A country is divided into three demographic regions. It is found that each year 5% of the residents of region 1 move to region 2, and 5% move to region 3. Of the residents of region 2, 15% move to region 1 and 10% move to region 3. And of the residents of region 3, 10% move to region 1 and 5% move to region 2. What is the probability matrix?

To construct the probability transition matrix for the movement of residents among the three demographic regions, we can denote the regions as \( R_1 \), \( R_2 \), and \( R_3 \). The matrix will represent the probabilities of moving from one region to another.

1. **Region 1**: 
   - 5% move to Region 2
   - 5% move to Region 3
   - Therefore, 90% stay in Region 1.
   
   The probabilities are: 
   \[
   \begin{bmatrix}
   0.90 & 0.05 & 0.05
   \end{bmatrix}
   \]

2. **Region 2**: 
   - 15% move to Region 1
   - 10% move to Region 3
   - Therefore, 75% stay in Region 2.

   The probabilities are: 
   \[
   \begin{bmatrix}
   0.15 & 0.75 & 0.10
   \end{bmatrix}
   \]

3. **Region 3**: 
   - 10% move to Region 1
   - 5% move to Region 2
   - Therefore, 85% stay in Region 3.

   The probabilities are: 
   \[
   \begin{bmatrix}
   0.10 & 0.05 & 0.85
   \end{bmatrix}
   \]

Putting these together, the transition probability matrix \( P \) is:

\[
P = \begin{bmatrix}
0.90 & 0.05 & 0.05 \\
0.15 & 0.75 & 0.10 \\
0.10 & 0.05 & 0.85
\end{bmatrix}
\]

### Summary
The matrix \( P \) summarizes the probabilities of movement among the three regions. Each row represents a region, and each column represents the destination region.

Do you want details on how to interpret this matrix or any other questions? Here are some related questions you might consider:

1. How can this matrix be used to predict future population distributions?
2. What happens if the percentages of movement change?
3. How would the transition matrix differ if there were a fourth region?
4. Can this model be applied to other scenarios, like cities or states?
5. How do we calculate steady-state distributions using this matrix?

**Tip:** Understanding transition matrices can help in modeling various systems, such as population dynamics, economics, or even game theory.

Explore the transition probability matrix for three demographic regions, where 5% of residents move between regions annually. This detailed solution illustrates key concepts in probability theory and Markov chains, suitable for advanced students in Grades 11-12.

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