Question

In a small town there are two places to eat: 1) a Chinese restaurant and 2) a pizza place. Everyone in town eats dinner at one of the these two places or eats dinner at home.

Assume the 20% of those who eat in the Chinese restaurant go to the pizza place the next time and 40% eat at home. From those who eat at the pizza place, 50% go to the Chinese restaurant and 30% eat at home the next time.  From those who eat at home, 20% go to the Chinese restaurant and 40% to the pizza place next time. We call this situation a system. This system can be modeled as a discrete-time Markov chainwith three states.

1. Define the three states and write down the one-step transition probability matrix.
2. If a family has decided to eat dinner in the Chinese restaurant with the probability of 0.05 or in the pizza place with the probability of 0.15 today, what is the probability that this family will eat dinner at home in two days
3. If a lady is having pizza today, how long will it take for her to have pizza again
4. Given that a man has been eating at the Chinese restaurant today and yesterday, on average, how long will it take for him to eat at the pizza place for the first time?
5. If a family plans to eat at home on Sunday, what is the probability that they will eat Chinese on Tuesday (same week) for the first timethen they will eat Pizza on Friday (same week) for the first time?

Answer 1

a. Let,

C - Eat at Chinese Restaurant,

P - Eat at Pizza Place

H - Eat at home

be the 3 states. Using shorthand , we have

Then we know that, . Therefore,

Also, we know that, . Therefore,

Also, we know that.. Therefore,

The probability matrix can then be written as, (C - first column and row, P - second column and row, H - third column and row)

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b. Given, Therefore,

Therefore, the probability vector,

Then, after two days, the probability vector will be,

Thus, the probability that the family will eat dinner at home in two days is 0.348.

c. Given, . We want to find,

, similarly, we define

Then ,

Now, . Therefore,

Similarly, we can construct,

Collecting the above three equations and rearranging their terms gives the following 3 equations

In matrix form

The solution to this equation is

Therefore, days.

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d. Given, and . Then we need to find, by Markov Property.

by Markov Property

Using similar reasoning as in Part (c), we get,

Rearranging, we get the matrix form

The solution to this equation is,

. Therefore, days.

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only 4 parts can be solved in 1 question.