Construct a model of population flows between cities suburbs

Construct a model of population flows between cities, suburbs, and nonmetropolitan areas of the United states. Their respective population in 2015 were 80 meters, 175 millions, and 52 millions, The stochastic matrix giving the probabilities of the move is predict the populations o city, suburban, and nonmetropolitan areas for 2016,2017and 2018. If a person was living in the city in 2015, what is the probability the person will be living in nonmetropolitan area in 2017?

Solution

Probability of moving from city to city = 0.96
Probability of moving from city to suburb = 0.03
Probability of moving from city to nonmetro = 0.01
Probability of moving from suburb to city = 0.01
Probability of moving from suburb to nonmetro = 0.98
Probability of moving from suburb to city = 0.01
Probability of moving from nonmetro to city = 0.015
Probability of moving from nonmetro to suburban = 0.005
Probability of moving from nonmetro to nonmetro = 0.98
Population of city =80
Population of suburban =175
Population of nonmetro = 52
For 2016
Population of city =0.96*80+0.01*175+0.015*52 = 79.33
Population of suburban =0.03*80+0.96*175+0.01*52 = 174.16
Population of nonmetro = 0.015*80+0.005*175+0.98*52 = 53.51
For 2017
Population of city =0.96*79.33+0.01*174.16+0.015*53.51 = 78.70105
Population of suburban =0.03*79.33+0.96*174.16+0.01*53.51 = 173.3243
Population of nonmetro = 0.015*79.33+0.005*174.16+0.98*53.51= 54.9747
For 2018
Population of city =0.96*78.70+0.01*17.3243+0.015*54.9747 = 78.11087
Population of suburban =0.03*78.70+0.96*173.3243+0.01*54.9747 = 172.4937
Population of nonmetro = 0.015*78.70+0.005*173.3243+0.98*54.9747= 56.39546
2) Person of city to nonmetro can move in 3 ways in 2015 to 2017
  a) City (2015 ) - City(2016)- Nonmetro(2017)
b) City (2015 ) - Nonmetro(2016)- Nonmetro(2017)
c) City (2015 ) - subUrban(2016)- Nonmetro(2017)

Calcualting sum of all above probabiltiy = 0.96*0.01+0.01*0.98+0.03*0.01 = 0.0197