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Date	May Specimen paper	Marks available	1	Reference code	SPM.2.AHL.TZ0.6
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Write down and Hence	Question number	6	Adapted from	N/A

Question

A city has two cable companies, X and Y. Each year 20 % of the customers using company X move to company Y and 10 % of the customers using company Y move to company X. All additional losses and gains of customers by the companies may be ignored.

Initially company X and company Y both have 1200 customers.

Write down a transition matrix T representing the movements between the two companies in a particular year.

[2]

Find the eigenvalues and corresponding eigenvectors of T.

[4]

Hence write down matrices P and D such that T = PDP⁻¹.

[2]

Find an expression for the number of customers company X has after $n$ years, where $n \in \mathbb{N}$ .

[5]

Hence write down the number of customers that company X can expect to have in the long term.

[1]

Markscheme

$\left( {\begin{array}{*{20}{c}} {0.8}&{0.1} \\ {0.2}&{0.9} \end{array}} \right)$ M1A1

[2 marks]

$\left| {\begin{array}{*{20}{c}} {0.8 - \lambda }&{0.1} \\ {0.2}&{0.9 - \lambda } \end{array}} \right| = 0$ M1

$\lambda = 1$ and 0.7 A1

eigenvectors $\left( {\begin{array}{*{20}{c}} 1 \\ 2 \end{array}} \right)$ and $\left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right)$ (M1)A1

Note: Accept any scalar multiple of the eigenvectors.

[4 marks]

EITHER

P = $\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)$ D = $\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{0.7} \end{array}} \right)$ A1A1

P = $\left( {\begin{array}{*{20}{c}} 1&1 \\ { - 1}&2 \end{array}} \right)$ D = $\left( {\begin{array}{*{20}{c}} {0.7}&0 \\ 0&1 \end{array}} \right)$ A1A1

[2 marks]

P⁻¹ = $\frac{1}{3}\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)$ A1

$\frac{1}{3}\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{{{0.7}^n}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&1 \\ 2&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {1200} \\ {1200} \end{array}} \right)$ M1A1

attempt to multiply matrices M1

so in company A, after $n$ years, $400\left( {2 + {{0.7}^n}} \right)$ A1

[5 marks]

400 × 2 = 800 A1

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 4—Statistics and probability » AHL 4.19—Transition matrices – Markov chains

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EXM.3.AHL.TZ0.4a.ii:
Explain why for any transition state diagram the sum of the out degrees of the directed edges from a vertex (state) must add up to +1.
SPM.2.AHL.TZ0.6b:
Find the eigenvalues and corresponding eigenvectors of T.
EXM.3.AHL.TZ0.4c.ii:
Explain which part of the transition state diagram confirms this.
22M.2.AHL.TZ2.5a.ii:
What does $b$ represent in this context?
22M.2.AHL.TZ2.5d.i:
when $n = 5$ .
22M.2.AHL.TZ2.5d.ii:
in the long term.
22M.2.AHL.TZ2.5a.i:
Write down the value of $b$ .
EXM.3.AHL.TZ0.4a.i:
Draw a transition state diagram for this Markov chain problem.
EXM.1.AHL.TZ0.17c:
Find the steady state probability vector for this Markov chain.
EXM.1.AHL.TZ0.17b:
We know that she went out for lunch on a particular Sunday, find the probability that she went out for lunch on the following Tuesday.
21M.2.AHL.TZ1.5a:
It is sunny today. Find the probability that it will be sunny in three days’ time.
21M.2.AHL.TZ1.5b:
Find the eigenvalues and eigenvectors of $T$ .
21M.2.AHL.TZ1.5c.ii:
Write down the matrix $D$ .
EXM.3.AHL.TZ0.4d:
Explain why having a steady state probability vector means that the matrix M must have an eigenvalue of $\lambda = 1$ .
EXM.1.AHL.TZ0.18a:
Show that $\lambda = 1$ is always an eigenvalue for M and find the other eigenvalue in terms of $a$ and $b$ .
EXM.3.AHL.TZ0.4e:
Find ${{\mathbf{v}}_1}{\text{,}}\,\,{{\mathbf{v}}_2}{\text{,}}\,\,{{\mathbf{v}}_3}{\text{,}}\,\,{{\mathbf{v}}_4}\,$ .
EXM.3.AHL.TZ0.4g:
Explain how your answer to part (f) fits with your answer to part (c).
EXM.1.AHL.TZ0.17a:
Write down the transition matrix for this Markov chain.
SPM.2.AHL.TZ0.6c:
Hence write down matrices P and D such that T = PDP⁻¹.
SPM.2.AHL.TZ0.6d:
Find an expression for the number of customers company X has after $n$ years, where $n \in \mathbb{N}$ .
EXM.1.AHL.TZ0.18b:
Find the steady state probability vector for M in terms of $a$ and $b$ .
EXN.1.AHL.TZ0.5a:
Find the transition matrix for the maze.
EXM.3.AHL.TZ0.4c.i:
Find the steady state probability vector for this Markov chain problem.
EXM.3.AHL.TZ0.4h:
Find the minimum number of tosses of the coin that Abi will have to make to be at least 95% certain of having finished the game by reaching state C.
SPM.2.AHL.TZ0.6a:
Write down a transition matrix T representing the movements between the two companies in a particular year.
21M.2.AHL.TZ1.5d:
Hence find the long-term percentage of sunny days in Vokram.
EXN.1.AHL.TZ0.5b:
A scientist sets up the robot and then leaves it moving around the maze for a long period of time.

Find the probability that the robot is in room $B$ when the scientist returns.
EXM.3.AHL.TZ0.4b:
Write down the transition matrix M, for this Markov chain problem.
EXM.3.AHL.TZ0.4f:
Hence, deduce the form of ${{\mathbf{v}}_n}$ .
21M.1.AHL.TZ2.13a:
Determine the transition matrix for this graph.
21M.1.AHL.TZ2.13c:
Comment on your answer to part (b), referring to at least one limitation of the model.
21M.1.AHL.TZ2.13b:
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21M.2.AHL.TZ1.5c.i:
Write down the matrix $P$ .
21N.1.AHL.TZ0.9a:
Complete the following transition diagram to represent this information.
21N.1.AHL.TZ0.9b:
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Topic 4—Statistics and probability

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