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Markov applications; Sony-Samsung 75/85

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%).

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?

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%).

✓ Answer checked by Claude Opus 5.5 (high effort)

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