---
title: "Honda-Yamaha Markov, key features"
subject: "Simulation and Modeling"
chapter: 5
key: "semester_8/simulation_and_modeling:markov-honda-yamaha"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/markov-honda-yamaha
site: BCT Engineering
---

# Honda-Yamaha Markov, key features

Simulation and Modeling · Chapter 5: Markov chains

- Asked in: 2072 Ashwin, 2+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/markov-honda-yamaha
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-5

## Question

What are the key features of Markov chain? Given that chance of a Honda Bike user to buy Honda Bike at next purchase is 70% and that his next purchase will be Yamaha Bike is 30% and chance of a Yamaha Bike user to buy Yamaha Bike at next purchase is 80% and chance that his next purchase will be Honda Bike is 20%. What is the probability to buy Yamaha Bike after three purchase of a current Honda Bike user?

## Short answer

Compute the three-step transition matrix P^3. The probability that a current Honda user buys a Yamaha after three purchases is 0.525 (52.5%).

## Answer

#### Key features of a Markov chain

1. A finite set of states (here Honda and Yamaha).
2. **Memoryless property**: the next state depends only on the present state.
3. The transition matrix is square, entries lie between 0 and 1, and each row sums to 1.
4. Probabilities stay constant with time (homogeneous chain).
5. n-step transition probabilities are the entries of P^n, and p(n) = p(0) P^n.
6. A regular chain reaches a unique steady-state distribution that does not depend on the starting state.

#### Numerical

**Assumption**: "after three purchases" means three transitions, P^3. States: H = Honda, Y = Yamaha.

P = [[0.7, 0.3], [0.2, 0.8]]

**P squared**

- Honda row: 0.7 x 0.7 + 0.3 x 0.2 = 0.55; 0.7 x 0.3 + 0.3 x 0.8 = 0.45
- Yamaha row: 0.2 x 0.7 + 0.8 x 0.2 = 0.30; 0.2 x 0.3 + 0.8 x 0.8 = 0.70

P^2 = [[0.55, 0.45], [0.30, 0.70]]

**P cubed = P^2 x P**, Honda row:

- Honda to Honda: 0.55 x 0.7 + 0.45 x 0.2 = 0.385 + 0.09 = 0.475
- Honda to Yamaha: 0.55 x 0.3 + 0.45 x 0.8 = 0.165 + 0.36 = **0.525**

**Answer**: probability that a current Honda user buys a Yamaha after three purchases = **0.525 (52.5%)**.

(Long-run share, for interest: pi(Yamaha) = 0.3 / (0.3 + 0.2) = 0.6.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is a key feature of a Markov chain?
   A. Memoryless property: the next state depends only on the present state
   B. It depends on the entire history
   C. Rows of the matrix sum to zero
   D. Probabilities change at every step
   Answer: A. This is the Markov property. (Key point: Key feature)

2. How is the distribution after n steps found?
   A. p(n) = n P
   B. p(n) = p(0) P^n
   C. p(n) = p(0) + nP
   D. p(n) = P / n
   Answer: B. n-step transition probabilities are entries of P^n. (Key point: n-step probabilities)

3. What is P with rows Honda, Yamaha?
   A. [[0.7, 0.2], [0.3, 0.8]]
   B. [[0.3, 0.7], [0.8, 0.2]]
   C. [[0.7, 0.3], [0.2, 0.8]]
   D. [[0.7, 0.3], [0.8, 0.2]]
   Answer: C. Honda: 0.7 stay, 0.3 to Yamaha; Yamaha: 0.8 stay, 0.2 to Honda. (Key point: Matrix)

4. What is the probability a Honda user buys Yamaha after three purchases?
   A. 0.475
   B. 0.450
   C. 0.600
   D. 0.525
   Answer: D. Honda-to-Yamaha in P^3 = 0.55x0.3 + 0.45x0.8 = 0.525. (Key point: Result)

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