Intelligent Systems II · TEE · mocks
Q1(d) Brand-switching Markov chain [5]
Final examination · 24 Nov 2025
100-mark timed practice using the actual Final 2025 paper. Q1 compulsory; any four of Q2–Q7. Worked solutions are below the questions.
Question. Users switch between Brand X and Brand Y each year: 25% of X users switch to Y and 75% stay; 40% of Y users switch to X and 60% stay. (a) Write the transition probability matrix. (b) Draw the state transition diagram. [5 marks]
Answer
Given
All transition values are given in the full question. State order: X, Y.
To find
The transition matrix and state diagram.
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.
Solution
Step 1. Convert every origin row to probabilities
| From / To | X | Y |
|---|---|---|
| X | 0.75 | 0.25 |
| Y | 0.4 | 0.6 |
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.
