OCR MEI FP3 2009 June — Question 5 24 marks

Exam BoardOCR MEI
ModuleFP3 (Further Pure Mathematics 3)
Year2009
SessionJune
Marks24
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TopicGroups
DifficultyModerate -0.5 This is a standard Markov chain question requiring routine matrix operations and probability calculations. While it has multiple parts, each part uses well-established techniques (transition matrices, matrix powers, geometric series for expected values) that are typical textbook exercises for this topic. The calculations are computational rather than requiring novel insight.
Spec4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar
5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.
  1. Write down the transition matrix.
  2. Find the probabilities that level 14 is set in each location.
  3. Find the probability that level 15 is set in the same location as level 14 .
  4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
  5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
  6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
  7. Find the new transition probabilities after a level set in Chiworld. }{www.ocr.org.uk}) after the live examination series.
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Question 5 (Pre-multiplication):
Part (i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{P} = \begin{pmatrix} 0 & 0.1 & 0 & 0.3 \\ 0.7 & 0.8 & 0 & 0.6 \\ 0.1 & 0 & 1 & 0.1 \\ 0.2 & 0.1 & 0 & 0 \end{pmatrix}\)B2 Give B1 for two columns correct
Total: 2
Part (ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{P}^{13}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}0.0810\\0.5684\\0.2760\\0.0746\end{pmatrix}\)M1, A2 Using \(\mathbf{P}^{13}\) (or \(\mathbf{P}^{14}\)). Give A1 for 2 probabilities correct (Max A1 if not at least 3dp). Tolerance \(\pm0.0001\)
Total: 3
Part (iii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(0.5684\times0.8 + 0.2760 = 0.731\)M1M1, A1 ft For \(0.5684\times0.8\) and \(0.2760\). Accept 0.73 to 0.7312
Total: 3
Part (iv):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{P}^{30}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}\cdot\\\cdot\\0.4996\\\cdot\end{pmatrix}\), \(\mathbf{P}^{31}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}\cdot\\\cdot\\0.5103\\\cdot\end{pmatrix}\)M1, A1 Finding \(P(C)\) for some powers of \(\mathbf{P}\). For identifying \(\mathbf{P}^{31}\)
Level 32A1
Total: 3
Part (v):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Expected number of levels including the next change of location is \(\frac{1}{0.2}=5\)M1, A1 For \(1/(1-0.8)\) or \(0.8/(1-0.8)\). For 5 or 4
Expected number of further levels in B is 4A1 For 4 as final answer
Total: 3
Part (vi):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{Q} = \begin{pmatrix}0&0.1&0&0.3\\0.7&0.8&0&0.6\\0.1&0&0.9&0.1\\0.2&0.1&0&0\end{pmatrix}\)B1 Can be implied
\(\mathbf{Q}^n \to \begin{pmatrix}0.0916&0.0916&0.0916&0.0916\\0.6183&0.6183&0.6183&0.6183\\0.1908&0.1908&0.1908&0.1908\\0.0992&0.0992&0.0992&0.0992\end{pmatrix}\)M1, M1 Evaluating powers of \(\mathbf{Q}\) or obtaining (at least) 3 equations from \(\mathbf{Q}\mathbf{p}=\mathbf{p}\). Limiting matrix with equal columns or solving to obtain one equilibrium prob
