sec 8 - gwinnett county public schools€¦ · sec 8.4 – matrices transformations voting methods...
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Sec 8.1 – Matrices Review
Voting methods & Matrix Applications Name:
Use the following matrices for the problems 1 - 13. Show ALL WORK by HAND and check with the calculator.
57
23A
37
25B
621
123C
23
34
02
D 5 3
0 7 1
xE
y
1. 2. TD
3. EC = 4. CD =
5. AB 32 = 6. DC =
7. B D = 8. B E =
What are the dimensions
of Matrix D?
Use the following matrices for the problems 1 - 13. Show ALL WORK by HAND and check with the
calculator.
57
23A
37
25B
621
123C
23
34
02
D 5 3
0 7 1
xE
y
9. C22 + D31 – B21 = 10. 2
A
11. T
D D
12. Create a 2x2, 3x3, and 5x 5 IDENTITY matrices
13. Are Matrices A and B inverses? Demonstrate how you can tell.
14. 2 2 2
2 1 6
a
b
14.
4 1 6 6 27
2 3 2 8
a
b c
15. The determinant of
35
24by hand and show work. 14. Find the determinant of
641
320
153
by hand and show work.
15. Find the inverse of
35
24by hand and show work. 16. Find the inverse of
24
36by hand and show work.
17. On one weekend, the Goxfield Theater reported the following ticket sales for three
first-run movies, as shown in the matrix at the right. If the ticket prices were $6 for
each adult and $4 for each child, what were the weekend sales for each movie?
Determine possible values for a, b, and c that would
make the following matrix statement true.
Determine possible values for a and b that would
make the following matrix statement true.
a =
b =
c =
a =
b =
Sec 8.2 – Informational Matrices
Voting methods & Matrix Applications Name:
1. A flu epidemic occurs in Middletown schools and each student is either susceptible, sick, or infected.
The percentage of students in each category by grade
level is given below.
The population distribution of the Middletown school district is
given below
Infected
Sick
eSusceptibl
S
%20%10%70
%20%15%65
%15%25%60
HighSchool
olMiddleScho
Elementary
P
11001115
795830
15801610
a. Describe the value of S3 2 and what it represents.
b. Describe the value of P2 2 and what it represents.
c. Find PS . (You can use your calculator if you would like).
INCLUDE LABELS
d. Describe the value of 12PS and what it represents.
e. Based on the flu epidemic in Middletown
schools, how many sick girls are there?
f. Based on the flu epidemic in Middletown
schools, how many boys are infected?
2. A study is being conducted by Time Warner about the network spending of its Network HBO and 2 online competitors.
The percentage of spending categories by each network
is shown below.
The amount of spending per month by each company is shown
below.
Movies
S Mini Series
CurrentTV
60% 35% 70%
30% 30% 30%
10% 35% 0%
NetFlix
P HULU
HBO
$85
$33
$912
million
million
million
a. Describe the value of S2 3 and what it represents.
b. Describe the value of P31and what it represents.
c. Find PS . (You can use your calculator if you would like).
INCLUDE LABELS
d. Describe the value of 1 1S P and what it represents.
e. Based on the study, how much do the 3 companies spend on
Mini Series?
f. Based on the study how much does HBO spend on
Movies and their licensing?
Girls Boys Elementary Middle High
School School School
Monthly Spending NetFlix HBO HULU
3. following matrices show information collected by an apparel store in Magic Kingdom park at Disney World
The matrix [N] shows the number of items sold on
Monday
The matrix [P] shows the Sale Price and Profit of each type of
item.
Adult
Youth
ToddlerInfant
N
/
251824
81530
42375
Hats
Shirts
Jackets
P
$75 $40
$30 $15
$16 $9
a. Describe the value of N23 and what it represents.
b. Describe the value of P2 1 and what it represents.
c. Find PN . (You can use your calculator if you would like).
INCLUDE LABELS
d. Describe the value of 23PN and what it represents.
e. Based on the information what was the total Sales on
Monday for Adult Apparel?
f. Based on the information what was the total
Profit made on Monday on Infant Apparel?
