Chapter 9.1, Problem 1MP

An insurance company classifies drivers as low-risk if they are accident-free for one year. Past records indicate that $98\%$ of the drivers in the low-risk category $\left(L\right)$ one year will remain in that category the next year, and $78\%$ of the drivers who are not in the low-risk category $\left(L\text{'}\right)$ one year will be in the low-risk category the next year.

(A) Draw a transition diagram.

(B) Find the transition matrix $P.$

(C) If $90\%$ of the drivers in the community are in the low-risk category this year, what is the probability that a driver chosen at random from the community will be in the low-risk category next year? Year after next?