Questions: The probability that a complex assembly line works correctly depends on whether the line worked correctly the last time it was used. There is a 0.6 chance that the line will work correctly if it worked correctly the time before, and a 0.2 chance that it will work correctly if it did not work correctly the time before. Set up a transition matrix with this information and find the long-range probability that the line will work correctly. Let the first state be the assembly line working correctly, and the second be that the assembly line does not work correctly. The transition matrix is P = . (Type an integer or decimal for each matrix element.) The long-range probability that the line will work correctly is . (Type an integer or a simplified fraction.)

Construct the transition matrix based on the given probabilities.

Define states and transition probabilities.

Let state 1 represent the assembly line working correctly, and state 2 represent the assembly line not working correctly. The transition matrix \( P \) will have the following form:

\( P = \begin{array}{cc} \begin{bmatrix} P(1 \rightarrow 1) & P(1 \rightarrow 2) \\ P(2 \rightarrow 1) & P(2 \rightarrow 2) \end{bmatrix} \end{array} \)

• \( P(1 \rightarrow 1) \) is the probability that the line works correctly given it worked correctly the previous time, which is 0.6.
• \( P(1 \rightarrow 2) \) is the probability that the line does not work correctly given it worked correctly the previous time, which is \( 1 - 0.6 = 0.4 \).
• \( P(2 \rightarrow 1) \) is the probability that the line works correctly given it did not work correctly the previous time, which is 0.2.
• \( P(2 \rightarrow 2) \) is the probability that the line does not work correctly given it did not work correctly the previous time, which is \( 1 - 0.2 = 0.8 \).

Construct the transition matrix.

Therefore, the transition matrix is:

\( P = \begin{array}{cc} \begin{bmatrix} 0.6 & 0.4 \\ 0.2 & 0.8 \end{bmatrix} \end{array} \)

\(\boxed{[[0.6, 0.4], [0.2, 0.8]]}\)

Calculate the long-range probability that the assembly line will work correctly.

Set up the equations for the stationary distribution.

To find the long-range probability, we need to find the stationary distribution \( \pi = [\pi_1, \pi_2] \) such that \( \pi P = \pi \) and \( \pi_1 + \pi_2 = 1 \). This means:

\( \begin{bmatrix} \pi_1 & \pi_2 \end{bmatrix} \begin{bmatrix} 0.6 & 0.4 \\ 0.2 & 0.8 \end{bmatrix} = \begin{bmatrix} \pi_1 & \pi_2 \end{bmatrix} \)

This gives us the following equations:

• \( 0.6\pi_1 + 0.2\pi_2 = \pi_1 \)
• \( 0.4\pi_1 + 0.8\pi_2 = \pi_2 \)
• \( \pi_1 + \pi_2 = 1 \)

Solve the equations for the stationary distribution.

From the first equation:

\( 0.2\pi_2 = 0.4\pi_1 \)
\( \pi_2 = 2\pi_1 \)

Substituting this into the third equation:

\( \pi_1 + 2\pi_1 = 1 \)
\( 3\pi_1 = 1 \)
\( \pi_1 = \frac{1}{3} \)

Then, \( \pi_2 = 2 \cdot \frac{1}{3} = \frac{2}{3} \)

The long-range probability that the line will work correctly is \( \pi_1 = \frac{1}{3} \).

\(\boxed{\frac{1}{3}}\)

The transition matrix is:
\(\boxed{[[0.6, 0.4], [0.2, 0.8]]}\)

The long-range probability that the line will work correctly is:
\(\boxed{\frac{1}{3}}\)