Douglas R. Anderson, Professor Fall 2015: MWF …faculty.cord.edu/andersod/105bookFall2015.pdf · Douglas R. Anderson, Professor Fall 2015: MWF 10:30-11 ... Excursions in Modern Mathematics,

Math 105 WorkbookExploring Mathematics

Douglas R. Anderson, Professor

Fall 2015: MWF 10:30-11:40, Old Main 102

Acknowledgment

First we would like to thank all of our former Math 105 students. Their successes,struggles, and suggestions have shaped how we teach this course in many importantways.

We also want to thank our departmental colleagues and several Cobber mathematicsmajors for many fruitful discussions and resources on the content of this course andthe makeup of this workbook.

Some of the topics, examples, and exercises in this workbook are drawn from otherworks. Most significantly, we thank Samantha Briggs, Ellen Kramer, and Dr. JessieLenarz for their work in Exploring Mathematics, as well as Dr. Dan Biebighauserand Dr. Anders Hendrickson. We have also used:

• Taxicab Geometry: An Adventure in Non-Euclidean Geometry by Eugene F.Krause,

• Excursions in Modern Mathematics, Sixth Edition, by Peter Tannenbaum.

• Introductory Graph Theory by Gary Chartrand,

• The Heart of Mathematics: An invitation to effective thinking by Edward B.Burger and Michael Starbird,

• Applied Finite Mathematics by Edmond C. Tomastik.

Finally, we want to thank (in advance) you, our current students. Your suggestionsfor this course and this workbook are always encouraged, either in person or overe-mail. Both the course and workbook are works in progress that will continue toimprove each semester with your help.

Let’s have a great semester this fall exploring mathematics together and fulfillingConcordia’s math requirement in 2015.

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Contents

1 Taxicab Geometry 3

1.1 Taxicab Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Taxicab Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Taxicab Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 Taxicab Minimizing Regions . . . . . . . . . . . . . . . . . . . . . . . 23

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.5 Taxicab Midsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.6 Taxicab Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.7 Chapter Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.8 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2 Counting and Probability 55

2.1 Introduction to Counting . . . . . . . . . . . . . . . . . . . . . . . . . 55

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.2 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.3 Introduction to Probability . . . . . . . . . . . . . . . . . . . . . . . . 71

iii

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4 Complements, Unions, and Intersections . . . . . . . . . . . . . . . . 79

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.5 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


2.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3 Graph Theory 103

3.1 Introduction to Graph Theory . . . . . . . . . . . . . . . . . . . . . . 103

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

3.2 Paths and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.3 Subgraphs and Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4 Graph Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.5 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.6 Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143


3.8 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4 Consumer Mathematics 155

4.1 Percentages and Simple Interest . . . . . . . . . . . . . . . . . . . . . 155

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

4.2 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.3 Effective Annual Yield . . . . . . . . . . . . . . . . . . . . . . . . . . 168

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Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4.4 Ordinary Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

4.5 Mortgages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180


4.7 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

5 Voting Theory 189

5.1 Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

5.2 Voting Paradoxes and Problems . . . . . . . . . . . . . . . . . . . . . 201

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

5.3 Weighted Voting Systems . . . . . . . . . . . . . . . . . . . . . . . . . 211

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

5.4 Banzhaf Power Index . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Homework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.5 Voting Theory Homework Set . . . . . . . . . . . . . . . . . . . . . . 222

5.6 Supplement for Voting Theory: Antagonists . . . . . . . . . . . . . . 224


5.8 Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

A Projects 235

B Syllabus 237

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Chapter 1

Taxicab Geometry

1.1 Taxicab Distance

1. Suppose, in the city shown below, that we want to ride in a taxicab along citystreets from the corner of 8th Street and 10th Avenue to the corner of 3rdStreet and 13th Avenue.

(a) How many blocks does it take to make such a trip?

(b) Does every route in the city grid from the corner of 8th Street and 10thAvenue to the corner of 3rd Street and 13th Avenue take the same dis-tance?

