IB Mathematics Applications & Interpretation Topic AHL 4.19—transition Matrices – Markov Chains Questionbank (HL)

IB Mathematics Applications & Interpretation Topic AHL 4.19—transition Matrices – Markov Chains Questionbank (HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic AHL 4.19—transition Matrices – Markov Chains with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for AHL 4.19—transition Matrices – Markov Chains and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

HLPaper 1

A particle moves between three states, $A$ , $B$ , and $C$ , according to a Markov chain. The transition matrix is:

$T = 0.5 0.3 0.2 0.4 0.4 0.2 0.2 0.4 0.4$

Verify that $T$ is a valid transition matrix.

[2]

Find the probability that the particle is in state $B$ after two transitions, given it starts in state $A$ .

[3]

Determine the eigenvalues of $T$ .

[3]

Question 2

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 1 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 3 July is calm.

[2]

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

[4]

Question 3

HLPaper 1

The day-to-day lunch choice of a student is modelled by the transition matrix $T = (0.8 0.2 0.3 0.7)$ , where the states are Cafeteria and Food Truck in that order. The eigenvalues of $T$ are $1$ and $0.5$ . If the student chooses the Cafeteria on one day, the probability of choosing the Food Truck the next day is $0.2$ . If the student chooses the Food Truck, the probability of choosing the Cafeteria the next day is $0.3$ .

Determine an eigenvector associated with the eigenvalue $1$ . Present your result as $(a b)$ , where $a, b \in Z$ .

[3]

Using your result from the previous part, or by another method, calculate the stationary (long-term) probability that the student chooses the Cafeteria. Express your final answer as $\frac{p}{q}$ , where $p, q \in Z^{+}$ .

[2]

Question 4

HLPaper 1

The day-to-day travel choice of a commuter is modelled by the transition matrix $T = (0.75 0.25 0.15 0.85)$ , where the states are Bridge and Tunnel in that order. The eigenvalues of $T$ are $1$ and $0.6$ . If the commuter uses the Bridge on one day, the probability of using the Tunnel the next day is $0.25$ . If the commuter uses the Tunnel, the probability of using the Bridge the next day is $0.15$ .

Determine an eigenvector associated with the eigenvalue $1$ . Present your result as $(a b)$ , where $a, b \in Z$ .

[3]

Using your result from the previous part, or by another method, calculate the stationary (long-term) probability that the commuter uses the Bridge route. Express your final answer as $\frac{p}{q}$ , where $p, q \in Z^{+}$ .

[2]

Question 5

HLPaper 1

The status of a library copy of a popular novel at the end of each week is modelled by the transition matrix $T = (0.65 0.35 0.45 0.55)$ , where the states are On shelf and On loan in that order. The eigenvalues of $T$ are $1$ and $0.2$ . If the book is On shelf in one week, the probability of being On loan the next week is $0.35$ . If the book is On loan, the probability of being On shelf the next week is $0.45$ .

Determine an eigenvector associated with the eigenvalue $1$ . Present your result as $(a b)$ , where $a, b \in Z$ .

[3]

Using your result from the previous part, or by another method, calculate the stationary (long-term) probability that the book is On shelf. Express your final answer as $\frac{p}{q}$ , where $p, q \in Z^{+}$ .

[2]

IB Mathematics Applications & Interpretation AHL 4.19—transition Matrices – Markov Chains Questionbank

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IB Mathematics Applications & Interpretation AHL 4.19—transition Matrices – Markov Chains Questionbank

IB Mathematics Applications & Interpretation (AI) Topic AHL 4.19—transition Matrices – Markov Chains Questionbank

Unlock all 16 questions with a free account