---
title: "Markov applications; Sony-Samsung 75/85"
subject: "Simulation and Modeling"
chapter: 5
key: "semester_8/simulation_and_modeling:markov-sony-samsung-75-85"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/markov-sony-samsung-75-85
site: BCT Engineering
---

# Markov applications; Sony-Samsung 75/85

Simulation and Modeling · Chapter 5: Markov chains

- Asked in: 2073 Magh, 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-sony-samsung-75-85
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-5

## Question

Write down the application of Markov chains? Given that chance of a Sony user to buy Sony at next purchase is 75% and that his next purchase will be Samsung is 25% and chance of a Samsung user to buy Samsung at next purchase is 85% and chance that his next purchase will be Sony is 15%. What is the probability to buy Sony user after three purchase of a current Samsung user?

## Short answer

Compute the three-step transition matrix P^3. The probability that a current Samsung user buys Sony after three purchases is 0.294 (29.4%).

## Answer

#### Applications of Markov chains

- Brand switching and market share prediction.
- Queue and inventory models in operations research.
- Weather forecasting, Google PageRank and web navigation.
- Reliability of machines (working, failed, repaired states).
- Speech recognition and text prediction (hidden Markov models).
- Random sampling methods (MCMC) and simulation of computer systems.

#### Numerical

**Assumption**: "after three purchases" means three transitions, P^3. States: Sony, Samsung.

P = [[0.75, 0.25], [0.15, 0.85]]

**P squared**

- Sony row: 0.75 x 0.75 + 0.25 x 0.15 = 0.6; 0.75 x 0.25 + 0.25 x 0.85 = 0.4
- Samsung row: 0.15 x 0.75 + 0.85 x 0.15 = 0.1125 + 0.1275 = 0.24; 0.15 x 0.25 + 0.85 x 0.85 = 0.0375 + 0.7225 = 0.76

P^2 = [[0.60, 0.40], [0.24, 0.76]]

**P cubed = P^2 x P** (only the Samsung row is needed for a Samsung user)

- Samsung to Sony: 0.24 x 0.75 + 0.76 x 0.15 = 0.18 + 0.114 = **0.294**
- Samsung to Samsung: 0.24 x 0.25 + 0.76 x 0.85 = 0.06 + 0.646 = 0.706

Check: 0.294 + 0.706 = 1.

**Answer**: the probability that a current Samsung user buys Sony after three purchases is **0.294 (29.4%)**.

## Check yourself

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

1. Which is an application of Markov chains?
   A. Brand switching and Google PageRank
   B. Sorting arrays
   C. Measuring resistance
   D. Compiling programs
   Answer: A. Markov chains model brand switching, PageRank, reliability and more. (Key point: Applications)

2. What is P with rows Sony, Samsung?
   A. [[0.75, 0.15], [0.25, 0.85]]
   B. [[0.75, 0.25], [0.15, 0.85]]
   C. [[0.25, 0.75], [0.85, 0.15]]
   D. [[0.75, 0.25], [0.85, 0.15]]
   Answer: B. Each row sums to 1. (Key point: Matrix)

3. Which row of P^2 is used to find Samsung to Sony in P^3?
   A. The Sony row [0.60, 0.40]
   B. The column sums
   C. The Samsung row [0.24, 0.76]
   D. Both rows averaged
   Answer: C. P^3 row Samsung = Samsung row of P^2 times P. (Key point: Row needed)

4. What is the probability a Samsung user buys Sony after three purchases?
   A. 0.706
   B. 0.240
   C. 0.600
   D. 0.294
   Answer: D. 0.24x0.75 + 0.76x0.15 = 0.18 + 0.114 = 0.294. (Key point: Result)

## Close variants set in other exams

- [Nokia-Samsung Markov, two purchases](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-nokia-samsung) — Simulation and Modeling, chapter 5, asked 2×

## Related in Simulation and Modeling

- [Sony-Samsung Markov, 80/85, 60% users](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-sony-samsung-80-85) — Simulation and Modeling, chapter 5
- [Honda-Yamaha Markov, key features](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-honda-yamaha) — Simulation and Modeling, chapter 5
- [Coke-Pepsi Markov model, 4th purchase](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-coke-pepsi-4th-purchase) — Simulation and Modeling, chapter 5
- [Pepsi user buying Coke, third purchase](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-pepsi-coke-third) — Simulation and Modeling, chapter 5