- 2074 Magh · 2×3 marks
Coke-Pepsi Markov model, 4th purchase
Short answer
Compute the four-step transition matrix P^4. (i) Probability a Pepsi user buys Coke in the 4th purchase is 0.5625. (ii) Probability a Pepsi user buys Pepsi is 0.4375. (iii) With 5250 Coke and 4950 Pepsi users initially, after four purchases about 4134 people buy Pepsi.
Design markov model and transition matrix for given data. Answer following questions based on the model. a. If a person purchase coke now the probability of purchase of coke next time is 80%. b. If a person purchase pepsi now the probability of purchasing pepsi next time is 70%. Problems: i. Find the probability of using coke for a current pepsi user in 4th purchases. ii. Find the probability for a person using pepsi now for a current pepsi purchaser after 4 purchases. iii. A population of 5250 (out of 10200) in a study of soft drink user uses coke now, find the population of purchasing pepsi in 4th purchases.
Assumption: "4th purchase" means four transitions from the current purchase, so P^4 is used. States: C = Coke, P = Pepsi.
Markov model
0.2
+----------->+
(C) (P)
0.8 loop 0.7 loop
+<-----------+
0.3
- Coke user: stays Coke 0.8, moves to Pepsi 0.2.
- Pepsi user: stays Pepsi 0.7, moves to Coke 0.3.
Transition matrix (rows = from, columns = to):
| From \ To | Coke | Pepsi |
|---|---|---|
| Coke | 0.8 | 0.2 |
| Pepsi | 0.3 | 0.7 |
Powers of P
- P^2: Coke row = [0.64 + 0.06, 0.16 + 0.14] = [0.70, 0.30]; Pepsi row = [0.24 + 0.21, 0.06 + 0.49] = [0.45, 0.55]
- P^3 = P^2 x P: Coke row = [0.70 x 0.8 + 0.30 x 0.3, 0.70 x 0.2 + 0.30 x 0.7] = [0.65, 0.35]; Pepsi row = [0.45 x 0.8 + 0.55 x 0.3, 0.45 x 0.2 + 0.55 x 0.7] = [0.525, 0.475]
- P^4 = P^3 x P: Coke row = [0.65 x 0.8 + 0.35 x 0.3, 0.65 x 0.2 + 0.35 x 0.7] = [0.625, 0.375]; Pepsi row = [0.525 x 0.8 + 0.475 x 0.3, 0.525 x 0.2 + 0.475 x 0.7] = [0.5625, 0.4375]
P^4 = [[0.625, 0.375], [0.5625, 0.4375]]
(i) Pepsi user, Coke in the 4th purchase
p(Pepsi to Coke) = 0.5625
(ii) Pepsi user, Pepsi in the 4th purchase
p(Pepsi to Pepsi) = 0.4375
(iii) Population of Pepsi buyers in the 4th purchase
Coke users now = 5250, so Pepsi users now = 10200 - 5250 = 4950.
Pepsi after 4 purchases = 5250 x p(C to P) + 4950 x p(P to P) = 5250 x 0.375 + 4950 x 0.4375 = 1968.75 + 2165.63 = 4134.4
About 4134 people will buy Pepsi (and 10200 - 4134 = 6066 will buy Coke).
✓ Answer checked by Claude Opus 5.5 (high effort)
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