Question

Teenager Mike wants to borrow the car. He can ask either parent for permission to take the car. If he asks his mom, there is a 20% chance she will say ”yes,” a 30% chance she will say ”no,” and a 50% chance she will say, ”ask your father.” Similarly, that chances of hearing ”yes”/”no”/”ask your mother” from his dad are 0.1, 0.2, and 0.7 respectively. Imagine Mike’s efforts can be modeled as a Markov chain with state (1) talk to Mom, (2) talk to Dad, (3) get the car (”yes”), (4) strike out (”no”). Assume that once either parent has said ”yes” or ”no,” Mike’s begging is done.

1. Construct the one-step transition matrix for this Markov chain.

2. Identify the absorbing state(s) of the chain.

3. Determine the mean times to absorption.

4. Determine the probability that Mike will eventually get the car if (1) he asks Mom fist and (2) he asks Dad first. Whom should he ask first?

Answer 1

P(yes | mom) = 0.2
P(no | mom) = 0.3
P(dad | mom) = 0.5

P(yes | dad) = 0.1,
P(no | dad) = 0.2,
P(mom | dad) = 0.7

a) As we are given here that once no or yes are done, the begging is done and so the transition probabilities from state no to no is 1 and from yes to yes is also 1. Therefore the transition probability matrix here is given as:

b) As already discussed the transition probabilities from No and Yes are always 1 to itself only, Therefore the third and the fourth states that is No and Yes are the absorbing states here.

c) Let the mean time to absorption be X from mom and from Dad be Y

From the first row, we have here:
X = 0.5*1 + 0.5*(1 + Y)
X = 1 + 0.5Y

From second row, we have here:
Y = 0.7*(X + 1) + 0.3*1
Y = 1 + 0.7X

Putting X = 1 + 0.5Y in the above equation, we get here:
Y = 1 + 0.7(1 + 0.5Y)
Y = 1.7 + 0.35Y
0.65Y = 1.7
Y = 1.7 / 0.65 = 2.62

X = 1 + 0.5Y = 2.31

Therefore the mean time to absorption from mom is 2.31 and from dad is 2.62.

d) Given that he asks mom first, let the probability that he gets the car be X. Also given that he is asking dad let the same probability be Y

Then from first row, we have here:
X = 0.5Y + 0.3*0 + 0.2*1
X = 0.5Y + 0.2

Also from second row, we have:
Y = 0.7X + 0.2*0 + 0.1*1
Y = 0.7X + 0.1

Putting the above equation in X = 0.5Y + 0.2 , we get here:
X = 0.5(0.7X + 0.1) + 0.2
0.65X = 0.25
X = 0.3846

Y = 0.7X + 0.1 = 0.3692

Therefore 0.3846 is the probability he gets car if he asks mom and 0.3692 is the probability that he gets car if he asks dad. Therefore he should ask mom first here.