---
title: "Sony-Samsung Markov, 80/85, 60% users"
subject: "Simulation and Modeling"
chapter: 5
key: "semester_8/simulation_and_modeling:markov-sony-samsung-80-85"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/markov-sony-samsung-80-85
site: BCT Engineering
---

# Sony-Samsung Markov, 80/85, 60% users

Simulation and Modeling · Chapter 5: Markov chains

- Asked in: 2074 Bhadra, 4+2 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-80-85
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-5

## Question

Given that chance of a Sony user to buy Sony at next purchase is 80% and that his next purchase will be Samsung is 20% 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 after three purchase of a current Samsung user? If 60% user uses Sony today, what percentage of user uses Samsung after three purchase?

## Short answer

Compute the three-step transition matrix P^3. The probability a Samsung user buys Sony after three purchases is 0.3109 (31.1%). With 60% Sony users initially, after three purchases 52.4% use Samsung.

## Answer

**Assumption**: each purchase is one transition of a Markov chain, so "after three purchases" means three transitions, P^3. States: 1 = Sony, 2 = Samsung.

#### Transition matrix

```text
           Sony   Samsung
 Sony   [  0.80    0.20  ]
 Samsung[  0.15    0.85  ]
```

P = [[0.80, 0.20], [0.15, 0.85]]

#### P squared

- Row Sony: 0.8 x 0.8 + 0.2 x 0.15 = 0.67; 0.8 x 0.2 + 0.2 x 0.85 = 0.33
- Row Samsung: 0.15 x 0.8 + 0.85 x 0.15 = 0.2475; 0.15 x 0.2 + 0.85 x 0.85 = 0.7525

P^2 = [[0.67, 0.33], [0.2475, 0.7525]]

#### P cubed = P^2 x P

- Row Sony: 0.67 x 0.8 + 0.33 x 0.15 = 0.5855; 0.67 x 0.2 + 0.33 x 0.85 = 0.4145
- Row Samsung: 0.2475 x 0.8 + 0.7525 x 0.15 = 0.198 + 0.112875 = 0.310875; 0.2475 x 0.2 + 0.7525 x 0.85 = 0.0495 + 0.639625 = 0.689125

P^3 = [[0.5855, 0.4145], [0.3109, 0.6891]] (rounded to 4 decimals)

#### (a) Samsung user buys Sony after three purchases

p21(3) = **0.3109** (about 31.1%).

#### (b) 60% use Sony today

Initial vector p(0) = [0.6, 0.4]. Then p(3) = p(0) P^3.

- Sony: 0.6 x 0.5855 + 0.4 x 0.310875 = 0.3513 + 0.12435 = 0.47565
- Samsung: 0.6 x 0.4145 + 0.4 x 0.689125 = 0.2487 + 0.27565 = 0.52435

After three purchases about **52.4%** of users use Samsung (and 47.6% use Sony). The two add up to 1, which checks the working.

## Check yourself

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

1. What does 'after three purchases' mean here?
   A. Three transitions, so P^3 is used
   B. One transition, P
   C. Two transitions, P^2
   D. Four transitions, P^4
   Answer: A. Each purchase is one step of the chain. (Key point: Interpretation)

2. What is the matrix with rows Sony, Samsung?
   A. [[0.80, 0.15], [0.20, 0.85]]
   B. [[0.80, 0.20], [0.15, 0.85]]
   C. [[0.20, 0.80], [0.85, 0.15]]
   D. [[0.80, 0.20], [0.85, 0.15]]
   Answer: B. Each row is a 'from' state and sums to 1. (Key point: Transition matrix)

3. What is the probability a Samsung user buys Sony after three purchases?
   A. 0.5855
   B. 0.6891
   C. 0.3109
   D. 0.1500
   Answer: C. This is the Samsung-to-Sony entry of P^3 = 0.3109. (Key point: Samsung to Sony)

4. With 60% Sony today, how is the distribution after three purchases found?
   A. p(3) = P^3 [0.6, 0.4], giving 47.6% Samsung
   B. p(3) = [0.6, 0.4] P, giving 52.4% Samsung
   C. p(3) = [0.6, 0.4] P^3, giving 41.5% Samsung
   D. p(3) = [0.6, 0.4] P^3, giving about 52.4% Samsung
   Answer: D. Sony 0.4757 and Samsung 0.5244, and they add to 1. (Key point: Initial vector)

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