Science:Math Exam Resources/Courses/MATH152/April 2016/Question B 04 (a)/Solution 1

Let $s_{1}=$city and $s_{2}=$suburbs be two states as in the hint and 1 year be one step.

Since 10% of city residents move to the suburbs every year, 90% of city residents remain in the city.

Using definition in the Hint, this implies that $p_{21}=0.1$ and $p_{11}=0.9$.

On the other hand, 20% of the residents of the suburbs move to cities every year, so that 80% of the suburbs residents stay in the suburbs.

Then, we have $p_{12}=0.2$ and $p_{22}=0.8$.

To sum, the transition probability matrix $P=\{p_{ij}\}$ is

$\color {blue}{P={\begin{pmatrix}0.9&0.2\\0.1&0.8\end{pmatrix}}}$.