Sale Profit Price Jackets Shirts Hats
Sec 8.3 – Basic Voting Methods Voting methods & Matrix Applications Name:
Consider the following preference schedules for an election.
1. How many preference schedules are possible (if ties are not permitted)?
2. Who is the plurality winner?
What is the percentage of 1st place votes each received?
3. How many first place votes would be needed in this example for there
to be a majority winner?
If there is a majority winner who is it?
4. Who is the ‘run off’ winner?
5. Who is the ‘sequential run off’ winner?
6. What is each candidates Borda count?
Who is the ‘Borda Count’ winner?
7. What is each candidates Condorcet winner?
#5)
#3)
#3)
#6)
Droid:
iP6:
Glxy5:
Z30:
#1)
#4)
#6)
Drd iP6 G5 Z30
Drd *
iP6 *
G5 *
Z30 *
#7)
The BIG QUESTION:
WHO REALLY WINS? SUPPORT YOUR REASONING.
#2)
***Demonstrate how this can be done with matrix multiplication****
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
8 5 6 7
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola) iPhone 6
(Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
iPhone 6 (Apple)
Galaxy 5 (Samsung)
Bold Z30 (Blackberry)
Droid 4 (Motorola)
8 5 6 7
Twenty-two Discrete Math students are arguing over which fast food restaurants and listed their preferences below.
1. In your own words, give a description of the plurality winner
What percentage of 1st place votes does each of the following choices have?
Choice Five Guys Chic-fil-a Zaxby’s Wendy’s
Percentage of 1st
place votes
Who is the plurality winner?
2. a.What is the minimum number of first place votes needed in this
example for there to be a majority winner?
b.If there is a majority winner who is it?
3. In your own words, give a description of the ‘run off’ winner :
Who is the ‘run off’ winner?
4. In your own words, give a description of the ‘sequential run off’ winner :
Who is the ‘sequential run off’ winner?
5. In your own words, give a description of the ‘Borda Count’ winner (on a separate page show
how this might be done using Materices):
a. Give the Borda Count for each letter:
b. Who is the ‘Borda Count’ winner?
6. Determine the Condorcet Winner.
#3a)
#3b)
#4)
5)
#6b)
#6a)
5G: C: Zx: W:
→ 5G C Zx W
5G *
C *
Zx *
W *
Chic-fil-a
Zaxby’s
Wendy’s
Five Guys
Wendy’s
Chic-fil-a
Zaxby’s
Five Guys
Chic-fil-a
Zaxby’s
Wendy’s
Five Guys
Wendy’s
Zaxby’s
Chic-fil-a
Five Guys
8 5 6 3
2)
•
•
•
• •
•
Sec 8.4 – Matrices Transformations
Voting methods & Matrix Applications Name:
Start by creating a picture on a Cartesian coordinate system (preferably
in the first quadrant)
The picture at the right would be represented by the matrix:
1447311
1144641
Enter this into the Matrix [A] in the calculator.
Press MATRX , ◄ , ENTER . Change the
dimensions of the matrix to match the points of
your picture. For our example we will need to
change the dimensions to 2 x 7
To return to the home screen press 2nd
MODE
To have the calculator show your original picture. Press MATRX , ►
, 8 . This will bring up Matr►list( on the calculator.
This function will enable us to put our matrix into the table of the
calculator which can be graphed.
Next, press MATRX , 1 . Before the Matrix can be transformed into
the table it has to be turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd ,
1 , , , 2nd , 2 , ) , ENTER .
Next hit 2nd
, Y= (Stat Plot) , 1.
Make sure your screen has the following options highlighed
Finally push ZOOM, 6.
Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar.
To have the calculator show your original picture. Press MATRX , ► , 8 . This will bring up
Matr►list( on the calculator. This function will enable us to put our matrix into the table of
the calculator which can be graphed.
Next, press, 2 , MATRX , 1 . Before the Matrix can be transformed into the table it has to be
turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd , 3 , , , 2nd
, 4 , ) , ENTER .
Next hit 2nd
, Y= (Stat Plot) , 2.
Make sure your screen has the following options highlighed
Finally push ZOOM, 6.
