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     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/
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