An Absorbing States Problem

Date: 11/28/2000 at 18:30:09
From: Nichole Goulding
Subject: Absorbing States

A mouse, after being placed in one of 4 rooms, will search for cheese 
in that room. If unsuccessful, after one minute it will exit to 
another room by selecting a door at random. (All the rooms connect to 
each other.) A mouse entering the room with the cheese will remain in 
that room. If the mouse begins in room 3, what is the probability that 
it will get trapped in the long run? Are there any absorbing states?

I know that you have to set up a transition matrix, but I am having 
trouble after that. Could you help me?

Date: 11/28/2000 at 19:32:23
From: Doctor Anthony
Subject: Re: Absorbing States

You have to assume that the cheese is in a particular room, say 
Room 1.

                              FROM
                Room(1)  Room(2)  Room(3)  Room(4)
        Room(1)[  1       1/3      1/3      1/3 ]
    TO  Room(2)[  0        0       1/3      1/3 ]
        Room(3)[  0       1/3       0       1/3 ]
        Room(4)[  0       1/3      1/3       0  ]

This is the transition matrix. Note that the sum of each column is 1.

As a probability matrix it ALWAYS has an eigenvector equal to 1 and 
the corresponding eigenvector gives the steady state of the system. 
This steady state will also be the shape of the columns of the matrix 
if it were raised to power infinity. The eigenvector in this case 
turns out to be

     [1]
     [0]
     [0]
     [0]

So if you take higher and higher powers of this matrix it converges to

     [ 1   1   1   1 ]
     [ 0   0   0   0 ]
     [ 0   0   0   0 ]
     [ 0   0   0   0 ]

which means that wherever you start you will always end up in Room 1. 
This is to be expected since Room 1 is an absorbing state - once 
there you never leave.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/