A mouse roams between the living room and the kitchen of a house. However, a cat also lives in the house, so the mouse is at great risk. For simplicity, we'll assume that the mouse can be at these 3 different places: in the living room, in the kitchen, or in the cat. Here are the transition probabilities:

If the mouse is in the living room, then one hour later the probability of being in the living room is 0.41 and the probability of being in the kitchen is 0.54. So the probability is 0.05 that the mouse will have been eaten by the cat. If the mouse is in the kitchen, then one hour later the probability is 0.71 that the mouse will be in the kitchen, 0.21 that the mouse will be in the living room, and 0.08 that the mouse will have been eaten.

Treat this as a 3-state Markov Chain with "in the cat" being an absorbing state (literally). Since absorbing states should be listed first, set this up with "in the cat" as state #1. The order of states 2 and 3 won't matter, since the other two states are non-absorbing states. Use the same order for row and column labels. Compute the matrix N = (I - Q)^-1. Recall from the videos that this matrix can be used to find expected values. Use your N matrix to answer the following questions. (Hint: Your answer for (a) will be the sum of a row of the matrix N. Your answer for (b) will be a single entry of the matrix N.)

(a) If the mouse is in the kitchen, what is the expected value for the number of hours that will elapse before the mouse is eaten by the cat? (Give your answer correct to 2 decimal places.) hours (b) If the mouse is in the kitchen, and if you check every hour to see if the mouse is in the kitchen, what is the expected value for the number of times you will find the mouse in the kitchen before the mouse finally gets eaten by the cat? (Give your answer correct to 2 decimal places.) hours