0.17 Markov chains: homework

Is the matrix given below a transition matrix for a Markov chain? Explain.

1. $\left[\begin{array}{ccc}\text{.}2& \text{.}3& \text{.}5\\ \text{.}3& -\text{.}2& \text{.}9\\ \text{.}3& \text{.}3& \text{.}5\end{array}\right]$

2. $\left[\begin{array}{ccc}\text{.}3& \text{.}3& \text{.}4\\ \text{.}3& \text{.}4& \text{.}4\\ 0& 0& 0\end{array}\right]$

A survey of American car buyers indicates that if a person buys a Ford, there is a 60% chance that their next purchase will be a Ford, while owners of a GM will buy a GM again with a probability of .80. The buying habits of these consumers are represented in the transition matrix below.

Find the following probabilities:

1. The probability that a present owner of a Ford will buy a GM as his next car.

2. The probability that a present owner of a GM will buy a GM as his next car.

3. The probability that a present owner of a Ford will buy a GM as his third car.

4. The probability that a present owner of a GM will buy a GM as his fourth car.

Professor Hay has breakfast at Hogee's every morning. He either orders an Egg Scramble, or a Tofu Scramble. He never orders Eggs on two consecutive days, but if he does order Tofu one day, then the next day he can order Tofu or Eggs with equal probability.

1. Write a transition matrix for this problem.

2. If Professor Hay has Tofu on the first day, what is the probability he will have Tofu on the second day?

3. If Professor Hay has Eggs on the first day, what is the probability he will have Tofu on the third day?

4. If Professor Hay has Eggs on the first day, what is the probability he will have Tofu on the fourth day?

A professional tennis player always hits cross-court or down the line. In order to give himself a tactical edge, he never hits down the line two consecutive times, but if he hits cross-court on one shot, on the next shot he can hit cross-court with .75 probability and down the line with .25 probability.

1. Write a transition matrix for this problem.

2. If the player hit the first shot cross-court, what is the probability that he will hit the third shot down the line?

The transition matrix for switching political parties in an election year is given below, where Democrats, Republicans, and Independents are denoted by the letters $D$ , $R$ , and $I$ , respectively.

1. Find the probability of a Democrat voting Republican.

2. Find the probability of a Democrat voting Republican in the second election.

3. Find the probability of a Republican voting Independent in the second election.

4. Find the probability of a Democrat voting Independent in the third election.

Determine whether the following matrices are regular Markov chains.

1. $\left[\begin{array}{cc}1& 0\\ \text{.}5& \text{.}5\end{array}\right]$

2. $\left[\begin{array}{cc}\text{.}6& \text{.}4\\ 0& 1\end{array}\right]$

3. $\left[\begin{array}{ccc}\text{.}6& 0& \text{.}4\\ \text{.}2& \text{.}4& \text{.}4\\ 0& 0& 0\end{array}\right]$

4. $\left[\begin{array}{ccc}\text{.}2& \text{.}4& \text{.}4\\ \text{.}6& \text{.}4& 0\\ \text{.}3& \text{.}2& \text{.}5\end{array}\right]$

Company I and Company II compete against each other, and the transition matrix for people switching from Company I to Company II is given below.

Find the following.

1. If the initial market share is 40% for Company I and 60% for Company II, what will the market share be after 3 steps?

2. If this trend continues, what is the long range expectation for the market?