| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Modelling with Recurrence Relations |
| Difficulty | Challenging +1.2 This is a structured Markov chain question with clear step-by-step parts. While it requires understanding of transition matrices and long-term behavior, most parts involve routine calculations (matrix powers, probability computations) and standard equilibrium analysis. The most challenging aspect is part (viii) requiring conditional probability reasoning, but overall this follows standard FP3 patterns with calculator support for matrix powers. |
| Answer | Marks |
|---|---|
| (i) Write down transition matrix P | [2] |
| (ii) Find \(\mathbf{P}^4\) and \(\mathbf{P}^7\) to 4 d.p. | [4] |
| (iii) Find probability 8th letter is C | [2] |
| (iv) Find probability 12th letter same as 8th | [4] |
| (v) Find probability \((n+1)\)th letter is A when n large even (A) and large odd (B) | [4] |
| (vi) Write down new transition matrix Q | [1] |
| (vii) Verify \(\mathbf{Q}^n\) approaches limit; write equilibrium probabilities | [4] |
| (viii) When n large, find probability \((n+1)\)th, \((n+2)\)th, \((n+3)\)th letters are DDD | [3] |
## Question 5 (Option 5: Markov chains)
**(i)** Write down transition matrix **P** | [2] |
**(ii)** Find $\mathbf{P}^4$ and $\mathbf{P}^7$ to 4 d.p. | [4] |
**(iii)** Find probability 8th letter is C | [2] |
**(iv)** Find probability 12th letter same as 8th | [4] |
**(v)** Find probability $(n+1)$th letter is A when n large even (A) and large odd (B) | [4] |
**(vi)** Write down new transition matrix **Q** | [1] |
**(vii)** Verify $\mathbf{Q}^n$ approaches limit; write equilibrium probabilities | [4] |
**(viii)** When n large, find probability $(n+1)$th, $(n+2)$th, $(n+3)$th letters are DDD | [3] |
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*Note: To obtain the actual mark scheme with worked solutions and marking criteria, you would need the official OCR mark scheme document for this paper.*5 A computer is programmed to generate a sequence of letters. The process is represented by a Markov chain with four states, as follows.
The first letter is $A , B , C$ or $D$, with probabilities $0.4,0.3,0.2$ and 0.1 respectively.\\
After $A$, the next letter is either $C$ or $D$, with probabilities 0.8 and 0.2 respectively.\\
After $B$, the next letter is either $C$ or $D$, with probabilities 0.1 and 0.9 respectively.\\
After $C$, the next letter is either $A$ or $B$, with probabilities 0.4 and 0.6 respectively.\\
After $D$, the next letter is either $A$ or $B$, with probabilities 0.3 and 0.7 respectively.
\begin{enumerate}[label=(\roman*)]
\item Write down the transition matrix $\mathbf { P }$.
\item Use your calculator to find $\mathbf { P } ^ { 4 }$ and $\mathbf { P } ^ { 7 }$. (Give elements correct to 4 decimal places.)
\item Find the probability that the 8th letter is $C$.
\item Find the probability that the 12th letter is the same as the 8th letter.
\item By investigating the behaviour of $\mathbf { P } ^ { n }$ when $n$ is large, find the probability that the ( $n + 1$ )th letter is $A$ when\\
(A) $n$ is a large even number,\\
(B) $n$ is a large odd number.
The program is now changed. The initial probabilities and the transition probabilities are the same as before, except for the following.
After $D$, the next letter is $A , B$ or $D$, with probabilities $0.3,0.6$ and 0.1 respectively.
\item Write down the new transition matrix $\mathbf { Q }$.
\item Verify that $\mathbf { Q } ^ { n }$ approaches a limit as $n$ becomes large, and hence write down the equilibrium probabilities for $A , B , C$ and $D$.
\item When $n$ is large, find the probability that the $( n + 1 )$ th, $( n + 2 )$ th and $( n + 3 )$ th letters are DDD.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP3 2007 Q5 [24]}}