Markov chains: IB Mathematics: Applications & Interpretation questions with answers
Free Markov chains questions for IB Mathematics: Applications & Interpretation (SL & HL), HL only, part of Statistics & probability. Each question has the answer and the marking points an examiner would look for.
Each day is either sunny (S) or rainy (R). If today is sunny, the probability that tomorrow is sunny is 0.9. If today is rainy, the probability that tomorrow is sunny is 0.4.
(a)
Write down the transition matrix T, with columns representing today’s weather.
[2]
(b)
Given that today is sunny, find the probability that it is sunny in two days’ time.
[2]
(c)
Find the long-term proportion of sunny days.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.9 0.40.1 0.6
correct structure (M1) 0.9 0.40.1 0.6A1
2
(b)
0.850
T2(M1) 0.850 A1
2
(c)
0.800
steady state: Tv = v (M1) 0.800 A1
2
Question 2Structured · 14 marks
Each year 20% of the customers of company X move to company Y, and 20% of the customers of company Y move to company X. Initially X has 1500 customers and Y has 1200.
(a)
Write down a transition matrix T for the movement of customers.
[2]
(b)
Find the eigenvalues and corresponding eigenvectors of T.
[4]
(c)
Hence write down matrices P and D such that T = PDP−1.
[2]
(d)
Find an expression for the number of customers of X after n years.
[5]
(e)
Write down the number of customers X can expect in the long term.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.8 0.20.2 0.8
columns sum to 1 (M1) 0.8 0.20.2 0.8A1
2
(b)
λ = 1, λ = 0.6
det(T − λI) = 0 (M1) λ = 1, λ = 0.6 A1 eigenvector for 1: 0.20.2 (or a multiple) A1 eigenvector for 0.6: 1−1A1
4
(c)
P, D
P = 0.2 10.2 −1A1 D = 1 00 0.6A1
2
(d)
1350 + 150(0.6)n
use of Tn = PDnP−1M1 correct P−1A1 xn = 1350 + 150(0.6)nA2 form A + B(λ)nA1
5
(e)
1350
1350 A1
1
Question 3Structured · 5 marks
If it is sunny today, the probability it is sunny tomorrow is 0.8. If it is rainy today, the probability it is sunny tomorrow is 0.5.
(a)
Write down the transition matrix.
[2]
(b)
Find the long-term probability that a day is sunny.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
T
0.8 0.50.200 0.500A2
2
(b)
0.714
steady state: Ts = s (M1) equation (A1) 0.714 A1
3
Question 4Structured · 6 marks
Each day is either sunny (S) or rainy (R). If today is sunny, the probability that tomorrow is sunny is 0.7. If today is rainy, the probability that tomorrow is sunny is 0.3.
(a)
Write down the transition matrix T, with columns representing today’s weather.
[2]
(b)
Given that today is sunny, find the probability that it is sunny in two days’ time.
[2]
(c)
Find the long-term proportion of sunny days.
[2]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.7 0.30.3 0.7
correct structure (M1) 0.7 0.30.3 0.7A1
2
(b)
0.580
T2(M1) 0.580 A1
2
(c)
0.500
steady state: Tv = v (M1) 0.500 A1
2
Question 5Structured · 14 marks
Each year 20% of the customers of company X move to company Y, and 15% of the customers of company Y move to company X. Initially X has 1200 customers and Y has 1800.
(a)
Write down a transition matrix T for the movement of customers.
[2]
(b)
Find the eigenvalues and corresponding eigenvectors of T.
[4]
(c)
Hence write down matrices P and D such that T = PDP−1.
[2]
(d)
Find an expression for the number of customers of X after n years.
[5]
(e)
Write down the number of customers X can expect in the long term.
[1]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
0.8 0.150.2 0.85
columns sum to 1 (M1) 0.8 0.150.2 0.85A1
2
(b)
λ = 1, λ = 0.65
det(T − λI) = 0 (M1) λ = 1, λ = 0.65 A1 eigenvector for 1: 0.150.2 (or a multiple) A1 eigenvector for 0.65: 1−1A1
4
(c)
P, D
P = 0.15 10.2 −1A1 D = 1 00 0.65A1
2
(d)
1286 − 85.7(0.65)n
use of Tn = PDnP−1M1 correct P−1A1 xn = 1286 − 85.7(0.65)nA2 form A + B(λ)nA1
5
(e)
1286
1286 A1
1
Question 6Structured · 5 marks
If it is sunny today, the probability it is sunny tomorrow is 0.7. If it is rainy today, the probability it is sunny tomorrow is 0.5.
(a)
Write down the transition matrix.
[2]
(b)
Find the long-term probability that a day is sunny.
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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in