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To determine the matrix \( M \), let's analyze the movement of residents among the four regions \( A \), \( B \), \( C \), and \( D \), based on the given probabilities:

1. **Region A**: 
   - 55% move to \( B \), so the probability of moving from \( A \) to \( B \) is \( 0.55 \).
   - 15% move to \( C \), so the probability of moving from \( A \) to \( C \) is \( 0.15 \).
   - 25% move to \( D \), so the probability of moving from \( A \) to \( D \) is \( 0.25 \).
   - The probability of staying in \( A \) itself is \( 0 \) (since all residents are moving to other regions).

2. **Region B**:
   - 45% move to \( C \), so the probability of moving from \( B \) to \( C \) is \( 0.45 \).
   - 20% move to \( D \), so the probability of moving from \( B \) to \( D \) is \( 0.20 \).
   - 15% move to \( A \), so the probability of moving from \( B \) to \( A \) is \( 0.15 \).
   - The probability of staying in \( B \) itself is \( 0.20 \) (the remaining percentage).

3. **Region C**:
   - Residents in \( C \) either stay in \( C \) or move to \( A \) with equal probability.
   - This means there’s a 50% probability of staying in \( C \), so \( P_{CC} = 0.5 \).
   - The probability of moving from \( C \) to \( A \) is also \( 0.5 \), so \( P_{CA} = 0.5 \).
   - No residents move from \( C \) to \( B \) or \( D \), so \( P_{CB} = 0 \) and \( P_{CD} = 0 \).

4. **Region D**:
   - 65% move to \( A \), so \( P_{DA} = 0.65 \).
   - 15% move to \( C \), so \( P_{DC} = 0.15 \).
   - 10% stay in \( D \), so \( P_{DD} = 0.10 \).
   - The probability of moving from \( D \) to \( B \) is \( 0.10 \), so \( P_{DB} = 0.10 \).

### Forming the Matrix \( M \)
The matrix \( M \) will be a \( 4 \times 4 \) matrix where each entry \( M_{ij} \) represents the probability of moving from region \( j \) to region \( i \).

\[
M = \begin{bmatrix}
0 & 0.15 & 0.5 & 0.65 \\
0.55 & 0.2 & 0 & 0.1 \\
0.15 & 0.45 & 0.5 & 0.15 \\
0.25 & 0.2 & 0 & 0.1 \\
\end{bmatrix}
\]

This matrix reflects the transition probabilities from one year to the next for each region.

Would you like further details on how these probabilities were used in forming the matrix, or do you have any other questions?

### Related Questions

1. How do we determine the long-term distribution of the population across these regions?
2. How can we verify that \( M \) is a stochastic matrix?
3. How can we calculate \( \mathbf{x}_{k+1} \) if given an initial population distribution \( \mathbf{x}_0 \)?
4. What would change in the matrix \( M \) if additional regions were added?
5. How does the Markov property apply to this population model?

### Tip
In Markov chain problems, ensure each column in the transition matrix sums to 1, as it represents total probability from one state to another.

A country is divided into four regions (A, B, C, and D). It is found that each year 55% of the residents in A move to B, 15% move to C, and 25% move to D. Of the residents in B, 45% move to C, 20% move to D, and 15% move to A. The residents of C will either stay where they are or move to A with equal probability. In D, 65% move to A, 15% move to C, and 10% stay in D. Suppose we model this system by \( x_{k+1} = M x_k \), where the vector \( x_k \) gives the populations in regions A, B, C, D respectively after k years. Determine the matrix \( M \).

Analyze the population movement in a four-region country with given yearly migration probabilities. Learn how to determine the transition matrix M in a Markov chain model where populations shift among regions A, B, C, and D. Suitable for Grades 11-12 and college students.

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