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Honda-Yamaha Markov, key features

Short answer

Compute the three-step transition matrix P^3. The probability that a current Honda user buys a Yamaha after three purchases is 0.525 (52.5%).

What are the key features of Markov chain? Given that chance of a Honda Bike user to buy Honda Bike at next purchase is 70% and that his next purchase will be Yamaha Bike is 30% and chance of a Yamaha Bike user to buy Yamaha Bike at next purchase is 80% and chance that his next purchase will be Honda Bike is 20%. What is the probability to buy Yamaha Bike after three purchase of a current Honda Bike user?

Key features of a Markov chain

  1. A finite set of states (here Honda and Yamaha).
  2. Memoryless property: the next state depends only on the present state.
  3. The transition matrix is square, entries lie between 0 and 1, and each row sums to 1.
  4. Probabilities stay constant with time (homogeneous chain).
  5. n-step transition probabilities are the entries of P^n, and p(n) = p(0) P^n.
  6. A regular chain reaches a unique steady-state distribution that does not depend on the starting state.

Numerical

Assumption: "after three purchases" means three transitions, P^3. States: H = Honda, Y = Yamaha.

P = [[0.7, 0.3], [0.2, 0.8]]

P squared

  • Honda row: 0.7 x 0.7 + 0.3 x 0.2 = 0.55; 0.7 x 0.3 + 0.3 x 0.8 = 0.45
  • Yamaha row: 0.2 x 0.7 + 0.8 x 0.2 = 0.30; 0.2 x 0.3 + 0.8 x 0.8 = 0.70

P^2 = [[0.55, 0.45], [0.30, 0.70]]

P cubed = P^2 x P, Honda row:

  • Honda to Honda: 0.55 x 0.7 + 0.45 x 0.2 = 0.385 + 0.09 = 0.475
  • Honda to Yamaha: 0.55 x 0.3 + 0.45 x 0.8 = 0.165 + 0.36 = 0.525

Answer: probability that a current Honda user buys a Yamaha after three purchases = 0.525 (52.5%).

(Long-run share, for interest: pi(Yamaha) = 0.3 / (0.3 + 0.2) = 0.6.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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