(c) Does every route in the city grid from the corner of 8th Street and 10thAvenue to the corner of 3rd Street and 13th Avenue that continues tomake progress at every point take the same distance?

3

4 CHAPTER 1. TAXICAB GEOMETRY

Definition: The taxicab distance between two points is the shortest possibledistance between the two points where we are only allowed to move horizontallyor vertically.

Examples:

1.1. TAXICAB DISTANCE 5

2. (a) Graph the points A = (1, 3), B = (1,−2), C = (−3,−1), and D = (0, 3).

(b) Now find the following distances in both Euclidean and taxicab geome-tries. Give a decimal approximation to 2 decimal places.

Euclidean distance Taxicab distance

from A to B

from B to C

from C to D

(c) If you know the Euclidean distance between two points, does that tellyou what the taxicab distance is? Why or why not?

(d) If you know the taxicab distance between two points, does that tell youwhat the Euclidean distance is? Why or why not?


3. (a) Consider the points in the following graph:

AB

C

D

E

Calculate the following distances in both Euclidean and taxicab geome-tries. Give a decimal approximation to 2 decimal places.

Euclidean distance Taxicab distance

from A to B

from A to C

from A to D

from A to E

(b) Is the Euclidean distance between two points always less than or equal tothe taxicab distance? If so, explain why. If not, give an example wherethe Euclidean distance is greater than the taxicab distance.

1.1. TAXICAB DISTANCE 7

4. One night the 911 dispatcher for Taxicab City receives a report of an accidentat X = (−1, 4). There are two police cars in the area, car C at (2, 1) andcar D at (−1,−1). Which car should be sent to the scene of the accident toarrive most quickly? (Since the cars must drive on the streets, we use taxicabgeometry to measure distances.)

5. Find the taxicab distance between A = (13, 32) and B = (8

3,−1

2).


Homework

Throughout this chapter, all taxicab pictures should be completed on graph paper.

1. (a) On a single large graph, plot the following points:

A = (5, 4) B = (1, 2) C = (4,−3)D = (−1, 5) E = (−5,−4) F = (1,−2)

(b) Find the Euclidean distance between A and B.

(c) Find the taxicab distance between A and B.

(d) Find the Euclidean distance between B and F .

(e) Find the taxicab distance between B and F .

(f) Find the Euclidean distance between F and C.

(g) Find the taxicab distance between F and C.

2. Let C = (1, 0).

(a) Find five different points that are a taxicab distance of 5 from C, but arenot a Euclidean distance of 5 from C.

(b) Graph all of the points that are a taxicab distance of 5 from C, includingthose that are a Euclidean distance of 5 from C. Be sure to include allpossible points, not just the ones with integer coordinates.

(c) Come up with a mathematically appropriate name for the answer to part2b.

3. Let A = (−1, 1) and B = (3, 3).

(a) Find a point C so that the taxicab distance between A and C is the sameas the taxicab distance between B and C.

(b) Find a different point D so that the taxicab distance between A and Dequals the taxicab distance between B and D.

(c) Graph all of the points P where the taxicab distance between A andP and the taxicab distance between B and P is the same. Be sure toinclude all possible points, not just the ones with integer coordinates.

1.2. TAXICAB CIRCLES 9

1.2 Taxicab Circles

Definition: The taxicab circle centered at a point C with a radius of r (where r isa number, r ≥ 0) is all of the points that are a taxicab distance of r from C.

1. Draw the taxicab circle of radius 5 around the point P = (3, 4).


2. Draw the taxicab circle of radius 6 around the point Q = (2,−1).

3. (a) On a single graph, draw taxicab circles around the point R = (1, 2) ofradii 1, 2, 3, and 4.

(b) What are the taxicab perimeters (circumferences) of the circles? Do yousee a pattern?

(c) How many grid squares are inside of each of the circles? Do you see apattern?


4. Describe a quick technique for drawing a taxicab circle of radius r around apoint P .

5. Tyrion Lannister has fled King’s Landing and now works in Taxicab City forthe 3M plant, located at M = (1, 2). He goes out to eat for lunch once a week,and out of company loyalty, he likes to walk exactly 3 blocks from the plantto do so. Where in the city are restaurants at which Tyrion can eat? Drawtheir locations on the graph.

