Rat and cat survival with independent Markov chain transitions

[Andrusko]

Homework Statement

Rat and Cat move between room 1 and 2 using different paths. Their motions are governed by their respective transition matrices:

[0.9, 0.1 ; 0.2, 0.8] [0.6, 0.4 ; 0.3, 0.7]

(semi colon is a new line in the matrix, like in matlab)

If they are ever in the same room, cat eats rat. What is the probability that rat will survive? How long on average will he survive? Hint: denote the state (i,j) where i is the location of the rat and j is the location of the cat.

Homework Equations

No idea what is even relevant to the problem.

The Attempt at a Solution

Well, I haven't got the foggiest idea where to start. I think that the Markov chains are independent of one another and haven't got a clue how to deal with that, because I've never seen a problem like this one before.

The hint sort of suggests to me that you only need one transition matrix, so my only idea is that you multiply the matrices together.

Any help appreciated.

 

[TMFKAN64]

If you denote the states (i,j), how many states are possible?

What would the size of your transition matrix be, given this number of states?

If you are in state (i,j), what would the probability of a transition to (k,l) be?

Once you have created the correct transition matrix, the rest of the problem should follow.