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

Absorbing States Problem


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

Room(1) ...

Solution Summary

This solution provides the step-by-step solution of how to solve the absorbing states problem using a transition matrix.