M


6. A developing company wants to construct an apartment building in TaxicabCity within six blocks of the mall at M = (−2, 1) and within four blocks ofthe tennis courts at T = (3, 3). Shade in the area of the graph that suits thebuilder’s requirements.


Homework

1. Graph all of the points that are a taxicab distance of exactly 4 from the pointA = (−2,−1).

2. Put your answers to these questions on separate graphs.

(a) Graph the taxicab circle that is centered at (1, 4) with a radius of 3.

(b) Graph the taxicab circle that is centered at (−1,−3) with a radius of 52.

3. Bernadette and Howard reside in Taxicab City, which is laid out like a perfectgrid centered on (0, 0). North-south and east-west streets join every pointwith integer coordinates. Bernadette works as an administrative assistant atan art school located at A = (−3,−1). Howard works as a bagel baker ina bakery located at B = (3, 3). Bernadette and Howard just got marriedand are looking for a house in the city. Bernadette has always dreamed of acozy little house on a corner lot, so they will only consider houses located atstreet corners. Because they will walk to their jobs along the city streets, theymeasure all their distances using taxicab geometry.

Put your answers to these questions on separate graphs.

(a) Is it possible for Bernadette and Howard to live exactly 5 blocks from Aand exactly 4 blocks from B? If so, find all locations that work and plotthem on a graph. If not, why is it impossible?

(b) Is it possible for Bernadette and Howard to live exactly 8 blocks from Aand exactly 6 blocks from B? If so, find all locations that work and plotthem on a graph. If not, why is it impossible?

(c) Is it possible for Bernadette and Howard to live 8 or fewer blocks fromA and 6 or fewer blocks from B? If so, find all locations that work andplot them on a graph. If not, why is it impossible?


4. Raj and Lalita Gupta also live in Taxicab City. Raj works at the cupcakeshop at C = (−2, 3) and Lalita works at the donut shop at D = (2,−1). Rajand Lalita are looking for places to live, but they do not necessarily have tolive at street corners.

Put your answers to these questions on separate graphs.

(a) Is it possible for Raj and Lalita to live within 5 blocks of C and within4 blocks of D? If so, find all locations that work and shade them on agraph. If not, why is it impossible?

(b) Raj and Lalita have a daughter, Priya, who goes to the elementary schoolat E = (3, 5). Is it possible for the family to live within 5 blocks of C,within 4 blocks of D, and within 3 blocks of E? If so, find all locationsthat work and shade them on a graph. If not, why is it impossible?

(c) Is it possible for the family to live within 5 blocks of C, within 4 blocksof D, and within 2 blocks of E? If so, find all locations that work andplot them on a graph. If not, why is it impossible?

5. Recall that a Euclidean square is a figure with four right angles and fourstraight sides of equal length, where we measure the length using Euclideandistance. A taxicab square is a figure with four right angles and four straightsides, and all four sides have the same length when measured using taxicabdistance. We’ve seen in class that every taxicab circle is actually a taxicabsquare. Is every taxicab square also a taxicab circle? If so, why? If not, givean example of a taxicab square that is not a taxicab circle.

1.3. TAXICAB APPLICATIONS 15

1.3 Taxicab Applications

1. Suppose we are at (2, 1) and we are severely wounded. There are hospitalsin our taxicab city at (−1, 1) and (4, 3). Where should we go, using taxicabgeometry? If we assume that we are out in a field and can use regular geometry,where should we go?

2. The mayor of Taxicab City promises to install drinking fountains so that everyperson living within the 12 × 12 square centered at the origin (0, 0) is withinthree blocks of a drinking fountain. There are three proposed plans from thecity council, shown below. Which plan should the mayor choose?


3. In the previous problem, you chose the better of the three given choices, buteven that one is not optimal in terms of the fewest drinking fountains (forexample, to contain costs). In the 12×12 grid below (one is for scratch work),place just 12 fountains. Hint: points for two fountains are provided, you placethe other 10 strategically.


