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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
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
Hafiz
what is Nano technology ?
Bob Reply
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold
Logan Reply
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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