Question

1). 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.

2) A deck of 72 cards contain either red or black cards. Suppose that it twice as likely to draw a black card than a red card. What is the probability of drawing 2 red cards and 2 black cards if you draw 4 cards with replacement?

Answer 1

We would be looking at the first question all parts here as:

Q1) a) We are given here that:
P(A --> B) = 0.2, Therefore P(A --> A) = 1 - 0.2 = 0.8
P(B --> A) = 0.35, therefore P(B --> B) = 1 - 0.35 = 0.65

a) Given that there are 20,000 people in town A and 15000 people in Town B initially, the transition probabilities for a 3 year period is computed here as:

• A --> A --> A --> A, Prob. = 0.8³ = 0.512
• A --> B --> A --> A, Prob. = 0.2*0.35*0.8 = 0.056
• A --> A --> B --> A, Prob. = 0.2*0.35*0.8 = 0.056
• A --> B --> B --> A, Prob. = 0.2*0.65*0.35 = 0.0455
• B --> B --> B --> B, Prob. = 0.65³ = 0.274625
• B --> A --> B --> B, Prob. = 0.35*0.2*0.65 = 0.0455
• B --> B --> A --> B, Prob. = 0.35*0.2*0.65 = 0.0455
• B --> A --> A --> B, Prob. = 0.35*0.8*0.2 = 0.056

Therefore the transition probabilities, here are obtained as:

P(A --> A) = 0.512 + 2*0.056 + 0.0455 = 0.6695
P(A --> B) = 1 - 0.6695 = 0.3305
P(B --> B) = 0.274625 + 0.0455*2 + 0.056 = 0.421625
Therefore, P(B --> A) = 1 - 0.421625 = 0.578375

Therefore, after 3 years, the number of people in each town here is computed as:
Town A: 20,000*0.6695 + 15,000*0.578375 = 13.39K + 8.675625K = 22,065.625

Therefore Town A: 22065.625
Town B: 35000 - 22065.625 = 12934.375

b) For a long term period, let the shares here be X and Y for towns A and B respectively here.

Therefore, 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, therefore X = 7/11

Therefore the long term distribution here are obtained as:
Town A: (4/11)*35000 = 12727.27

Therefore, Town B: 35000 - 12727.27 = 22272.73