Question

A mouse is placed in room 5 of the maze shown to the right. At each step the mouse exits the particular room that it is​ in, and is equally likely to choose any of the available doors to leave.

a. Construct a transition matrix and an initial probability vector for the​ mouse's travels.
b. What are the probabilities that the mouse is in each of the rooms after 5 moves?

Answer 1

To construct a transition matrix for the mouse's travels, divide 1 by the number of doors in the current room. The initial probability vector has a 1 in the room where the mouse starts. To find the probabilities after 5 moves, multiply the initial probability vector by the transition matrix raised to the power of 5.

Step-by-step explanation:

To construct a transition matrix for the mouse's travels, we need to determine the probabilities of the mouse moving from one room to another. Since the mouse is equally likely to choose any available door, the probability of moving from one room to another is 1 divided by the number of doors in the current room. The transition matrix will have rows representing the current room and columns representing the next room. The values in the matrix will be the probabilities of moving from the current room to the next room. The initial probability vector will have a 1 in the row corresponding to the room where the mouse starts and 0 in all other rows.

For example, if the maze has 4 rooms and room 3 has doors to rooms 1, 2, and 4, the transition matrix would be:

From → To ↓
Room 1
Room 2
Room 3
Room 4


Room 1
0
0
0
0


Room 2
0
0
0
0


Room 3
1/3
1/3
0
1/3


Room 4
0
0
0
1

If the mouse starts in room 3, the initial probability vector would be [0, 0, 1, 0].

b. To find the probabilities that the mouse is in each room after 5 moves, we can multiply the initial probability vector by the transition matrix raised to the power of 5.