4. In the same 12× 12 grid, the city decided to install fire hydrants so that everyresident is within 4 blocks of a fire hydrant. What is the fewest number ofhydrants needed, and where should they be located? (A few grids are providedfor practice below.)


5. Suppose now that the city is a 14×14 grid, and we still want every resident tobe within 4 blocks of a fire hydrant. What is the fewest number of hydrantsneeded, and where should they be located?

6. A furniture company wants to build a factory F in Taxicab City. They storeunfinished tables in their warehouse at W = (−3, 2); they want to ship theirtables from there to the new factory F , and then from F to their retail storeat S = (4, 0). If they want to minimize the total distance they ship the tables,where should they put their factory F? Shade in all locations for F on thegraph below.


7. Leonard is moving to Taxicab City to look for dark matter at D = (4,−2),so he is looking for an apartment. He walks to work along the city blocks.For various reasons, Leonard cannot live more than 5 blocks from work. On agraph, shade in all the places Leonard can live.

D

8. Now suppose Leonard wants to look for dark matter at D = (4,−2) and livenear Caltech at C = (0, 1). He is looking for an apartment A so that thedistance from A to D plus the distance from A to C is at most 9 blocks.Shade in all the places Leonard can live.

D

C


9. Acme Industrial Parts wants to build a factory in Taxicab City. It needs toreceive shipments from the railroad depot at R = (−5,−3) and ship partsout by plane, so it wants the factory to be located so that the total distancefrom the depot to the factory to the airport at A = (5,−1) is at most 16blocks. However, a city noise ordinance prohibits any factories from beingbuilt within 3 blocks of the public library at L = (−4, 2). Where can Acmebuild its factory?

R

A

L


Homework

1. (a) The city council of Taxicab City has decided to build parks on streetcorners so that every resident of Taxicab City is within 6 blocks of apark. If Taxicab City is currently a 14 × 14 grid, what is the minimumnumber of parks needed and where should they be located?

(b) Suppose the city already has a park located at (1, 2). Does that changethe minimum number of parks needed? If it does, what is the new numberof parks needed and where are they located? If it doesn’t change, whynot?

2. Bernadette and Howard reside in Taxicab City. Bernadette works as an ad-ministrative assistant at an art school located at A = (−3,−1). Howard worksas a bagel baker in a bakery located at B = (3, 3). Bernadette and Howardjust got married and are looking for a house in the city. Bernadette has alwaysdreamed of a cozy little house on a corner lot, so they will only consider houseslocated at street corners. Because they will walk to their jobs along the citystreets, they measure all their distances using taxicab geometry.

Put your answers to the following questions on different graphs.

(a) The newlyweds decide to find a house located so that the number ofblocks Bernadette has to walk to work plus the number of blocks Howardhas to walk to work is as small as possible. Where should they look fora house?

(b) Now Howard decides to be chivalrous and insist that Bernadette shouldnot have to walk any farther than he does, but they still want the totalamount of walking to be minimal. Now where should they look for ahouse?

(c) Bernadette decides to be generous in return, and wants both her husbandand herself to walk exactly the same distance to work. They still wantthe total amount of walking to be minimal. Now where should they lookfor a house?


(d) They still haven’t found a house! Having decided to widen their search,Bernadette and Howard keep only the requirement that they both walkthe same distance to work. (So now the total amount of walking doesnot need to be minimal.) Where should they look for a house?

(e) Suppose Bernadette and Howard only consider the following criterion:they want the total number of blocks they have to walk—Bernadette plusHoward together—to be at most 12 blocks. Where should they look?

3. Suppose that Jim and Pam live in Taxicab City, and that Jim works at (1,−1)while Pam works at (−3, 5). (You might be surprised to see that Jim and Pamwork at different places!) Suppose that Jim and Pam are looking for a houseand they want to minimize the total combined distance that they will walk towork. Pam must walk at least as far as Jim but no more than twice as faras Jim. They can only live on street corners. Where should they look for ahouse?