Translation To translate an object we have to set up a translation matrix. We can enter this in matrix [B]
5555555
7777777
Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C]
01
10
Fill in all of the points all the way
down and make certain you finish
with the point you
started with (4,7)
to connect the last
line.
Shift left 7 and down 5
Rotation by 90
General Rotation To rotate an object we have to set up a rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.)
cos 160 sin 160
sin 160 cos 160
Combination Transformations (Rotate 200° and Translate Right 6) To translate an object we have to set up the rotation matrix. We can enter this in matrix [C] (be sure you are in DEGREE mode.) and a
translation matrix in [B].
5 5 5 5 5 5 5
0 0 0 0 0 0 0B
cos 200 sin 200
sin 200 cos 200C
Programmers often use matrices to write visual code even for the mouse cursor For
example let’s suggest that the points A,B,C,D,E,F,G represent a mouse cursor.
Programmers would use the matrix:
a. Draw the original mouse cursor on the graph using a blue colored
pencil.
b. Usually mouse cursors are translated. To do this
programmers add by a translation matrix. Add the
following translation matrix to [S] and draw the new
image using a red pen.
4444444
8888888S
c. To rotate an object counter clockwise programmers
multiply a rotation matrix and [S]. Multiply the
following and graph using a black or gray pencil.
(should be degree mode)
3455232
2325543S
GFEDCBA
S
130cos130sin
130sin130cos
Rotation by 160
Press the MODE key and
switch to DEGREE mode
Rotation by 200
Press the MODE key and
switch to DEGREE mode
Shift right 5
Matrices are used to describe most graphics on computer (including
computer/video games). Consider the following Phoenix bird. As a matrix it
would be described as:
4 4 0 5 5 2 1 3 0 1 4
7 4 7 4 7 0 3 3 4 0 7
A B C D E F G H I J A
Once entered as a matrix several graphical transformations can be performed using matrix operations.
DILATIONS:
To dilate (bigger or smaller) from the origin point, you would only need to multiply the original matrix by a scalar multiple. e. g.
4 4 0 5 5 2 1 3 0 1 42
7 4 7 4 7 0 3 3 4 0 7
……would make the picture twice as big
12
4 4 0 5 5 2 1 3 0 1 4
7 4 7 4 7 0 3 3 4 0 7
……would make the picture shrink the picture to half size
TRANSLATION Matrices:
4 4 0 5 5 2 1 3 0 1 4 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0
7 4 7 4 7 0 3 3 4 0 7 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1h k
In the above example all you would need to do is change ‘h’ to the a number to translate the picture left or right OR change ‘k’ to a number to translate
the picture up or down.
ROTATION Matrices:
cos sin 4 4 0 5 5 2 1 3 0 1 4
sin cos 7 4 7 4 7 0 3 3 4 0 7
In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin.
ROTATION Matrices:
cos sin 4 4 0 5 5 2 1 3 0 1 4
sin cos 7 4 7 4 7 0 3 3 4 0 7
In the above example all you would need to do is change theta to an angle you wish to rotate the shape about the origin.
A
B
Start by creating a picture on a Cartesian coordinate system (preferably
in the first quadrant)
The picture at the right would be represented by the matrix:
Enter this into the Matrix [A] in the
calculator.
Press MATRX , ◄ , ENTER . Change the
dimensions of the matrix to match the points
of your picture.
For our example we will need to change the
dimensions to 2 x 26
To return to the home screen press 2nd
MODE
To have the calculator show your original picture. Press MATRX , ► , 8 . This will bring up
Matr►list( on the calculator. This function will enable us to put our matrix into the table
of the calculator which can be graphed.
Next, press MATRX , 1 . Before the Matrix can be transformed into the table it has to be turned
vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd
, 1 , , , 2nd
, 2 , ) ,
ENTER .
Next hit 2nd
, Y= (Stat Plot) , 1.
Make sure your screen has the following options highlighed
Finally push ZOOM, 6.
You can turn the axes on or off by pressing 2nd
, ZOOM and selecting the option AxesOff .
Dilations To make the object dilate, using (0,0) as the center of dilation, multiply the matrix by a scalar.