A: 0.0916, B: 0.6183, C: 0.1908, D: 0.0992A2 Give A1 for two correct (Max A1 if not at least 3dp). Tolerance \(\pm0.0001\)
Total: 5
Part (vii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\begin{pmatrix}0&0.1&a&0.3\\0.7&0.8&b&0.6\\0.1&0&c&0.1\\0.2&0.1&d&0\end{pmatrix}\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}=\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}\)M1, A1 Transition matrix and \(\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}\)
\(0.075+0.04a+0.03=0.11\); \(0.077+0.6+0.04b+0.06=0.75\); \(0.011+0.04c+0.01=0.04\); \(0.022+0.075+0.04d=0.1\)M1 Forming at least one equation, or \(a+b+c+d=1\)
\(a=0.125,\ b=0.325,\ c=0.475,\ d=0.075\)A2 Give A1 for two correct
Total: 5
Question 5 (Post-multiplication):
Part (i):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{P} = \begin{pmatrix}0&0.7&0.1&0.2\\0.1&0.8&0&0.1\\0&0&1&0\\0.3&0.6&0.1&0\end{pmatrix}\)B2 Give B1 for two rows correct
Total: 2
Part (ii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{13} = (0.0810\ \ 0.5684\ \ 0.2760\ \ 0.0746)\)M1, A2 Using \(\mathbf{P}^{13}\) (or \(\mathbf{P}^{14}\)). Give A1 for 2 probabilities correct (Max A1 if not at least 3dp). Tolerance \(\pm0.0001\)
Total: 3
Part (iii):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(0.5684\times0.8+0.2760=0.731\)M1M1, A1 ft For \(0.5684\times0.8\) and \(0.2760\). Accept 0.73 to 0.7312
Total: 3
Part (iv):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\((0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{30}=(\ldots\ \ \ldots\ \ 0.4996\ \ \ldots)\)M1, A1 Finding \(P(C)\) for some powers of \(\mathbf{P}\). For identifying \(\mathbf{P}^{31}\)
\((0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{31}=(\ldots\ \ \ldots\ \ 0.5103\ \ \ldots)\)
Level 32A1
Total: 3
Part (v):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
Expected number of levels including the next change of location is \(\frac{1}{0.2}=5\)M1, A1 For \(1/(1-0.8)\) or \(0.8/(1-0.8)\). For 5 or 4
Expected number of further levels in B is 4A1 For 4 as final answer
Total: 3
Part (vi):
AnswerMarks Guidance
Answer/WorkingMarks Guidance
\(\mathbf{Q} = \begin{pmatrix}0&0.7&0.1&0.2\\0.1&0.8&0&0.1\\0&0&0.9&0.1\\0.3&0.6&0.1&0\end{pmatrix}\)B1 Can be implied
\(\mathbf{Q}^n \to \begin{pmatrix}0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\end{pmatrix}\)M1, M1 Evaluating powers of \(\mathbf{Q}\) or obtaining (at least) 3 equations from \(\mathbf{p}\mathbf{Q}=\mathbf{p}\). Limiting matrix with equal rows or solving to obtain one equilibrium prob
A: 0.0916, B: 0.6183, C: 0.1908, D: 0.0992A2 Give A1 for two correct (Max A1 if not at least 3dp). Tolerance \(\pm0.0001\)
Total: 5
Question (vii):
AnswerMarks Guidance
Answer/WorkingMark Guidance Notes
\((0.11 \quad 0.75 \quad 0.04 \quad 0.1)\begin{pmatrix} 0 & 0.7 & 0.1 & 0.2 \\ 0.1 & 0.8 & 0 & 0.1 \\ a & b & c & d \\ 0.3 & 0.6 & 0.1 & 0 \end{pmatrix}\)M1 Transition matrix and \((0.11 \quad 0.75 \quad 0.04 \quad 0.1)\)
\(= (0.11 \quad 0.75 \quad 0.04 \quad 0.1)\)A1
\(0.075 + 0.04a + 0.03 = 0.11\)M1 Forming at least one equation
\(0.077 + 0.6 + 0.04b + 0.06 = 0.75\)
\(0.011 + 0.04c + 0.01 = 0.04\)
\(0.022 + 0.075 + 0.04d = 0.1\) *or* \(a + b + c + d = 1\)
\(a = 0.125, \quad b = 0.325, \quad c = 0.475, \quad d = 0.075\)A2 Give A1 for two correct
Total: 5 marks 
# Question 5 (Pre-multiplication):

