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

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-2025-april/q1d

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-2025-april--q1d/

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



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



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



$$
P=\begin{bmatrix}0.60&0.40\\0.30&0.70\end{bmatrix}_{(B,C)}
$$



| Current → next | BestTV | CableCast | Row total |
| --- | --- | --- | --- |
| BestTV | 0.6 | 0.4 | 1 |
| CableCast | 0.3 | 0.7 | 1 |

### 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.](https://deep-learning-study-room.vercel.app/is2/fig/legacy25-cable.c15b41f741.svg)

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.