Question

Each American family is classified as living in an urban, rural, or suburban location. During a given year, 10% of all urban families move to a suburban location, and 5% move to a rural location; also, 18% of all suburban families move to an urban location, and 4% move to a rural location; finally, 4% of all rural families move to an urban location, and 6% move to a suburban location.

1- If a family now lives in an urban location, what is the probability that it will live in an urban area two years from now? A suburban area? A rural area?

2- Suppose that at present, 45% of all families live in an urban area, 35% live in a suburban area, and 20% live in a rural area. Two years from now, what percentage of American families will live in an urban area?

Answer 1

The probability matrix, in a sequence of Urban, Suburban, Rural is given below

Where

(1)

A family lives in an urban location. After 2 years, the family can still be living an urban location following ways

(a)

(b)

(c)

The probability that it will live in an urban area 2 years from now

A family lives in an urban location. After 2 years, the family can still be living an Suburban location following ways

(a)

(b)

(c)

The probability that it will live in a Suburban area 2 years from now

The probability that it will live in a rural area 2 years from now is

(2)

Present distribution of population after 1 year is given by

The distribution of population after 2 years is given by

Hence, the required percentage of American families who would be living in urban areas is