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Coke-Pepsi Markov diagram, 3rd purchase

Short answer

Using the transition matrix P and computing P^3, a current Coke purchaser becomes a Pepsi purchaser on the third purchase with probability 0.525. If 55% use Coke today, after three purchases about 41.9% use Coke, from p(0) times P^3.

Given figure shows Coke and Pepsi purchaser (state diagram: Coke to Coke 70%, Coke to Pepsi 30%, Pepsi to Pepsi 80%, Pepsi to Coke 20%). a) If currently Coke purchaser, what is the probability of Pepsi purchaser in 3rd purchase? b) If 55% of people use Coke today, what percentage of people will use Coke after 3 purchases?

From the diagram the transition matrix is (states Coke, Pepsi):

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

Assumption: "3rd purchase" means three transitions, so P^3 is used.

P squared

  • Coke 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
  • Pepsi 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 cubed = P^2 x P

  • Coke row: [0.55 x 0.7 + 0.45 x 0.2, 0.55 x 0.3 + 0.45 x 0.8] = [0.475, 0.525]
  • Pepsi row: [0.30 x 0.7 + 0.70 x 0.2, 0.30 x 0.3 + 0.70 x 0.8] = [0.350, 0.650]

P^3 = [[0.475, 0.525], [0.350, 0.650]]

(a) Coke user now, Pepsi in the 3rd purchase

p(Coke to Pepsi) in P^3 = 0.525

(b) 55% use Coke today

p(0) = [0.55, 0.45]; p(3) = p(0) P^3.

  • Coke: 0.55 x 0.475 + 0.45 x 0.35 = 0.26125 + 0.1575 = 0.41875
  • Pepsi: 0.55 x 0.525 + 0.45 x 0.65 = 0.28875 + 0.2925 = 0.58125

After three purchases about 41.9% of people use Coke (and 58.1% use Pepsi).

✓ Answer checked by Claude Opus 5.5 (high effort)

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