This student never eats the same kind of food for 2 consecutive weeks. If she eats a Chinese restaurant one week, then she is four 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 three 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. **Exploring Transition Matrices in Markov Processes**In this activity, you are tasked with finding the transition matrix \( P \) for a given Markov process. ### Understanding the Transition MatrixA transition matrix is a square matrix used to describe the transitions of a Markov chain. Each element \( p_{ij} \) in the matrix represents the probability of transitioning from state \( i \) to state \( j \). ### Matrix StructureThe matrix \( P \) provided here is shown as a 3x3 matrix, with each element being a placeholder for these probabilities. - **Rows and Columns**: Each row and column of the matrix corresponds to a state in the Markov process.- **Matrix Elements**: Each cell in the matrix is a probability value that the process will transition from one state to another.### Filling in the MatrixTo complete this task, calculate the probability of transitioning from each state \( i \) to each state \( j \) and fill these probabilities into the corresponding positions in the matrix. Remember that the sum of probabilities in each row must equal 1, which ensures that the transition from a given state to any other state is certain.**Challenge**: Analyze the given Markov process and determine each state’s transition probability to complete the matrix \( P \) accurately.

If X(t), t∈T is a stochastic process such that, given the value X(s), the value of X(t), t>s, do not depend on the values of X(u), u<s, then the process is said to be a Markov process. A definition of such a process is given by, If, for, t1< t2< . . . <tn<t Pra≤X(t)≤b X(t1)=x1, . . . , X(tn)=xn = Pra≤X(t)≤b X(tn)=xn The process X(t), t∈T is a Markov process. A discrete parameter Markov process is known as a Markov chain. The definition for transition matrix is given by, The transition probabilities pjk satisfy pjk≥0, ∑kpjk=1 for all j These matrix can be written in the matrix form P=p11p12p13...p21p22p23...p31p32p33............... This is called the transition probability matrix of the Markov chain.Assume that state 1 is Chinese , state 2 is Greek and state 3 is Italian. The resulting transition matrix P is 3×3 matrix P=p11p12p13p21p22p23p31p32p33 Here, the transition probabilities are pjk , where j represent the food eaten in the present week and k represent the food eaten in the next week and the total probability of each row is 1. That is p11 represent eating Chinese food in one week and eating Chinese food in the next week, p12 represent eating Chinese food in one week and eating Greek food in next week and p13 represent eating Chinese food in one week and eating Italian food in the next week and so on... Given that " The student never eats the same kind of food for 2 consecutive weeks", then the transition probabilities p11=p22=p33=0 And "If she eats a Chinese restaurant one week, then she is four times as likely to have Greek as Italian food the next week", then p12=45 and p13=15 ( since the total probability of each row is equal to one, p11+p12+p13=1 and also p11=0) And "If she eats a Greek restaurant one week, then she is likely to have chinese as Italian food the next week", then p21=12 and p23=12 ( since the total probability of each row is equal to one, p21+p22+p23=1 and also p22=0) And similarly " If she eats a Italian restaurant one week, then she is three times as likely to have Chinese as Greek food the next week" then p31=34 and p32=14( since the total probability of each row is equal to one, p31+p32+p33=1 and also p33=0) Thus the transition matrix for this Markov process is given by P=045151201234140