# Absorbing States Probability Problem

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

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

I have attached the solution to this problem for best formatting (also presented below). I hope you find this helpful.

RESPONSE:

Absorbing States Problem

Solution:

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.

BEST OF LUCK!

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