A car insurance company classifies drivers in three categories: bad, neutral and good. The

reclassification is done in January of each year and the probabilities for transitions between

different categories is given by

$P=\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{5}& \frac{2}{5}& \frac{2}{5}\\ \frac{1}{5}& \frac{1}{5}& \frac{3}{5}\end{array}\right]$

where the first row$/$column corresponds to the bad category, the second to neutral and the third

to good.

a$)$ The company started in January $1990$ with $1400$ drivers in each category. Estimate the

number of drivers in each category in $2090.$ Assume that the total number of drivers does

not change in time.

b$)$ A yearly premium charged to a driver depends on his$/$her category; it is $$1000$ for bad

drivers, $$500$ for neutral drivers and $$200$ for good drivers. A typical driver has a total of

$56$ years of driving experience. Estimate the total amount he$/$she will pay in insurance

premiums in those $56$ years. You may assume that $56$ years is long enough for the limiting

theorems to provide an accurate approximation.