State Transition Matrix Application

A city classifies its neighborhoods as Developing, Stable, or Declining. The annual transition probabilities are as follows: a Developing neighborhood has a 75% probability of remaining Developing, 20% of becoming Stable, and 5% of becoming Declining; a Stable neighborhood has a 10% probability of becoming Developing, 80% of remaining Stable, and 10% of becoming Declining; and a Declining neighborhood has a 5% probability of becoming Developing, 15% of becoming Stable, and 80% of remaining Declining. Given an initial distribution of 50 Developing, 100 Stable, and 30 Declining neighborhoods, what is the expected number of Stable neighborhoods after 2 years?