Final answer:
In this situation, we can find the stochastic matrix, calculate probabilities of going to McDonald's in future visits, determine the probability of visiting Krusty's on the third visit, and find the steady-state vector for the Markov chain.
Step-by-step explanation:
(a) To find the stochastic matrix, we need to determine the probabilities of transitioning from one state to another. Let's label Krusty's as state 1 and McDonald's as state 2. Based on the given information, the transition probabilities are:
P(1 -> 1) = 1 - 0.65 = 0.35
P(1 -> 2) = 0.65
P(2 -> 1) = 0.84
P(2 -> 2) = 1 - 0.84 = 0.16
Therefore, the stochastic matrix is:
Krusty's 0.35 0.84
McDonald's 0.65 0.16
(b) i. To find the probability that two Sundays from now the customer will go to McDonald's, we need to calculate the 2-step transition probability from Krusty's to McDonald's. We can raise the stochastic matrix to the power of 2 and find the entry in the first row and second column: 0.35 * 0.84 + 0.65 * 0.16 = 0.411.
ii. To find the probability that three Sundays from now the customer will go to McDonald's, we raise the stochastic matrix to the power of 3 and find the entry in the first row and second column: 0.35 * (0.84 * 0.35 + 0.16 * 0.65) + 0.65 * (0.84 * 0.16 + 0.16 * 0.65) = 0.3623.
(c) To find the probability that the consumer's third fast food experience will be at Krusty's, we need to calculate the 3-step transition probability from Krusty's to Krusty's. We can raise the stochastic matrix to the power of 3 and find the entry in the first row and first column. Using the same method as in part b, we get a probability of approximately 0.3819.
(d) To find the steady-state vector for this Markov chain, we need to find the eigenvector corresponding to the eigenvalue 1. We can solve this by setting up a system of equations where each component of the eigenvector is multiplied by the corresponding entry of the stochastic matrix. Solving the system, we find that the steady-state vector is approximately [0.58, 0.42].