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

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.

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?

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

           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.

✓ Answer checked by Claude Opus 5.5 (high effort)

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