To have the calculator show your original picture. Press MATRX , ► , 8 . This will bring up
Matr►list( on the calculator. This function will enable us to put our matrix into the table
of the calculator which can be graphed.
Next, press, 2 , MATRX , 1 . Before the Matrix can be transformed into the table it has to be
turned vertically or “TRANSPOSED”. Press MATRX , ► , 2 , , , 2nd
, 1 , , , 2nd
, 2 ,
) , ENTER .
Finally push ZOOM, 6.
6 10 10 8 8 6 6 8 8 0 5 8 5 7 7 4 3 5 7 5 5 3 3 3 0 5
10 10 4 4 10 8 3 4 10 0 10 10 6 6 2 1 1 2 2 2 6 4 1 4 0 10
If each screen was viewed in rapid succession it would give the appearance of flying into the picture.
Create your own version of a drawing that you would like to animate using transformational matrices
Using up to 10 frame of animation, show rough sketches of your anticipated slides to create your still animation.
Attempt #1 Attempt #2
Slide # 1 Slide # 2 Slide # 3 Slide # 4
Slide # 5 Slide # 6 Slide # 7 Slide # 8
Slide # 9 Slide # 10
Summary: With a couple of similar pictures we can make an animation of a fish that blows bubbles, an eye that’s winking, or just
about anything you wish to make move. Animations for cartoons, movies, and flip books are created by making several similar
pictures each with a slight change showing where the object has moved too. Creating a fish that moves to the right would require
creating the following pictures:
If these pictures were played in rapid succession on the TI-83 the fish would appear to move to the right. As you create each picture on
the TI-83 you will need to store the picture.
If these pictures were played in rapid succession on the TI-83 the heart would appear to beat. As you create each picture on the TI-83
you will need to store the picture.
Storing your Pictures:
First display the picture you would like to store as your first picture in your animation. You can turn on
a plot by selecting it from the Y= button.
o Press Y = . Highlight Plot1 by pressing the cursor up. Pressing ENTER , will either “turn on”
or “turn off” the plot (if the plot is highlighted then it is turned on). For the “Heart” example
turn Plot1 on and Plot2 off .
o Press GRAPH
With the picture you wish to store on the graph screen, press: 2nd
, PRGM , ◄ , 1 . The
calculator should say StorePic. The TI-82/83/84 allows you to save up to 10 different pictures. Next,
we will need to select where we would like to store the picture. Press: VARS , 4 . Now select
where you would like to store the picture in Pic1 through Pic9. Select Pic1 if this is the first picture.
After selecting the appropriate place to store the picture press ENTER . After pressing enter the
picture that you are storing should re-appear.
DRAW
Next display the picture you would like to store as your second picture in your animation.
o Press Y = . Highlight Plot2 by pressing the cursor up. Pressing ENTER , will either “turn on” or “turn
off” the plot (if the plot is highlighted then it is turned on). For the “Heart” example turn Plot1 off and
Plot2 on .
o Press GRAPH
With the picture you wish to store on the graph screen, press: 2nd
, PRGM , ◄ , 1 . The calculator
should say StorePic. The TI-82/83/84 allows you to save up to 10 different pictures. Next, we will need to
select where we would like to store the picture. Press: VARS , 4 . Now select where you would like to
store the picture in Pic1 through Pic9. After selecting the appropriate place to store the picture press ENTER .
After pressing enter the picture that you are storing should re-appear.
Repeat this process until all of the pictures are stored.
Next, we will need to create a program that displays each of the pictures in rapid succession to give the illusion
of movement. It may help at this point to turn off all of the plots under Y =.
Creating an Animation Program:
Start by creating a new program to do this press: PRGM and select NEW by pressing the ► , ► ,
ENTER and then typing in a name such as “A” , “N” ,“I” ,“M” ENTER .
(FnOff) Now, we should have an almost blank screen ready to be programmed. The first thing we will need to do is force the
calculator to be set up correctly. The first line we will put in a command to turn off any graphed equations. Press VARS , ,
4 ,and 2 ENTER.
(LBL 1) Next, we will need to set up a label so that the program can loop continuously through the
animated sequence. Press: PRGM , 9 , 1 , ENTER.