1.4. TAXICAB MINIMIZING REGIONS 23

1.4 Taxicab Minimizing Regions

Let’s revisit 6. from last time again:

1. A furniture company wants to build a factory F in Taxicab City. They storeunfinished tables in their warehouse at W = (−3, 2); they want to ship theirtables from there to the new factory F , and then from F to their retail storeat S = (4, 0). If they want to minimize the total distance they ship the tables,where should they put their factory F? Shade in all locations for F on thegraph below.

Definition: Given a collection of points A,B,C, . . . , the point or points Pfor which the total taxicab distance from P to A, from P to B, from P toC, . . . , is as small as possible is called the minimizing region of the pointsA,B,C, . . . .


2. Draw the minimizing region for the two points (1, 2) and (3, 4).

3. Draw the minimizing region for the two points (1, 2) and (1, 4).

4. Draw the minimizing region for the three points (1, 2), (1, 4), and (3, 3).

1.4. TAXICAB MINIMIZING REGIONS 25

Homework

1. Draw the minimizing region for each of the following sets of points. Draw eachminimizing region on a new graph. Be sure to include all possible points inthe minimizing region, not just the points with integer coordinates.

(a) A = (−2, 3), B = (1,−4)

(b) A = (1,−3), B = (4, 0)

(c) A = (2, 4), B = (7,−1), C = (−3, 1)

(d) A = (−3, 4), B = (4, 3), C = (0,−2)

(e) A = (−6, 0), B = (2, 4), C = (0, 4), D = (−1,−2)

(f) A = (−4, 0), B = (−1, 3), C = (3,−1), D = (1,−3)

(g) A = (−4, 0), B = (−3, 3), C = (0, 2), D = (3,−2), E = (−1,−2)

(h) A = (1, 1), B = (1, 4), C = (6, 1)

(i) A = (1, 1), B = (3, 1), C = (6, 1)

(j) A = (0, 0), B = (2, 2), C = (0, 4), D = (−5, 2)

(k) A = (0, 1), B = (1, 2), C = (2, 0), D = (4,−2), E = (1,−1)

2. In Taxicab City, Butter King has butter stands at eight corners: (−5, 5),(−2, 4), (1, 1), (2, 6), (5,−2), (3,−4), (−2,−1), and (−4,−4). Throughoutthis problem, assume that buildings can only be located at corners with inte-ger coordinates, and that warehouses and butter stands can be at the samelocation.

(a) Butter King wants to build a central supply warehouse so that the sumof the distances from the warehouse to the eight butter stands is as smallas possible. Where could the warehouse be located?

(b) What is the total distance from the warehouse(s) in your answer to part(a) to the eight butter stands?

(c) Suppose now that the butter stand at (2, 6) does so much business thatit requires twice as many deliveries from the central warehouse as eachof the other stands. Now where should the warehouse be located?

3. If a collection of points consists of an odd number of points, what can you sayabout its minimizing region?


1.5 Taxicab Midsets

1. Using taxicab geometry, consider the points A = (−3, 2) and B = (3, 0).

A

B

(a) Is the point (−2,−3) closer to A or to B?

(b) Is the point (1,−2) closer to A or to B?

(c) Find one point that is exactly the same distance from A as it is from B.Mark it on the graph.

(d) Find another such point. Mark it on the graph.

(e) Mark all points on the graph that are equally distant from A and fromB. (Remember, this includes points with non-integer coordinates.)

1.5. TAXICAB MIDSETS 27

Definition: Given two points A and B, the midset of these points is thecollection of all points (not just the ones with integer coordinates) that arethe same distance from A as from B.

Example from Euclidean geometry:

Consider A = (2, 0) and B = (0, 4).

A

B


2. Consider the points in the following diagram.

A

B

C

D

E

F

(a) Find the taxicab midset of A and B.

(b) Find the taxicab midset of C and D.

(c) Find the taxicab midset of E and F .