## Part (i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P} = \begin{pmatrix} 0 & 0.1 & 0 & 0.3 \\ 0.7 & 0.8 & 0 & 0.6 \\ 0.1 & 0 & 1 & 0.1 \\ 0.2 & 0.1 & 0 & 0 \end{pmatrix}$ | B2 | Give B1 for two columns correct |
| **Total: 2** | | |

## Part (ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P}^{13}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}0.0810\\0.5684\\0.2760\\0.0746\end{pmatrix}$ | M1, A2 | Using $\mathbf{P}^{13}$ (or $\mathbf{P}^{14}$). Give A1 for 2 probabilities correct (Max A1 if not at least 3dp). Tolerance $\pm0.0001$ |
| **Total: 3** | | |

## Part (iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.5684\times0.8 + 0.2760 = 0.731$ | M1M1, A1 ft | For $0.5684\times0.8$ and $0.2760$. Accept 0.73 to 0.7312 |
| **Total: 3** | | |

## Part (iv):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P}^{30}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}\cdot\\\cdot\\0.4996\\\cdot\end{pmatrix}$, $\mathbf{P}^{31}\begin{pmatrix}0.6\\0.4\\0\\0\end{pmatrix} = \begin{pmatrix}\cdot\\\cdot\\0.5103\\\cdot\end{pmatrix}$ | M1, A1 | Finding $P(C)$ for some powers of $\mathbf{P}$. For identifying $\mathbf{P}^{31}$ |
| Level 32 | A1 | |
| **Total: 3** | | |

## Part (v):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Expected number of levels including the next change of location is $\frac{1}{0.2}=5$ | M1, A1 | For $1/(1-0.8)$ or $0.8/(1-0.8)$. For 5 or 4 |
| Expected number of further levels in B is 4 | A1 | For 4 as final answer |
| **Total: 3** | | |

## Part (vi):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{Q} = \begin{pmatrix}0&0.1&0&0.3\\0.7&0.8&0&0.6\\0.1&0&0.9&0.1\\0.2&0.1&0&0\end{pmatrix}$ | B1 | Can be implied |
| $\mathbf{Q}^n \to \begin{pmatrix}0.0916&0.0916&0.0916&0.0916\\0.6183&0.6183&0.6183&0.6183\\0.1908&0.1908&0.1908&0.1908\\0.0992&0.0992&0.0992&0.0992\end{pmatrix}$ | M1, M1 | Evaluating powers of $\mathbf{Q}$ or obtaining (at least) 3 equations from $\mathbf{Q}\mathbf{p}=\mathbf{p}$. Limiting matrix with equal columns or solving to obtain one equilibrium prob |
| A: 0.0916, B: 0.6183, C: 0.1908, D: 0.0992 | A2 | Give A1 for two correct (Max A1 if not at least 3dp). Tolerance $\pm0.0001$ |
| **Total: 5** | | |

## Part (vii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix}0&0.1&a&0.3\\0.7&0.8&b&0.6\\0.1&0&c&0.1\\0.2&0.1&d&0\end{pmatrix}\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}=\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}$ | M1, A1 | Transition matrix and $\begin{pmatrix}0.11\\0.75\\0.04\\0.1\end{pmatrix}$ |
| $0.075+0.04a+0.03=0.11$; $0.077+0.6+0.04b+0.06=0.75$; $0.011+0.04c+0.01=0.04$; $0.022+0.075+0.04d=0.1$ | M1 | Forming at least one equation, or $a+b+c+d=1$ |
| $a=0.125,\ b=0.325,\ c=0.475,\ d=0.075$ | A2 | Give A1 for two correct |
| **Total: 5** | | |

---

# Question 5 (Post-multiplication):

## Part (i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P} = \begin{pmatrix}0&0.7&0.1&0.2\\0.1&0.8&0&0.1\\0&0&1&0\\0.3&0.6&0.1&0\end{pmatrix}$ | B2 | Give B1 for two rows correct |
| **Total: 2** | | |

## Part (ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{13} = (0.0810\ \ 0.5684\ \ 0.2760\ \ 0.0746)$ | M1, A2 | Using $\mathbf{P}^{13}$ (or $\mathbf{P}^{14}$). Give A1 for 2 probabilities correct (Max A1 if not at least 3dp). Tolerance $\pm0.0001$ |
| **Total: 3** | | |