(ClrDraw) Now, we will need to clear the current screen to begin the animation.
Press: 2nd
, PRGM , 1 , ENTER.
(RecallPic Pic1) Then, we will need to recall the first picture in the animation sequence. Press: 2nd
,
PRGM , ◄ , 2 , VARS , 4 , 1 , ENTER.
(For) If we were to immediately clear this picture and display the next picture the animation would take place too quickly. So,
we will need to set up some type of delay while this picture is being displayed. This can be done with a quick “FOR – Loop” as
shown in example at the right The FOR command is found by pressing PRGM , 4 , X,T,,n , , ,1 , , ,3 ,0 , , ,1 , ) ,
ENTER. The (X, 1, 30, 1) shown in Example 2 stands for (Variable, Beginning Count Number, Ending Count Number, Count
By). PRGM , 7 (End) ENTER . Try changing the Ending Number (30) for different delays.
(ClrDraw) Now, we will need to clear the first picture. Press: 2nd
, PRGM , 1 , ENTER.
(RecallPic Pic2) Then, we will need to recall the first picture in the animation sequence. Press: 2nd
, PRGM , ◄ , 2 ,
VARS , 4 , 2 , ENTER.
Again, if we were to immediately clear this picture & display the next the animation would take place too quickly. We need to
create another “FOR-Loop”. PRGM , 4 , X,T,,n , , ,1 , , ,3 ,0 , , ,1 , ) , ENTER, PRGM , 7 ,ENTER .
If there are additional pictures to include in the animation the procedure is the same for adding more pictures.
(Goto 1) Finally, we need to loop the animation back to the beginning to set up a continuous animation sequence.
Press: PRGM , 0 , 1 . Go back to the HOME SCREEN 2nd
, MODE and Execute PRGM the ANIM program.
Example
DRAW
Press the ON button to interrupt the animation.
PROEJCT – SHEET (INITIAL PICTURES) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
PROEJCT – SHEET (TRANSFORMATIONS) NAME:
Sec 8.5 – Population & Leslie Matrices
Voting methods & Matrix Applications Name:
Suppose a new species of pig Sus-Gigantis in a small region had the following population
a. Calculate how many 0-2 year olds there will be in the next 5 year cycle.
b. Complete the Table for the next cycle and show your steps.
c. Complete the Table for the next cycle and show your steps.
Age Groups (in years) Year: 2008 0-2 2-4 4-6 6-8 8-10 10-12
# of Pigs in
initial
population
30
25
26
28
22
15
Age Groups
Birth
Rate
Survival
Rate
0-2 0.00 0.60
2-4 0.00 0.85
4-6 1.20 0.90
6-8 0.70 0.80
8-10 0.60 0.50
10-12 0.05 0.00
Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12
in initial pop
(Year:2008)
30
25
26
28
22
15
Population
(Year: 2010)
z Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12
in initial pop
(Year:2010)
Population
(Year: 2012)
DO THIS PROBLEM AGAIN BUT USING MATRICES………….
Suppose a species of pig, Sus-Gigantis, has characteristics as described below:
a. Create the initial population matrix and the corresponding LESLIE matrix.
b. How many Leslie Cycles would have been completed by the year 2010?
c. Using your calculator and the Matrices created in “part a” of this question determine the population of monkeys for
each age group in the year 2010.
d. How would you suggest estimating how many pigs were in each age group in the year 2011?
e. What would be your estimate for the year 2011 using your suggestion from part d?
f. Why is it necessary for the survival rate of the last group to be 0 for this model to work?
Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12
Population
(Year: 2010)
Age Groups (in years) 0-2 2-4 4-6 6-8 8-10 10-12
Population
(Year: 2011)
Age Groups
Birth
Rate
Survival
Rate
0-2 0.00 0.60
2-4 0.00 0.85
4-6 1.20 0.90
6-8 0.70 0.80
8-10 0.60 0.50
10-12 0.05 0.00
Age Groups (in years) Year: 2008 0-2 2-4 4-6 6-8 8-10 10-12
# of Pigs in
initial
population
30
25
26
28
22
15
2. Suppose a certain species of Elephants has characteristics as described:
a. Calculate how many 0-5 year olds there will be in the next 5 year cycle.
b. Complete the Table for the next cycle and show your steps.
c. Create the initial population matrix and the corresponding LESLIE matrix.
d. How many Leslie Cycles would have been completed by the year 2025?
e. Using your calculator and the Matrices created in “part a” of this question determine the population of elephants for
each age group in the year 2025.
f. It appears that the overall population is declining. If this trend continues how long (or in what year) would you estimate until the
population of all of the subgroup categories finally fall below 1 elephant.
Age Groups (in years) Year: 2000 0-5 5-10 10-15 15-20 20-25 25-30 30-35
# of elephants in
initial
population
20
30
10
30
12
10
6
Age Groups
Birth
Rate
Survival
Rate
0-5 0.00 0.50
5-10 0.10 0.70
10-15 0.40 0.80
15-20 0.90 0.70
20-25 1.10 0.60
25-30 0.30 0.40
30-35 0.00 0
Age Groups (in years) 0-5 5-10 10-15 15-20 20-25 25-30 30-35
in initial pop
(Year:2000)
20
30
10
30
12
10
6
Population
(Year: 2005)
Age Groups (in years) 0-5 5-10 10-15 15-20 20-25 25-30 30-35
Population
(Year: 2025)
g. BONUS: How would you suggest estimating elephant populations in each age group in 2026?
h. What would be your estimate for the year 2026 using your suggestion from part e?
Age Groups (in years) 0-5 5-10 10-15 15-20 20-25 25-30 30-35
Population
(Year: 2026)
Sec 8.6 – Markov Chains
Voting methods & Matrix Applications Name:
A Markov chain is a process that arises naturally in problems that involve a finite number of events or states that change over time.
A company is doing a research study in the metro-Atlanta area about smart phones (either android or
iPhone) and determined that 56% of the sample owned iPhones and the other 44% owned Android based
phones. After following the customers for a while, the researcher determined that each year after that 80%
of the iPhone owners purchased a new iPhone again when it came time to renew their contract. The
remaining 20% switched to Android. The researcher also determined that 65% of android owners purchased
a new Android phone again when it came time to renew their contract. The remaining 35% of android
owners switched to an iPhone.
a) Create a tree diagram showing the percentages of each owner after one cycle of renewals.
b) Create the initial distribution matrix Do
c) Create a Transition Matrix.
d) In the study there were initially 224 iPhone owners and 176 Android owners. If each participant
purchases a new phone every 2 years, determine how many of the participants would own iPhones and
how many would own Androids 8 years after the study began?
Suppose that in the Northwest region of the country, Ford dealers make up 40% of the automobile sales,
GM dealers make up 45% and foreign car dealers 15%. The foreign car dealers decide to offer a no-
interest-for-6-months-after-purchase incentive plan and during the next year of business, the following
transition matrix evolves:
Ford GM Foreign
Ford 0.50 0.20 0.30
GM 0.30 0.40 0.30
Foreign 0.15 0.10 0.75
Based on this matrix, what percentage of GM drivers will switch to Ford each year?
After year 3, what percentage of drivers will be driving Fords?
What percentage of drivers will be driving Fords long-term?
FROM
TO
A food service director for a local high school conducted a survey in hopes of predicting the number of
students who will eat in the cafeteria in the future. The results of the survey are as follows:
If a student eats in the cafeteria on a given day, the probability that he or she will eat there
again the next day is 70% and the probability that he or she will not eat there is 30%.
If a student does not eat in the cafeteria on a given day, the probability that he or she will eat
in the cafeteria the next day is 40% and the probability that he or she will not eat there is 60%.
On Monday, 70% ate at the cafeteria and 30% at somewhere else
a) Create a tree diagram showing the percentages of each owner after one cycle of renewals.
b) Create the initial distribution matrix Do
c) Create a Transition Matrix.
d) Find D3
e) How many students could be expected to eat at
the cafeteria on Friday if there were 350 students
that ate at the cafeteria and 150 that did not on
Monday?
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