In Oak Hill, Florida, 90% of all sunny days are followed by sunny days while 80% of cloudy days are followed by cloudy days. A local hotdog vendor makes, on average, $300 on sunny days and $100 on cloudy days. Assume you want to build a Markov model that you can use to answer questions b., c., d., and e. below.

a. Define the states of the system and give the one-step transition probability matrix.

b. If today is sunny, what is the probability it is sunny 5 days from now?

c. What is the long run probability a day is cloudy?

d. What is the long run expected revenue each day for the hotdog vendor (i.e. the gain)?