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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?

This figure shows the layout of the rooms that the rat can navigate through.

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

10 / 19 size 12{"10"/"19"} {}
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Chapter review

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

  1. . 1 . 4 . 5 . 5 . 3 . 8 . 3 . 4 . 3 size 12{ left [ matrix { "." 1 {} # "." 4 {} # "." 5 {} ##"." 5 {} # - "." 3 {} # "." 8 {} ## "." 3 {} # "." 4 {} # "." 3{}} right ]} {}

  2. . 2 . 6 . 2 0 0 0 . 3 . 4 . 5 size 12{ left [ matrix { "." 2 {} # "." 6 {} # "." 2 {} ##0 {} # 0 {} # 0 {} ## "." 3 {} # "." 4 {} # "." 5{}} right ]} {}

  1. No
  2. No
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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.

This matrix shows the probable buying habits of consumers from Apple to IBM.

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
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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 size 12{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. 0 1 2 / 3 1 / 3 size 12{ left [ matrix { 0 {} # 1 {} ##2/3 {} # 1/3{} } right ]} {}
  2. 1
  3. 2 / 3 size 12{2/3} {}
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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 size 12{S} {} , B size 12{B} {} , and A size 12{A} {} , respectively.

This matrix shows the probability for students to switch between these three majors.

  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
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Determine whether the following matrices are regular Markov chains.

  1. 1 0 . 3 . 7 size 12{ left [ matrix { 1 {} # 0 {} ##"." 3 {} # "." 7{} } right ]} {}
  2. . 2 . 4 . 4 . 6 . 4 0 . 3 . 2 . 5 size 12{ left [ matrix { "." 2 {} # "." 4 {} # "." 4 {} ##"." 6 {} # "." 4 {} # 0 {} ## "." 3 {} # "." 2 {} # "." 5{}} right ]} {}
  1. No
  2. Yes
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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 size 12{P} {} represents a pass and H size 12{H} {} a handoff.

This matrix shows the probability that of the next play being a pass or a hand off.

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 size 12{P=2/3} {} , H = 1 / 3 size 12{H=1/3} {}
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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.

This matrix shows the probability that customers will switch between the three companies.

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. . 36 . 34 . 30 size 12{ left [ matrix { "." "36" {} # "." "34" {} # "." "30"{}} right ]} {}
  2. 3 / 7 9 / 28 1 / 4 size 12{ left [ matrix { 3/7 {} # 9/"28" {} # 1/4{}} right ]} {}
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Given the following absorbing Markov chain.

This matrix depicts the probability of moving from one sate to the other.

  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. S1 size 12{S1} {} and S2 size 12{S2} {}
  2. This matrix shows the probability of switching from S3 or S4 to S1 or S2
  3. 26 / 35
  4. 19 / 35
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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 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?

This figure shows the layout of the rooms that the rat can navigate through.

2 / 7 size 12{2/7} {}
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In [link] , what is the probability that the rat will end up in room 6 if it was initially in room 2?

3 / 7 size 12{3/7} {}

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
William Reply
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
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At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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