3. Given two points X and Y , describe a rule for finding the taxicab midset ofX and Y .


4. Find the taxicab midset of G and H.

G

H

5. When, if ever, is the taxicab midset the same as the Euclidean midset?


6. (a) Taxicab City has two fire stations: Firehouse North located at N = (1, 6)and Firehouse South at S = (4,−3). Where should the fire departmentdraw a district boundary line so that all homes within the district areserved by the closest fire station?

N

S

(b) Taxicab City has grown so much that the city council decided to builda third fire station. Firehouse West will be located at W = (−2,−1).Where should the fire district boundary lines be now?

N

S

W

(c) The fire department decides to build a training facility in Taxicab City.Where should they put the training facility if they want it to be equallydistant from all three fire stations?


Homework

1. Find the midset of A = (1, 3) and B = (−3, 5). (Recall that midsets caninclude points that do not have integer coordinates.)

2. Find the midset of C = (0,−2) and B = (−3, 5).

3. Find the midset of A = (1, 3) and D = (−3,−1).

4. Taxicab City has two high schools, Taxicab West at (−4, 3) and Taxicab Eastat (2, 1). (Put your answers to the following questions on different graphs!)

(a) Where should they draw the school district boundary line so that eachstudent attends the high school nearest their home? (As always, wemeasure distances in taxicab geometry.)

(b) The city builds a third high school, Taxicab South, at (−1,−6). Nowwhere should the boundary lines be drawn?

(c) The owner of Pizza Palace wants to set up a pizzeria P that is equallydistant from all three high schools. Where should the owner build therestaurant?

(d) Wow! Our city is really growing! When Taxicab North High School isbuilt at (2, 5), where should the school district boundary lines be drawn?

5. Bernadette and Howard have adopted a boy named Raj, who attends Chester-ton Elementary School at C = (0,−3). Bernadette still works at the art schoolA = (−3,−1) and Howard at the bagel bakery at B = (3, 3). The family doesnot necessarily have to live at street corners.(Put your answers to the following questions on different graphs!)

(a) Where should they live so that each of the three of them has the samedistance to walk to work or school?

(b) Where should they live if they have decided that Raj should have theshortest walk, Bernadette the second shortest walk, and Howard thelongest walk? Shade the appropriate region of your graph.

6. Given two points A and B, we know that the minimizing region for these twopoints is either a line segment joining A and B or a rectangle.

(a) If the minimizing region for A and B is a line segment, what can we sayabout the midset of A and B?

(b) Of course, some rectangles are squares, and others are not squares. If theminimizing region for A and B is a square, what can we say about themidset of A and B?


1.6 Taxicab Lines

1. In nearby Omnibus City, a river runs through town on a line running through(0,−1) and (2, 2) as shown.

J

(a) Josephine currently lives in an apartment at J = (−3, 2). What point onthe river is closest to her apartment (in taxicab geometry, of course)?

(b) How far is Josephine’s apartment from the river (in taxicab geometry, ofcourse)?

(c) Josephine wants to move to a scenic apartment within three blocks’ walkof the river. Where should Josephine look for an apartment?

1.6. TAXICAB LINES 33

2. The bike path in Taxicab City runs on a line through (−5,−3) and (3,−1),as shown. Hilda lives at H = (3, 3).

H

(a) Hilda is recuperating from an accident and can’t walk very far. She wantsto know where she can go if she only walks two blocks. Draw the taxicabcircle representing the places she can visit.

(b) Hilda’s recovery is proceeding well from week to week. Draw circles rep-resenting how far she can go if she walks 3 blocks, 4 blocks, or 5 blocks.

(c) How far is Hilda’s house from the bike path?

3. The bike path in Taxicab City runs on a line through (−5,−3) and (3,−1),as shown. How far is City Hall C = (−1, 1) from the bike path?

C


4. Fred is a city engineer preparing to hook up the electricity to Taxicab City’snew football stadium at F = (3,−1). He needs to hook into a main power linethat runs along a line from (−7,−5) to (5, 7).

