Given that a person's last cola purchase was Coke, there is a $90 \%$ chance that his/her next cola purchase will also be Coke. If a person's last cola purchase was Pepsi, there is an $80 \%$ chance that his/her next cola purchase will also be Pepsi. Given that a person is currently a Pepsi purchaser, what is the probability that s/he will purchase Pepsi four purchases from now? 0.124 0.351 0.698 0.493 0.215
Solution
Step 1 ：This problem can be modeled as a Markov Chain with two states: Coke and Pepsi. The transition probabilities are given in the problem statement.
Step 2 ：We are asked to find the probability that a person will purchase Pepsi four purchases from now given that the person is currently a Pepsi purchaser. This is equivalent to finding the (1,1) entry of the 4th power of the transition matrix.
Step 3 ：The transition matrix P is given by \[P = \begin{bmatrix} 0.9 & 0.1 \\ 0.2 & 0.8 \end{bmatrix}\]
Step 4 ：The 4th power of the transition matrix P is given by \[P_4 = \begin{bmatrix} 0.7467 & 0.2533 \\ 0.5066 & 0.4934 \end{bmatrix}\]
Step 5 ：The (1,1) entry of the 4th power of the transition matrix is 0.4934, which is the probability that a person will purchase Pepsi four purchases from now given that the person is currently a Pepsi purchaser.
Step 6 ：Final Answer: The probability that a person will purchase Pepsi four purchases from now given that the person is currently a Pepsi purchaser is \(\boxed{0.493}\).
