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Nokia-Samsung Markov, two purchases

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%).

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?

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%).

✓ Answer checked by Claude Opus 5.5 (high effort)

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