Each American family is classified as living in an urban, rural, or suburban loc

Question

Each American family is classified as living in an urban, rural, or suburban location. During a given year, 12% of all urban families move to a suburban location, and 8% move to a rural location; also, 6% 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. a 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? b Suppose that at present, 40% of all families live in an urban area, 35% live in a suburban area, and 25% live in a rural area. Two years from now, what percentage of American families will live in an urban area? c What problems might occur if this model were used to predict the future population distribution of the United States?

Explanation / Answer

(A) Family in Urban can move to Sub-urban or Rural and then the family can move from Sub-urban to Urban or Rural to Urban or it remains in Urban only Thus, there are three ways in which family now living in Urban can move to Urban two years from now.

Urban -> Sub urb -> Urban OR Probability = 0.12 * 0.06 = 0.0072

Urban -> Rural ->Urban OR Probability = 0.08 * 0.04 = 0.0032

Urban ->Urban -> Urban Probability = 0.80 *0.80 (because 80% families remain in Urban itself) =0.64

P(urban 2 years from now) = 0.6504

Urban -> Sub urb -> Sub urb OR Probability = 0.12 * 0.90 = 0.108

Urban -> Rural ->Sub Urb OR Probability = 0.08 * 0.06 = 0.0048

Urban ->Urban -> Sub Urb Probability = 0.80 *0.12 (because 80% families remain in Urban itself) = 0.096

P(sub urban 2 years from now) = 0.2088

Urban -> Sub urb ->Rural OR Probability = 0.12 * 0.04 = 0.0048

Urban -> Rural ->RuralOR Probability = 0.08 * 0.90 = 0.072

Urban ->Urban -> Rural Probability = 0.80 *0.08 (because 80% families remain in Urban itself) = 0.064

P(rural 2 years from now) = 0.1408

(B) 40% families in Urban now, there is a probability that 2 years from now 0.6504 will live in Urban.

35% in Sub Urban, p(starting from sub urban, in urban 2 years from now) =? 0.1036

25% in rural, p(starting from rural, in urban 2 years from now)=? 0.0716

p(starting from sub urban, in urban 2 years from now) =p(sub urb -> urb -> urb)+ p(sub urb-> rural -> urb) + p(sub urb-> sub urb -> urb) = (0.06*0.80) + (0.04*0.04)+ (0.90*0.06) = 0.048 + 0.0016 + 0.054 = 0.1036

p(starting from rural, in urban 2 years from now) =p(rural -> urb -> urb)+ p(rural-> rural -> urb) + p(rural-> sub urb -> urb) = (0.04*0.80) + (0.90*0.04) + (0.06*0.06) = 0.032 +0.036 + 0.0036 = 0.0716

So, 40 * 0.6504 = 26% people

35 * 0.1036 = 3.626%

25 * 0.0716 = 1.79%

Total people in urban two years from now = 26%+ 3.626% + 1.79% = 31.416%

(c) Problem that might occur is that if the percent of people that move from one region to another region keeps changing year to year, this model may not suffice.

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