OCR MEI FP3 2008 June — Question 5 24 marks

Exam BoardOCR MEI
ModuleFP3 (Further Pure Mathematics 3)
Year2008
SessionJune
Marks24
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Mark schemeDownload PDF ↗
TopicSequences and Series
TypeModelling with Recurrence Relations
DifficultyChallenging +1.2 This is a standard Markov chain question requiring matrix operations and steady-state analysis. While it involves multiple parts and matrix powers, the techniques are routine for Further Maths students: writing transition matrices, computing powers, finding steady states by solving linear equations, and interpreting probabilities. The most challenging aspect is part (vi) requiring reverse-engineering transition probabilities from desired steady states, but this follows a standard method. Overall, this is a typical FP3 examination question requiring competent application of learned techniques rather than novel insight.
Spec4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar
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.
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.\\
(i) Write down the transition matrix $\mathbf { P }$.\\
(ii) Find the probability that route $B$ is used on the 7th day.\\
(iii) Find the probability that the same route is used on the 7th and 8th days.\\
(iv) Find the probability that the route used on the 10th day is the same as that used on the 7th day.\\
(v) 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.\\
(vi) Find the new transition probabilities after route $C$.\\
(vii) 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.

\hfill \mbox{\textit{OCR MEI FP3 2008 Q5 [24]}}
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