5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).

• Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
• Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
• Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.

This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probabilities that the 10th vehicle is in each of the three categories.
3. Find the probability that the 12th and 13th vehicles are in the same category.
4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
   • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
   • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
   • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
   In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
8. Find the transition matrix for the new stretch of road.