Consider a mouse in the maze pictured below. The maze has a set of "rooms" $($the circles in the

picture$)$ connected by halls $($the lines in the picture$).$ Assume that we record the room that the

mouse is in each time it moves. Assume also that the mouse is equally likely to leave via any

of the halls leaving its current location. For example, if the mouse is currently in room $1$ then it

is equally likely to go next to room $2$ or room $6.$ Notice that there is a looping hallway leaving

room $4$ that return directly to room $4.$

a$.$ Develop a model that can be used to determine the long run expected fraction of the time

the mouse is in each room. Be sure to define a state space, any assumptions, and the one$-$

step state transition probability matrix, $P.$

b$.$ Solve for the steady state probabilities $(\text{'}$s