## Part (iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.5684\times0.8+0.2760=0.731$ | M1M1, A1 ft | For $0.5684\times0.8$ and $0.2760$. Accept 0.73 to 0.7312 |
| **Total: 3** | | |

## Part (iv):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{30}=(\ldots\ \ \ldots\ \ 0.4996\ \ \ldots)$ | M1, A1 | Finding $P(C)$ for some powers of $\mathbf{P}$. For identifying $\mathbf{P}^{31}$ |
| $(0.6\ \ 0.4\ \ 0\ \ 0)\mathbf{P}^{31}=(\ldots\ \ \ldots\ \ 0.5103\ \ \ldots)$ | | |
| Level 32 | A1 | |
| **Total: 3** | | |

## Part (v):

| Answer/Working | Marks | Guidance |
|---|---|---|
| Expected number of levels including the next change of location is $\frac{1}{0.2}=5$ | M1, A1 | For $1/(1-0.8)$ or $0.8/(1-0.8)$. For 5 or 4 |
| Expected number of further levels in B is 4 | A1 | For 4 as final answer |
| **Total: 3** | | |

## Part (vi):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{Q} = \begin{pmatrix}0&0.7&0.1&0.2\\0.1&0.8&0&0.1\\0&0&0.9&0.1\\0.3&0.6&0.1&0\end{pmatrix}$ | B1 | Can be implied |
| $\mathbf{Q}^n \to \begin{pmatrix}0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\\0.0916&0.6183&0.1908&0.0992\end{pmatrix}$ | M1, M1 | Evaluating powers of $\mathbf{Q}$ or obtaining (at least) 3 equations from $\mathbf{p}\mathbf{Q}=\mathbf{p}$. Limiting matrix with equal rows or solving to obtain one equilibrium prob |
| A: 0.0916, B: 0.6183, C: 0.1908, D: 0.0992 | A2 | Give A1 for two correct (Max A1 if not at least 3dp). Tolerance $\pm0.0001$ |
| **Total: 5** | | |

## Question (vii):

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $(0.11 \quad 0.75 \quad 0.04 \quad 0.1)\begin{pmatrix} 0 & 0.7 & 0.1 & 0.2 \\ 0.1 & 0.8 & 0 & 0.1 \\ a & b & c & d \\ 0.3 & 0.6 & 0.1 & 0 \end{pmatrix}$ | M1 | Transition matrix and $(0.11 \quad 0.75 \quad 0.04 \quad 0.1)$ |
| $= (0.11 \quad 0.75 \quad 0.04 \quad 0.1)$ | A1 | |
| $0.075 + 0.04a + 0.03 = 0.11$ | M1 | Forming at least one equation |
| $0.077 + 0.6 + 0.04b + 0.06 = 0.75$ | | |
| $0.011 + 0.04c + 0.01 = 0.04$ | | |
| $0.022 + 0.075 + 0.04d = 0.1$ | | *or* $a + b + c + d = 1$ |
| $a = 0.125, \quad b = 0.325, \quad c = 0.475, \quad d = 0.075$ | A2 | Give A1 for two correct |
| **Total: 5 marks** | | |
5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location.

In the first version of the game, the initial and transition probabilities are as follows.\\
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.\\
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities $0.7,0.1,0.2$ respectively.\\
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities $0.1,0.8,0.1$ respectively.\\
After a level set in Chiworld, the next level will also be set in Chiworld.\\
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities $0.3,0.6,0.1$ respectively.

The situation is modelled as a Markov chain with four states.\\
(i) Write down the transition matrix.\\
(ii) Find the probabilities that level 14 is set in each location.\\
(iii) Find the probability that level 15 is set in the same location as level 14 .\\
(iv) Find the level at which the probability of being set in Chiworld first exceeds 0.5.\\
(v) Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location.

In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities $0.9,0.1$ respectively.\\
(vi) By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations.

In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.\\
(vii) Find the new transition probabilities after a level set in Chiworld.

}{www.ocr.org.uk}) after the live examination series.\\
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1PB.\\
OCR is part of the

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