AHL 4.19—Transition matrices – Markov chains Questionbank

Practice AHL 4.19—Transition matrices – Markov chains with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

A system transitions between states $X, Y$ , and $Z$ with transition matrix:

T=\left(\begin{array}{lll} 0.4 & 0.3 & 0.3 \\ 0.2 & 0.5 & 0.3 \\ 0.4 & 0.2 & 0.4 \end{array}\right)

Find an eigenvector corresponding to the eigenvalue $\lambda=1$ .

[3]

Using part (a), determine the long-term probability of being in state Y.

[2]

Question 2

HLPaper 2

A network of paths connects four nodes, A, B, C, and D, as shown:

Dashed lines indicate paths that are twice as likely to be chosen.

Construct the transition matrix for a particle moving between the nodes.

[3]

Find the steady-state probability distribution using a graphic display calculator.

[3]

Explain one limitation of this model in the context of path selection.

[2]

Question 3

HLPaper 2

A zoo has three habitats, H, I, and J, connected as shown:

An animal moves between habitats with equal probability for each available path.

Write down the transition matrix.

[3]

If the animal starts in habitat H , find the probability it is in habitat J after three moves.

[3]

Find the long-term probability of the animal being in habitat I.

[2]

Question 4

HLPaper 1

A Markov chain models a system with states $\mathrm{K}, \mathrm{L}$ , and M , with transition matrix:

T=\left(\begin{array}{lll} 0.7 & 0.2 & 0.2 \\ 0.1 & 0.6 & 0.3 \\ 0.2 & 0.2 & 0.5 \end{array}\right)

Find the steady-state distribution.

[3]

If the system starts in state K , find the expected number of steps to reach state M for the first time.

[4]

Question 5

HLPaper 1

A Markov chain models a process with states $\mathrm{E}, \mathrm{F}$ , and G , with transition matrix:

T=\left(\begin{array}{ccc} 0.5 & 0.2 & 0.3 \\ 0.3 & 0.4 & 0.3 \\ 0.2 & 0.4 & 0.4 \end{array}\right)

Verify that $T$ is a valid transition matrix.

[2]

Express $T$ as $P D P^{-1}$ , finding matrices $P$ and $D$ .

[5]

Question 6

HLPaper 2

A game involves a player moving between states $\mathrm{A}, \mathrm{B}, \mathrm{C}$ , and D , with transitions shown:

The player moves with equal probability to each connected state.

Write down the transition matrix.

[3]

Find the steady-state probability vector.

[3]

Find the minimum number of moves for the player to be at least $90 \%$ certain of reaching state D , starting from A .

[4]

Question 7

HLPaper 2

A biologist models animal migration between three regions, $\mathrm{K}, \mathrm{L}$ , and M , with some paths preferred (dashed lines, twice as likely):

Construct the transition matrix.

[3]

Find the steady-state distribution.

[3]

If the animal starts at K , find the probability it is at M after two moves.

[2]

Question 8

HLPaper 2

A robot navigates a grid with states $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ , and S , connected as follows:

The robot moves with equal probability to each connected state.

Write down the transition matrix.

[3]

Find the probability of being in state S after four transitions, starting from P.

[3]

Determine the expected number of steps to return to P , starting from P.

[3]

Question 9

HLPaper 2

John likes to go sailing every day in July. To help him make a decision on whether it is safe to go sailing he classifies each day in July as windy or calm. Given that a day in July is calm, the probability that the next day is calm is 0.9. Given that a day in July is windy, the probability that the next day is calm is 0.3. The weather forecast for the 1st July predicts that the probability that it will be calm is 0.8.

Draw a tree diagram to represent this information for the first three days of July.

[3]

Find the probability that the 3rd July is calm.

[2]

Find the probability that the 1st July was calm given that the 3rd July is windy.

[4]

Question 10

HLPaper 2

A wildlife reserve is tracking the movement of a certain species of bird between three regions: Region X, Region Y, and Region Z. Each month, 20% of the birds in Region X move to Region Y, 10% move to Region Z, 15% of the birds in Region Y move to Region X, 5% move to Region Z, 25% of the birds in Region Z move to Region X, and 10% move to Region Y.

Construct the transition matrix for the movement of birds between the three regions.

[2]

If initially there are 100 birds in Region X, 150 in Region Y, and 200 in Region Z, calculate the distribution of birds after one month.

[3]

Draw the transition diagram representing the movement of birds between the three regions.

[2]