Question

Consider a Markov system modelling the migration of people with 2 states: a person can either be in town A or in town B. Every year, a person from town A has 20% chance of moving to town B, and a person from town B has a 35% chance of moving to town A. (a) If there were 20000 of people in Town A and 15000 of people in Town B initially, find the number of people in Town A and B after 3 years. (b) Suppose that these people are immortal. Find the approximate number of people in Town A and B respectively after 1000000000000000 years.

Answer 1

Answer:

P(A - > B) = 0.2,

P(A - > A) = 1 - 0.2

= 0.8

P(B - > A) = 0.35,

P(B - > B) = 1 - 0.35

= 0.65

a)

Given that there are 20,000 individuals around An and 15000 individuals around B at first, the progress probabilities for a multi year time span is processed here as:

For, A - > A - > A - > A,

Probability = 0.8^3

= 0.512

For, A - > B - > A - > A,

Probability = 0.2*0.35*0.8 = 0.056

For, A - > A - > B - > A,

Probability = 0.2*0.35*0.8 = 0.056

For, A - > B - > B - > A,

Probability = 0.2*0.65*0.35 = 0.0455

For, B - > B - > B - > B

Probability = 0.65^3 = 0.274625

For, B - > A - > B - > B,

Probability = 0.35*0.2*0.65 = 0.0455

For, B - > B - > A - > B,

Probability = 0.35*0.2*0.65 = 0.0455

For, B - > A - > A - > B,

Probability = 0.35*0.8*0.2 = 0.056

Along these lines the change probabilities, here are gotten as:

For, P(A - > A) = 0.512 + 2*0.056 + 0.0455 = 0.6695

For, P(A - > B) = 1 - 0.6695 = 0.3305

For, P(B - > B) = 0.274625 + 0.0455*2 + 0.056

= 0.421625

Along these lines,

P(B - > A) = 1 - 0.421625

= 0.578375

Along these lines, following 3 years, the quantity of individuals in every town here is registered as:

Town A: 20,000*0.6695 + 15,000*0.578375

= 13.39K + 8.675625K

= 22,065.625

Along these lines Town A: 22065.625

Town B:

35000 - 22065.625 = 12934.375

b)

For a drawn out period, let the offers here be X and Y for towns A and B individually here.

Along these lines, X + Y = 1

X = 0.8X + 0.35Y

0.2X = 0.35Y

0.2 - 0.2Y = 0.35Y

0.2 = 0.55Y

Y = 4/11, along these lines X = 7/11

Along these lines the drawn out dissemination here are acquired as:

Town A:

(4/11)*35000 = 12727.27

Along these lines, Town B:

35000 - 12727.27 = 22272.73