Intelligent Systems II · TEE · mocks

Q5(b) Hotel booking channels (Markov chain) [10]

Re-examination · 31 Jan 2026

100-mark timed practice using the actual Re-exam 2026 paper. Q1 compulsory; any four of Q2–Q7. Worked solutions are below the questions.

Question. Bookings come via Website (W), App (A) or Counter (C). From W: 50% W, 30% A, 20% C. From A: 40% A, 40% W, 20% C. From C: 20% W, 30% A, 50% C. 1. Develop the transition matrix. 2. Interpret retention and gain/loss. 3. Draw the diagram. 4. If now 40% W, 35% A, 25% C, estimate next week. 5. Which channel grows or shrinks in the long term? [10 marks]

Answer

Given

All transition values are given in the full question. State order: W, A, C. Initial vector [0.4, 0.35, 0.25].

To find

The matrix, diagram, next-step vector and long-run comparison.

Formula and why it applies

Rows mean current state; columns mean next state. For each destination, multiply each origin amount by its transition probability, then add.

x_{next}=x_{now}P\qquad\pi=\pi P,\quad\sum_i\pi_i=1

Solution

Step 1. Convert every origin row to probabilities

From / To	W	A	C
W	0.5	0.3	0.2
A	0.4	0.4	0.2
C	0.2	0.3	0.5

Each row sums to 1 because it lists all possible destinations for that origin. A percentage is divided by 100: 25%=.25.

Step 2. Interpret entries and draw the transitions

Diagonal entries are the fractions staying at their current state; off-diagonal entries are fractions moving to a different state. Each arrow uses its origin row and destination column. These fractions have different origin populations, so adding raw column probabilities does not give the number arriving.

Swipe across the diagram to read its labels ↔

Diagram: Each arrow is labelled with the probability of that transition.. Labels: Weekly booking channel: Website, App, Counter | 0.3 | 0.2 | 0.4 | 0.2 | 0.2 | 0.3 | 0.5 | 0.4 | 0.5 | W | A | C

Each arrow is labelled with the probability of that transition.

Step 3. Next step: collect all contributions to W

Origin	Current count/share	Transition to W	Contribution
W	0.4	0.5	0.4 × 0.5 = 0.2
A	0.35	0.4	0.35 × 0.4 = 0.14
C	0.25	0.2	0.25 × 0.2 = 0.05

W_{new}=0.2+0.14+0.05=0.39

Step 4. Next step: collect all contributions to A

Origin	Current count/share	Transition to A	Contribution
W	0.4	0.3	0.4 × 0.3 = 0.12
A	0.35	0.4	0.35 × 0.4 = 0.14
C	0.25	0.3	0.25 × 0.3 = 0.075

A_{new}=0.12+0.14+0.075=0.335

Step 5. Next step: collect all contributions to C

Origin	Current count/share	Transition to C	Contribution
W	0.4	0.2	0.4 × 0.2 = 0.08
A	0.35	0.2	0.35 × 0.2 = 0.07
C	0.25	0.5	0.25 × 0.5 = 0.125

C_{new}=0.08+0.07+0.125=0.275

Step 6. Stationarity: find Counter first

c=0.2w+0.2a+0.5c

Since w+a+c=1, replace w+a with 1−c.

c=0.2(1-c)+0.5c=0.2+0.3c

0.7c=0.2\quad c=2/7

Step 7. Find App in terms of Website

a=0.3w+0.4a+0.3c

0.6a=0.3w+0.3c\quad a=0.5w+0.5c

Step 8. Use the sum to isolate Website

w+(0.5w+0.5c)+c=1

1.5w=1-1.5c=1-3/7=4/7

w=(4/7)/1.5=8/21

a=1-w-c=1-8/21-6/21=7/21=1/3

Step 9. Compare the stationary shares with the initial shares

State	Initial share	Stationary share	Change in share	Conclusion
W	0.4	0.380952381	-0.019047619	Loses
A	0.35	0.333333333	-0.016666667	Loses
C	0.25	0.285714286	0.035714286	Gains

Compare like with like: divide counts by their total before comparing with stationary probabilities. “Stationary” means one more step leaves the shares unchanged.

Final answer. Next-step vector [0.39, 0.335, 0.275]. Stationary shares ≈ [0.380952381, 0.333333333, 0.285714286]. W loses, A loses, C gains.