The current population has 80% in the city and 20% in the suburb. Each year, 3% of the city dwellers will move to the suburbs while 4% of the suburbanites will move to the city. We are asked to find the population percentages after 1 and 2 years.

One way to look at this is using a tree diagram. If 3% leave the city each year, then 97% remain, and if 4% leave the suburbs, then 96% remain. After 1 year, you have CC(start and remain in city)=(.8)(.97)=.776; SC=(.2)(.04)=.008, so after **1 year, the percentage in the city is .776+.008=.784.** SS=(.2)(.96)=.192 and CS=(.8)(.03)=.024, **so the percentage in the suburbs after 1 year is .216.**

As the number of years increases, this becomes tedious. An easier computation is to use a transition matrix.

The original population can be described with the matrix A=(.8,.2). The transition matrix then takes into account the 4 possible moves: CC, CS, SS, SC. The transition matrix is `T=([.97.03],[.04.96])` .

Now if we take AT, we get `([.8,.2])([.97,.03],[.04,.96])=([.784,.216])`, as before.

The true power comes as we compute more years. Instead of computing the probabilities for each of CCC, CCS, CSC, SCC, SSC, SCS, CSS, and SSS and then adding CCC+CSC+SCC+SSC to find the probability that someone is in the city after two years and a similar computation for the suburbs, we instead take `AT^2` .

Thus, `([.8,.2])([.97,.03],[.04,.96])^2=([.76912,.23088])`, **so the percentage in the city after 2 years is .769 or 76.9% and the percentage in the suburbs will be .231 or 23.1%**.

The true power comes if this ends up in a steady state. After 200 years or so, we end up with 57.1% of the population in the city and 42.9% in the suburbs with virtually no change after that.

**Further Reading**