Question

Suppose that there are two schools in a region. 7% of students will move from school A to school B at the end of each year, while 11% of students will move from school B to school A. Suppose that initially 30% of students attend school A and 70% attend school B.

a) Determine the transition matrix and the initial probability vector which can be used to represent this information.

b) Find the percentage of students attending each school at the end of 3 years ( give answers to four decimal places)

Answer 1

a)

Let the states of Markov chain be A and B denoting the students are in school A and B respectively.

The transition probability from State A to State B is 0.07

The transition probability from State A to State A is 1-0.07 = 0.93

The transition probability from State B to State A is 0.11

The transition probability from State B to State B is 1-0.11 = 0.89

The transition matrix is,

The initial probability vector is,

b)

The transition matrix after 3 years is,

The percentage of students attending each school at the end of 3 years are,

= [0.3 * 0.8255 + 0.7 * 0.2742 , 0.3 * 0.1745 + 0.7 * 0.7258]

= [0.4396, 0.5604]

Thus,

Percentage of students attending school A at the end of 3 years is 43.96%

Percentage of students attending school B at the end of 3 years is 56.04%