Consumers in Shelbyville have a choice of one of two fast food restaurants, Krus

Question

Consumers in Shelbyville have a choice of one of two fast food restaurants, Krustyas and McDonaldas. Both have trouble keeping customers. Of those who last went to Krustyas, 58% will go to McDonaldas next time, and of those who last went to McDonaldas, 76% will go to Krustyas next time. Find the transition matrix describing this situation. A customer goes out for fast food every Sunday, and just went to Krustyas. What is the probability that two Sundays from now she will go to McDonaldas? What is the probability that three Sundays from now she will go to McDonaldas? Suppose a consumer has just moved to Shelbyville, and there is a 42% chance that he will go to Krustyas for his first fast food outing What is the probability that his third fast food experience will be at Krustyas? Find the steady-state vector.

Explanation / Answer

Answer

Just for ease in presentation, we will use the following codes:

1. K for Krushtya's M for McDonalds

2. K1K2M3 for first visit to K, second visit to K and third visit to M etc to represent sequential visits

3. K2M3/M1 for 'given first visit to M' second visit is to K and third visit is to M

Also, since there are only two options, P(visit to one restaurant) = 1 - P(visit to the other restaurant)

With this codification,

Part (a)

Transition Probabilty Matrix

Part (bi)

We want P(-M3/K1) = P(K2M3/K1) + P(M2M3/K1)

= (0.42x0.58) + (0.58x0.24) = 0.3828 ANSWER

Part (bii)

We want P(- -M4/K1) = P(K2K3M4/K1) + P(K2M3M4/K1) + P(M2K3M4/K1) + P(M2M3M4/K1)

= (0.42x0.42X0.58) + (0.42X0.58x0.24) + (0.58X0.76X0.58) + (0.58X0.24X0.24 )= 0.449848 ANSWER

Part (C)

We want P(- -M3) = P(-M3/K1) + P(-M3/M1) = = P(K2M3/K1) + P(M2M3/K1) + P(K2M3/M1) + P(M2M3/M1)

= (0.42X0.58) + (0.58X0.24) + (0.76X0.58) + (0.24X0.24) = 0.8884 ANSWER

K M K 0.42 0.58 M 0.76 0.24

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