Intelligent Systems II · TEE · papers

Q5(b) Brand-switching transition matrix [10]

Final examination · 17 Apr 2024

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

Question. Q5(b) [10 marks · 2024-04-17]

Two cold-drink brands A and B are sold. An A buyer has 80% chance of buying A next. A B buyer has 90% chance of buying B next. (I) Develop the transition matrix. (II) Interpret retention and loss, and retention and gain. (III) Draw the transition diagram.

Answer

Given

A→A=.80; B→B=.90. Two exhaustive states A and B.

To find

Transition matrix, diagram, and correct flow interpretation.

Formula and why it applies

Every row lists all possible destinations from one origin; its probabilities sum to one. Missing switch probability is 1 minus retention.

P_{ij}=P(\text{next}=j\mid\text{current}=i)

Solution

Step 1. Find the two switch probabilities

P_{AB}=1-0.8=0.2,\quad P_{BA}=1-0.9=0.1

Step 2. Put origins in rows

P=\begin{bmatrix}0.8&0.2\\0.1&0.9\end{bmatrix}

Swipe across the diagram to read its labels ↔

Diagram: Redrawn transition diagram. Each arrow goes from the current state to the next state; labels are probabilities.. Labels: legacy24-brands | 0.8 | 0.2 | 0.1 | 0.9 | A | B

Redrawn transition diagram. Each arrow goes from the current state to the next state; labels are probabilities.

Step 3. Interpret retention, loss and gain with consistent populations

A retains 80% of its own current buyers and loses 20% to B. B retains 90% of its own buyers and loses 10% to A. A gains 10% of B’s current buyers; B gains 20% of A’s current buyers. These percentages have different denominators.

A_{next}=0.8A+0.1B,\quad B_{next}=0.2A+0.9B

For an illustrative 100 buyers of each brand, next counts are A=80+10=90 and B=20+90=110. Without initial counts, “A loses 10% overall” is not justified.

Final answer. P=[[0.8,0.2],[0.1,0.9]], with the four labeled arrows above.