Intelligent Systems II · TEE · papers

Q1(d) Cable-company Markov matrix and diagram [5]

Final / re-examination · 28 Apr 2025

Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.

Question. Q1(d) [5 marks · 2025-04-28]

A city is served by two cable TV companies, BestTV and CableCast. Due to their aggressive sales tactics, each year 40% of BestTV customers switch to CableCast; the other 60% of BestTV customers stay with BestTV. On the other hand, 30% of the CableCast customers switch to Best TV. The two states in this example are BestTV and CableCast.

  • Express the information above as a transition matrix which displays the probabilities of going from one state into another state.
  • Draw the State diagram for the above information.

Answer

Given

Order the states as B=BestTV and C=CableCast.

To find

The one-year transition matrix and all directed state transitions.

Formula and why

P_{ij}=P(\text{next state}=j\mid\text{current state}=i),\qquad\sum_jP_{ij}=1LaTeX: P_{ij}=P(\text{next state}=j\mid\text{current state}=i),\qquad\sum_jP_{ij}=1

Rows mean current state; columns mean next state. Each row includes every possible destination.

Step 1. Complete the missing stay probability

BestTV stays =1−0.40=0.60. CableCast stays =1−0.30=0.70.

Step 2. Place each probability in its row

P=\begin{bmatrix}0.60&0.40\\0.30&0.70\end{bmatrix}_{(B,C)}LaTeX: P=\begin{bmatrix}0.60&0.40\\0.30&0.70\end{bmatrix}_{(B,C)}
Current → nextBestTVCableCastRow total
BestTV0.60.41
CableCast0.30.71

Step 3. Draw the state diagram

Swipe across the diagram to read its labels ↔
BestTV → CableCast:0.40; CableCast → BestTV:0.30. Self-loops:0.60 and 0.70.
Diagram: BestTV → CableCast:0.40; CableCast → BestTV:0.30. Self-loops:0.60 and 0.70.. Labels: Cable-company transition probabilities | BestTV | CableCast | 0.40 | 0.30 | 0.60 | 0.70
BestTV → CableCast:0.40; CableCast → BestTV:0.30. Self-loops:0.60 and 0.70.