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The problem you've uploaded is about finding the steady-state vector for a stochastic matrix. It involves two games: "Hide and Seek" and "Tag," represented by a matrix \( M \).

Given the stochastic matrix \( M \):
\[
M = \begin{bmatrix}
0.6 & 0.2 \\
0.4 & 0.8
\end{bmatrix}
\]
We are tasked with finding the solution to the matrix equation \( (M - I) x = 0 \) and identifying the steady-state vector.

Let's go through the steps:

1. **Define the steady-state vector**: \( x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \), where \( x_1 \) and \( x_2 \) are the probabilities of being in each state (Hide and Seek, and Tag, respectively) after a long time.

2. **Set up the equation**:
   The matrix equation is \( M x = x \), which implies \( (M - I)x = 0 \), where \( I \) is the identity matrix.

   Substituting the values:
   \[
   \begin{bmatrix}
   0.6 & 0.2 \\
   0.4 & 0.8
   \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}
   \]

   This leads to the system of linear equations.

Let me solve for the steady-state vector using these equations.The steady-state vector is approximately:

\[
x = \begin{bmatrix} 0.3333 \\ 0.6667 \end{bmatrix}
\]

This means that in the long run, approximately 33.33% of children will be playing "Hide and Seek," and 66.67% will be playing "Tag."

### Conclusion:
Most children will be playing **Tag** after a long period of time.

Would you like more details on the process or have any questions?

Here are some related questions to further explore:

1. How does the initial state affect the time taken to reach the steady state?
2. What is the significance of a stochastic matrix in probability theory?
3. Can this method be applied to larger state spaces?
4. How would you interpret the transition probabilities in this problem?
5. How is the steady-state solution related to eigenvectors of the matrix?

**Tip**: The sum of the probabilities in the steady-state vector must always equal 1, ensuring all possible states are accounted for.

Given the stochastic matrix M = [[0.6, 0.2], [0.4, 0.8]], find the solution vector to the matrix equation (M - I)x = 0 and the steady-state vector. Which game will most children be playing after a long period of time?

Learn how to find the steady-state vector of a stochastic matrix using linear algebra techniques. This problem involves solving the matrix equation (M - I)x = 0 and determining which game—Hide and Seek or Tag—most children will be playing after a long period. Suitable for university-level students studying linear algebra and Markov chains.

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