Final answer:
To construct a Markov chain for the rat movements among the rooms, define the state space and transition probabilities. The long-term distribution of rats is found using the stationary distribution that is calculated from the transition matrix. The long-term probability that a given marked rat is in room 4 is the corresponding element of the stationary distribution vector.
Step-by-step explanation:
To construct a Markov chain for the movement of rats between the four rooms, we need to define the state space and the transition probabilities between the states. Assuming each door has an equal chance of being chosen by a rat, and if there are k doors in a given room, the probability of moving to any of the adjacent rooms is 1/k. For example, from room 1, a rat can move to room 2 or 3, so the transition probabilities are 1/2 for each. Similarly, we can define the transition probabilities for rooms 2, 3, and 4. The Markov chain will include these probabilities between the states (rooms).
To predict the long-term distribution of rats, we need to find the stationary distribution of the Markov chain. This is the vector that satisfies the equation πP = π, where P is the transition matrix and π is the stationary distribution. Once the stationary distribution is calculated, it gives us the long-term probabilities of finding any rat in a specific room. For the marked rat, we are interested in the element of the stationary distribution vector corresponding to room 4.
The probability that a randomly chosen (marked) rat is in room 4 in the long run is the element of the stationary distribution corresponding to room 4. To find this probability, we solve for the stationary distribution and extract the appropriate element of the distribution-vector.