Help with determining the transition matrix for a markov chain

[subopolois]

Homework Statement

well i have my algebra exam coming up and my teacher told us that there is going to be a markov chain problem. the only problem i have is that i don't know how to get the initial transition matrix, which is crucial in getting full marks. can someone help me in determining how to get it from this problem:
a wolf pack hunts in one of 3 regions, its habits are as follows
1. if it hunts in one region one day it is as likely not to hunt there the next day
2. if it hunts in the first region, it never hunts in the second region the next day
3. if it hunts in the 2nd or 3rd, it is equally likely to hunt in each of the other regions the next day

Homework Equations

na

The Attempt at a Solution

this is an example from the textbook so they give the transition matrix as
0.5 0.25 0.25
0 0.5 0.25
0.5 0.25 0.5
given the above habits, i have trouble finding the transition matrix not only for this but other problems also, if someone can explain it, I am sure i can pick up on it pretty easily.

 

[Nino MMP]

The transition matrix is a table that contains the probabilities of transitioning from one state to another. In this case, the states are Regions 1, 2, and 3. To start, we need to determine the probability of transitioning from each state to the other two. For example, if the wolf pack is currently in Region 1, what is the probability of it transitioning to Region 2 or 3?From the problem statement, we know that if the wolf pack is in Region 1, it is equally likely to stay in Region 1 or transition to Region 3. Thus, the probability of transitioning from Region 1 to Region 2 is 0, and the probability of transitioning from Region 1 to Region 3 is 0.5. Following a similar logic, we can determine the probabilities of transitioning from Region 2 or 3 to the other states. If the wolf pack is in Region 2, then it never transitions to Region 1 the next day, so the probability of transitioning from Region 2 to Region 1 is 0. On the other hand, the probability of transitioning from Region 2 to Region 3 is 0.25.Similarly, if the wolf pack is in Region 3, it is equally likely to transition to either Region 1 or Region 2, so the probability of transitioning from Region 3 to Region 1 is 0.5, and the probability of transitioning from Region 3 to Region 2 is 0.25.Using the above information, we can construct the transition matrix:0.5 0.00 0.500.00 0.50 0.250.50 0.25 0.50