Solving Markov Chain Problem for Proportions in Areas A, B & C

[subopolois]

Homework Statement

i have a scenario which i have to find the proportion of time spent in each area by a person using markov chains. i was given a word problem, which i have put into a matrix and the question asks what the proportion of time is spent in each area A, B and C.

Homework Equations

na

The Attempt at a Solution

i have the final matrix as:
0 0.34 0.67
0 0 0.34
1 0.34 0

im stuck at how to determine how to do this, can someone help?

 

[Pere Callahan]

Find the stationary distribution of the Markov chain (that is an eigen vector to the transition matrix)

 

[subopolois]

Find the stationary distribution of the Markov chain (that is an eigen vector to the transition matrix)

we haven't been taught eigen vectors yet

 

[D H]

i have the final matrix as:
0 0.34 0.67
0 0 0.34
1 0.34 0

That is not a valid transition matrix. Each row must sum to 1.

 

[subopolois]

That is not a valid transition matrix. Each row must sum to 1.

sorry, i made a typo, its:
0 0.67 0.67
0 0 0.34
1 0.33 0

 

[D H]

That's still not valid. In fact, it's worse; now the first row is also invalid. The sum of each row must be identically one. What is the word problem that led to this matrix?

 

[subopolois]

heres the word problem:
A fox hunts in three territories A, B and C. He
never hunts in the same territory on two successive days.
If he hunts in A, then he hunts in C the next day. If he
hunts in B or C, he is twice as likely to hunt in A the next
day as in the other territory.

i only gave the scenerio, i don't need help with the actual questions, just how to get the initial transition mtrix. its just confusing for me reading the problem and transfering it to a mathematica matrix. i am going by the examples in the textbook in terms of the rows adding up to 1, in my textbook it has the columns adding to 1

 

[D H]

Sorry for the misdirection. Your corrected matrix is fine.

Suppose the probabilities that the fox at the n^th time step t_n is in state A, B, or C are P_A(t_n), P_B(t_n), and P_C(t_n). Let P(t_n) be the column vector formed from these individual state probabilities. The state probabilities at the next time step are given by the transition matrix S: P(t_n+1)=S×P(t_n).

The system is in steady-state if P(t_n+1)=P(t_n). Try to find this steady-state probability vector (the components must add to one).