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Date	May 2022	Marks available	1	Reference code	22M.2.AHL.TZ2.5
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Write down	Question number	5	Adapted from	N/A

Question

A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.

The model is of the form

$(\begin{matrix} X_{n + 1} \\ Z_{n + 1} \end{matrix}) = M (\begin{matrix} X_{n} \\ Z_{n} \end{matrix})$

where $X_{n}$ is the probability of the gene being in its normal state after dividing for the $n th$ time, and $Z_{n}$ is the probability of it being in another state after dividing for the $n th$ time, where $n \in ℕ$ .

Matrix $M$ is found to be $(\begin{matrix} 0.94 & b \\ 0.06 & 0.98 \end{matrix})$ .

The gene is in its normal state when $n = 0$ . Calculate the probability of it being in its normal state

Write down the value of $b$ .

[1]

a.i.

What does $b$ represent in this context?

[1]

a.ii.

Find the eigenvalues of $M$ .

[3]

Find the eigenvectors of $M$ .

[3]

when $n = 5$ .

[2]

d.i.

in the long term.

[2]

d.ii.

Markscheme

$0.02$ A1

[1 mark]

a.i.

the probability of mutating from ‘not normal state’ to ‘normal state’ A1

Note: The A1 can only be awarded if it is clear that transformation is from the mutated state.

[1 mark]

a.ii.

$det (\begin{matrix} 0.94 - λ & 0.02 \\ 0.06 & 0.98 - λ \end{matrix}) = 0$ (M1)

Note: Award M1 for an attempt to find eigenvalues. Any indication that $det (M - λ I) = 0$ has been used is sufficient for the (M1).

$(0.94 - λ) (0.98 - λ) - 0.0012 = 0$ OR $λ^{2} - 1.92 λ + 0.92 = 0$ (A1)

$λ = 1, 0.92 (\frac{23}{25})$ A1

[3 marks]

$(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = (\begin{matrix} x \\ y \end{matrix})$ OR $(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix}) (\begin{matrix} x \\ y \end{matrix}) = 0.92 (\begin{matrix} x \\ y \end{matrix})$ (M1)

Note: Award M1 can be awarded for attempting to find either eigenvector.

$0.02 y - 0.06 x = 0$ OR $0.02 y + 0.02 x = 0$

$(\begin{matrix} 1 \\ 3 \end{matrix})$ and $(\begin{matrix} 1 \\ - 1 \end{matrix})$ A1A1

Note: Accept any multiple of the given eigenvectors.

[3 marks]

${(\begin{matrix} 0.94 & 0.02 \\ 0.06 & 0.98 \end{matrix})}^{5} (\begin{matrix} 1 \\ 0 \end{matrix})$ OR $(\begin{matrix} 0.744 & 0.0852 \\ 0.256 & 0.915 \end{matrix}) (\begin{matrix} 1 \\ 0 \end{matrix})$ (M1)

Note: Condone omission of the initial state vector for the M1.

$0.744 (0.744311 \dots)$ A1

[2 marks]

d.i.

$(\begin{matrix} 0.25 \\ 0.75 \end{matrix})$ (A1)

Note: Award A1 for $(\begin{matrix} 0.25 \\ 0.75 \end{matrix})$ OR $(\begin{matrix} 0.25 & 0.25 \\ 0.75 & 0.75 \end{matrix})$ seen.

$0.25$ A1

[2 marks]

d.ii.

Examiners report

There was some difficulty in interpreting the meaning of the values in the transition matrix, but most candidates did well with the rest of the question. In part (d) there was frequently evidence of a correct method, but a failure to identify the correct probabilities. It was surprising to see a significant number of candidates diagonalizing the matrix in part (d) and this often led to errors. Clearly this was not necessary.

a.i.

[N/A]

a.ii.

[N/A]

d.i.

[N/A]

d.ii.

Syllabus sections

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

Show 36 related questions

22M.1.AHL.TZ1.11b:
Using your answer to (a), or otherwise, find the long-term probability of the switch being in state $A$ . Give your answer in the form $\frac{c}{d}$ , where $c, d \in ℤ^{+}$ .
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.
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.
SPM.2.AHL.TZ0.6e:
Hence write down the number of customers that company X can expect to have in the long term.
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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