# 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?

$3/7$

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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LITNING
scanning tunneling microscope
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Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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