on any day depends only on the weather the day before, find the probability that in the given city a rainy day in May is followed by two more rainy days, then a sunny day, and finally another rainy day.

Answer 1

We assume that the events of being rainy or sunny are exclusive and independent.

Let us say on Day 1 was rainy. 

Now Probability that Day 2 is rainy = 0.80  - as day 1 was rainy
Again Probability that Day 3 is rainy = 0.80 * 0.80      - as Day2 was rainy

Probability that a Sunny day follows a rainy day = 1 - 0.80 = 0.20

Hence probability that Day 4 is Sunny = 0.80 * 0.80 * 0.20

Finally probability that Day 5 is a Rainy day: 0.80*0.80 *0.20 * 0.60
       (as day 4 was Sunny)

Answer = 768/10000   = 48/625 = 0.0768