---
title: "Nokia-Samsung Markov, two purchases"
subject: "Simulation and Modeling"
chapter: 5
key: "semester_8/simulation_and_modeling:markov-nokia-samsung"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/markov-nokia-samsung
site: BCT Engineering
---

# Nokia-Samsung Markov, two purchases

Simulation and Modeling · Chapter 5: Markov chains

- Asked in: 2070 Magh, 6 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-nokia-samsung
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-5

## Question

Explain Markov Chains? Given that chance of a Nokia user to buy Nokia 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 Nokia is 15%. What is the probability to buy Samsung after two purchase of a current Nokia user?

## Short answer

A Markov chain has the memoryless property: the next state depends only on the current state, with transition probabilities given in a row-stochastic matrix. For the Nokia-Samsung matrix, the probability that a current Nokia user buys Samsung after two purchases is 0.33 (33%).

## Answer

#### Markov chain

A **Markov chain** is a stochastic process with a finite set of states where the next state depends only on the current state (memoryless property). It is described by a transition matrix with non-negative entries and each row summing to 1, and the state distribution after n steps is p(n) = p(0) P^n. It is widely used for brand-switching, weather, queue and reliability models.

#### Numerical

States: N = Nokia, S = Samsung. "After two purchases" means two transitions, P^2.

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

Nokia row of P^2:

- To Nokia: 0.80 x 0.80 + 0.20 x 0.15 = 0.64 + 0.03 = 0.67
- To Samsung: 0.80 x 0.20 + 0.20 x 0.85 = 0.16 + 0.17 = **0.33**

Check by paths (Nokia to Samsung in two steps):

- N - N - S: 0.8 x 0.2 = 0.16
- N - S - S: 0.2 x 0.85 = 0.17
- Total = 0.33

**Answer**: a current Nokia user buys a Samsung after two purchases with probability **0.33 (33%)**.

## Check yourself

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

1. What is the defining (memoryless) property of a Markov chain?
   A. The next state depends on all previous states equally
   B. The next state depends only on the current state
   C. The next state is always the same as the current state
   D. The states must be infinite in number
   Answer: B. In a Markov chain the future depends only on the present state, not on how it was reached. (Key point: Memoryless property)

2. To find the Samsung probability after two purchases for a Nokia user, which matrix is used?
   A. P (one step)
   B. P^3
   C. P^2 (two steps)
   D. The sum P + P
   Answer: C. Two purchases means two transitions, so the Nokia row of P^2 is used (p(2) = p(0) P^2). (Key point: Meaning of two purchases)

3. Which pair of paths gives Nokia to Samsung in exactly two steps?
   A. N-N-S and N-S-S
   B. N-S-N and S-S-N
   C. N-N-N and N-S-N
   D. N-S-S and S-N-S
   Answer: A. Start at N and end at S: N-N-S (0.8 x 0.2) or N-S-S (0.2 x 0.85). (Key point: Path calculation)

4. What is the probability that a current Nokia user buys Samsung after two purchases?
   A. 0.67
   B. 0.20
   C. 0.37
   D. 0.33
   Answer: D. 0.8 x 0.2 + 0.2 x 0.85 = 0.16 + 0.17 = 0.33. The value 0.67 is the chance of Nokia. (Key point: Final result)

## Close variants set in other exams

- [Markov applications; Sony-Samsung 75/85](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Amarkov-sony-samsung-75-85) — Simulation and Modeling, chapter 5, asked 2×

## Related in Simulation and Modeling

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