Chapter 8, Problem 12RE

Mouse in a House Figure 1 gives the layout of a house with four rooms connected by doors. Room I contains a mousetrap, and room II contains cheese. A mouse, after being placed in one of the rooms, will search for cheese; if unsuccessful after one minute, it will exit to another room by selecting one of the doors at random. (For instance, if the mouse is in room III, after one minute, it will go to room II with probability 13 and to room IV with probability 23.) A mouse entering room I will be trapped and therefore no longer move. Also, a mouse entering room II will remain in that room.

Figure 1

a. Set up the $4\times 4$ absorbing stochastic matrix that describes this situation.

b. If a mouse begins in room IV, what is the probability that it will find the cheese after 2 minutes?

c. If a mouse begins in room IV, what is the probability that it will find the cheese in the long run?

d. For a mouse beginning in room III, determine the expected number of minutes that will elapse before the mouse either finds the cheese or is trapped.