The professor of an IOE class may end the lecture early, on time, or late on any given day. Students have noticed that they can forecast the probability of ending the class on time based on the ending time of the previous lecture. Experience shows that after ending a lecture early, the Professor tries to make up the time by running late the next class 9 times out of 10, the rest of the time she follows early lectures with on-time lectures (she never follows an early lecture with another early lecture). On the days that classes end on time, students believe that the next class will get out early with a probability of 0.6; otherwise there is equal probability the next class will end on time or late. Following days when the class runs late, 2 out of 10 classes are late again. The Professor never lets the class out early following a late class. (a) (5 points) Construct a Markov Chain describing the ending time of the classes. Define the states and the one-step transition probability matrix.
(b) (5 points) Indicate the period of each state.
(c) (5 points) Assume that the third class ends late. What is the probability that the fifth class will end late?
(d) (5 points) Now assume that for the second class of the semester, the professor is equally likely to end the class on time or late, but twice as likely to end the class early (compared to ending on time or late). On the day of the fourth class, you are meeting friends for lunch right after class. What is the probability that you will be late?
2. (20 noints) A meteorolocical groun is monitoring which days it snows/does not snow in Ann