The probability of the mouse going into any of the rooms is 1/3.

denote your 3 rooms by i = 1,2,3 and the starting(initial) point by i=0.

let E(i) be the expected time till the mouse finds cheese if it has arrived at i.

now set up your equations as:

try to figure out your equations for E(2) and E(3) and using the four equations, you can find E(0)