Question

This student never eats the same kind of food for 2 consecutive weeks. If she eats a Chinese restaurant one week, then she is three times as likely to have Greek as Italian food the next week. If she eats a Greek restaurant one week, then she is four times as likely to have Chinese as Italian food the next week. If she eats a Italian restaurant one week, then she is five times as likely to have Chinese as Greek food the next week.

Answer 1

The Question is incomplete. Here is the complete Question:

This student never eats the same kind of food for 2 consecutive weeks. If she eats a Chinese restaurant one week, then she is three times as likely to have Greek as Italian food the next week. If she eats a Greek restaurant one week, then she is equally likely to have Chinese as Italian food the next week. If she eats a Italian restaurant one week, then she is four times as likely to have Chinese as Greek food the next week.

Assume that state 1 is Chinese and that state 2 is Greek, and state 3 is Italian.

Find the transition matrix for this Markov process. P=[]

Answer:

        C        G        I

C      0        3/4     1/4

G     1/2      0        1/2

I      4/5     1/5       0

Step-by-step explanation:

In Markov chain/matrix, it is important to understand that it deals with the probability of the event in future given the present condition.

Let's say C = chinese, G = Greek, I = Italian.

The row of the transition matrix will represent FIRST WEEK , and the column will represent NEXT WEEK (future)

To make this simple and understandable, we build the matrix as shown below:

        C        G        I

C     C-C    C-G    C-I

G     G-C    G-G    G-I

I      I-C      I-G      I-I

note that C-C represent eating Chinese in one week, and also chinese in the next week, C-G represent eating Chinese in one week, and Greek in the next week, and so on...

- 'This student never eats the same kind of food for 2 consecutive weeks' tells us the node of C-C, G-G, I-I will be zero

- 'If she eats a Chinese restaurant one week, then she is three times as likely to have Greek as Italian food the next week' means C-G = 3/4 and C-I = 1/4. You should have understand this since C-C = 0 and total probability of each row is equal to 1.

- Similarly, the sentence 'If she eats a Greek restaurant one week, then she is equally likely to have Chinese as Italian food the next week' gives us G-C = 1/2 and G-I = 1/2

- Lastly 'If she eats a Italian restaurant one week, then she is four times as likely to have Chinese as Greek food the next week' tells us I-C = 4/5 and I-G =1/5

From these we construct the transition matrix based on the data given to us as follows:

        C        G        I

C      0        3/4     1/4

G     1/2      0        1/2

I      4/5     1/5       0

To confirm our answer, each row must have total probability of 1.