a Suppose the mouse is initially placed in room 1 What is the probability the

A suppose the mouse is initially placed in room 1

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(a) Suppose the mouse is initially placed in room 1. What is the probability the mousewill be in room 9 one minute later?
{u}whereu=14314353431431.Thus, the nullspace ofP-Iis one-dimensional and everything in the nullspace is amultiple ofu. Unfortunately,uis not a probability vector since the sum of its entriesis11. But, if we multiplyuby111, then the result is a probability vector which is inthe nullspace ofP-I. Therefore, the steady state vector forPis111u=111433111433533433111433111.Thus, after many observations, this vector gives the probability that the mouse will beobserved in any given room. Notice that each room’s probability depends only on thenumber of doors ito that room (i.e., the steady state probability for any room isn+111wherenis the number of doors).

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