0.17 Markov chains: homework  (Page 3/3)

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A rat is placed in the maze shown below, and it moves from room to room randomly. From any room, the rat will choose a door to the next room with equal probabilities. Once it reaches room 1, it finds food and never leaves that room. And when it reaches room 5, it is trapped and cannot leave that room. What is the probability the rat will end up in room 5 if it was initially placed in room 3?

In [link] , what is the probability the rat will end up in room 1 if it was initially placed in room 2?

$\text{10}/\text{19}$

Chapter review

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

1. $\left[\begin{array}{ccc}\text{.}1& \text{.}4& \text{.}5\\ \text{.}5& -\text{.}3& \text{.}8\\ \text{.}3& \text{.}4& \text{.}3\end{array}\right]$

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

1. No
2. No

A survey of computer buyers indicates that if a person buys an Apple computer, there is an 80% chance that their next purchase will be an Apple, while owners of an IBM will buy an IBM again with a probability of .70. 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 an Apple will buy an IBM as his next computer.

2. The probability that a present owner of an Apple will buy an IBM as his third computer.

3. The probability that a present owner of an IBM will buy an IBM as his fourth computer.

1. 0.2
2. 0.3
3. 0.475

Professor Trayer either teaches Finite Math or Statistics each quarter. She never teaches Finite Math two consecutive quarters, but if she teaches Statistics one quarter, then the next quarter she will teach Statistics with a $1/3$ probability.

1. Write a transition matrix for this problem.

2. If Professor Trayer teaches Finite Math in the Fall quarter, what is the probability that she will teach Statistics in the Winter quarter.

3. If Professor Trayer teaches Finite Math in the Fall quarter, what is the probability that she will teach Statistics in the Spring quarter.

1. $\left[\begin{array}{cc}0& 1\\ 2/3& 1/3\end{array}\right]$
2. 1
3. $2/3$

The transition matrix for switching academic majors each quarter by students at a university is given below, where Science, Business, and Liberal Arts majors are denoted by the letters $S$ , $B$ , and $A$ , respectively.

1. Find the probability of a science major switching to a business major during their first quarter.

2. Find the probability of a business major switching to a Liberal Arts major during their second quarter.

3. Find the probability of a science major switching to a Liberal Arts major during their third quarter.

1. 0.3
2. 0.31
3. 0.28

Determine whether the following matrices are regular Markov chains.

1. $\left[\begin{array}{cc}1& 0\\ \text{.}3& \text{.}7\end{array}\right]$
2. $\left[\begin{array}{ccc}\text{.}2& \text{.}4& \text{.}4\\ \text{.}6& \text{.}4& 0\\ \text{.}3& \text{.}2& \text{.}5\end{array}\right]$
1. No
2. Yes

John Elway, the football quarterback for the Denver Broncos, calls his own plays. At every play he has to decide to either pass the ball or hand it off. The transition matrix for his plays is given in the following table, where $P$ represents a pass and $H$ a handoff.

Find the following.

1. If John Elway threw a pass on the first play, what is the probability that he will handoff on the third play?

2. Determine the long term play distribution.

1. 0.32
2. $P=2/3$ , $H=1/3$

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

Find the following.

1. If the initial market share is 20% for Company I, 30% for Company II and 50% for Company III, what will the market share be after the next year?

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

1. $\left[\begin{array}{ccc}\text{.}\text{36}& \text{.}\text{34}& \text{.}\text{30}\end{array}\right]$
2. $\left[\begin{array}{ccc}3/7& 9/\text{28}& 1/4\end{array}\right]$

Given the following absorbing Markov chain.

1. Identify the absorbing states.

2. Write the solution matrix.

3. Starting from state 4, what is the probability of eventual absorption in state 1?

4. Starting from state 3, what is the probability of eventual absorption in state 2?

1. $\mathrm{S1}$ and $\mathrm{S2}$
2. $26/35$
3. $19/35$

A rat is placed in the maze shown below, and it moves from room to room randomly. From any room, the rat will choose a door to the next room with equal probabilities. Once it reaches room 1, it finds food and never leaves that room. And when it reaches room 6, it is trapped and cannot leave that room. What is the probability that the rat will end up in room 1 if it was initially placed in room 3?

$2/7$

In [link] , what is the probability that the rat will end up in room 6 if it was initially in room 2?

$3/7$

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