5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.

1. Write down the transition matrix \(\mathbf { P }\).
2. Find the probability that route \(B\) is used on the 7th day.
3. Find the probability that the same route is used on the 7th and 8th days.
4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
6. Find the new transition probabilities after route \(C\).
7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }