A company has 2 machines. On any day, each machine that is working has a 1/3 chance of breaking down. If a machine breaks down during the day, it is sent to a repair facility and will be working 2 days after it breaks down. Letting the state of the system be the number of machines working at the beginning of the day, formulate a transition probability matrix for this situation.
P =
[1/3, 2/3, 0, 0]
[0, 1/3, 2/3, 0]
[0, 0, 1/3, 2/3]
[0, 0, 0, 1]
To formulate a transition probability matrix for this situation, we will consider the possible states of the system and the probabilities of transitioning between these states.
In this scenario, we have two machines, and at the beginning of the day, there are three possible states:
1. State 0: Both machines are broken.
2. State 1: One machine is working, and the other is broken.
3. State 2: Both machines are working.
Now, let's calculate the probabilities of transitioning between these states:
1. From State 0:
- The probability of staying in State 0 is (1/3) * (1/3) = 1/9. This means that both machines may break down again during the day.
- The probability of transitioning to State 1 is (1/3) * (2/3) = 2/9. This occurs when one machine starts working again.
2. From State 1:
- The probability of transitioning to State 0 is (1/3) * (2/3) = 2/9. This happens when the working machine breaks down.
- The probability of staying in State 1 is (1/3) * (1/3) = 1/9. This means that the broken machine remains broken.
- The probability of transitioning to State 2 is (2/3) * (2/3) = 4/9. This occurs when the broken machine gets repaired.
3. From State 2:
- The probability of transitioning to State 1 is (2/3) * (1/3) = 2/9. This happens when one machine breaks down.
- The probability of staying in State 2 is (2/3) * (2/3) = 4/9. Both machines continue to work.
Now, we can construct the transition probability matrix:
[ P(0→0) P(0→1) P(0→2) ]
[ ]
[ P(1→0) P(1→1) P(1→2) ]
[ ]
[ P(2→0) P(2→1) P(2→2) ]
Substituting the calculated probabilities:
[ 1/9 2/9 0 ]
[ ]
[ 2/9 1/9 4/9 ]
[ ]
[ 0 2/9 4/9 ]
This is the transition probability matrix for the given situation. Each element in the matrix represents the probability of transitioning from one state to another.
To formulate a transition probability matrix for this situation, we need to consider the possible states of the system and the probabilities of transitioning between these states.
In this case, the possible states of the system are:
1. Both machines working (state 0)
2. One machine working (state 1)
3. Both machines broken (state 2)
Given that each machine has a 1/3 chance of breaking down on any day, we can determine the probabilities of transitioning between these states.
For state 0:
- There is a 1/3 chance that both machines will break down: P(0 → 2) = 1/3
- There is a 2/3 chance that both machines will continue working: P(0 → 0) = 2/3
For state 1:
- There is a 1/3 chance that the remaining working machine will break down: P(1 → 2) = 1/3
- There is a 2/3 chance that the other machine will be repaired and both machines will be working: P(1 → 0) = 2/3
For state 2:
- Both machines are not working, so they need to be repaired. After 2 days, both machines will be working again: P(2 → 0) = 1
Therefore, the transition probability matrix for this situation is:
| 0 1 2
——————————
0 | 2/3 0 1/3
1 | 2/3 0 1/3
2 | 1 0 0
In this matrix, each row represents the current state, and each column represents the next state. The values in the matrix represent the probabilities of transitioning from the current state to the next state.