An auto insurance company classifies drivers as low risk if they are accident-free for one year. Historically 98% of the drivers in the low-risk category for one year will remain in that category for the next year, and 78% of the drivers who are not low-risk one year will be in the low-risk category the next year.

i) write a transition matrix with this information. Ninety percent of the drivers in a community are in the low-risk category this year. Write the initial probability vector for this community. Then determine the probability that a driver-selected at random from the low-risk category will be in the low-risk category next year 

ii) Determine the steady-state vector for this Markov chain. 

iii) Determine the percentage of the drivers in the low-risk category this year that will remain in the low-risk category.