F

His cable needs to be buried along the city streets. How many blocks’ worthof cable does he need to reach from the stadium to the power line? Whatroute(s) could the cable take?

(Challenge: How many different minimal routes could the cable take?)


Definition: Consider a point P and a line `. The taxicab distance from P to` is the taxicab distance from P to the closest point (out of all points) on `.

5. Come up with a general rule for deciding how far a point is from a line intaxicab distance.

P

A

B

P

B

A

P

A′

A

Comparison to Euclidean geometry:

P

A

P

A


6. Sheldon and Amy are moving to town. Sheldon got a job at the Synchrotronat S = (−3,−1), whereas Amy will be working for the city light rail line ` thatruns through the city as shown. One of Amy’s fringe benefits is that when shecomes to work she can just get on the train wherever is closest to her home.They measure all of their distances using taxicab geometry, and they do notneed to live at street corners!

(a) Sheldon and Amy want to live where the distance Sheldon has to walk towork plus the distance Amy has to walk to work is a minimum. Whereshould they look?

`

S

(b) They change their minds and decide to live where they both walk thesame distance to work. Where should they look?

`

S


(c) Where should they look if all that matters is that Sheldon have a shorterdistance to walk than Amy?

`

S

The shape in part (b) is called a taxicab parabola.


Homework

1. Find the taxicab distance between the point A = (1, 2) and the line that passesthrough the points (3, 0) and (4, 2).

2. (a) Draw the line that passes through (0, 3) and (−4, 1). Call it `.

(b) Find two different points that are a taxicab distance of 2 from the line `.Name their coordinates.

(c) Draw a picture of all the points that are a taxicab distance of 2 from theline `. (Points with non-integer coordinates are allowed!)

3. (a) Draw the line that passes through (0, 0) and (1, 3). Call it `.

(b) Find two different points that are a taxicab distance of 4 from the line `.Name their coordinates.

(c) Draw a picture of all the points that are a taxicab distance of 4 from theline `. (Points with non-integer coordinates are allowed!)

4. Find a line ` carefully chosen so that the collection of points that are a taxicabdistance of 3 from ` and the collection of points that are a Euclidean distanceof 3 from ` are the same. In other words, every point that is a taxicab distanceof 3 from ` is also a Euclidean distance of 3 from `, and vice versa.

5. Sheldon got a job at the Synchrotron at S = (−3,−1), whereas Amy will beworking for the city light rail line ` that runs through the city as shown. Oneof Amy’s fringe benefits is that when she comes to work she can just get onthe train wherever is closest to her home. Sheldon and Amy do not necessarilyhave to live at street corners.

`

S

(a) Sheldon and Amy want to live where the distance Sheldon has to walkto work is no more than 2 blocks and the distance Amy has to walk to


work is no more than 3 blocks. Where should they look? Shade in youranswers on a graph.

(b) They change their minds and decide to live where they both walk exactlythree blocks to work. Where should they live? List the coordinates ofthe points.

6. Alex lives at the point A = (−2, 2) in Taxicab City, and Bonnie lives at thepoint B = (2,−5). A railroad track has been constructed along the line joining(0, 0) and (3, 1) as shown, and there are only three places to cross the railroad.The crossings are at C1 = (−3,−1), C2 = (0, 0), and C3 = (3, 1). If Alex wantsto walk to Bonnie’s house as quickly as possible, which crossing should he use,and how many blocks does it take him in total to get to Bonnie’s house?

A

B

C1

C2

C3


7. Find the taxicab distance from the point A to the shaded region in the figurebelow. Round your answer to the nearest whole number. (One way to findthe answer is to draw concentric taxicab circles centered at A until they touchthe shaded region.)

8. Hazel and Gus are moving to town. Hazel is an engineer for the city light railline ` that runs through the city as shown. One of Hazel’s fringe benefits isthat when she goes to work she can just get on the train at the point closestto her house. Gus is a dentist with a practice located at the point D = (2, 2).

They decide to live where they both walk the same distance to work. Whereshould they look?

`

D

1.7. CHAPTER PROJECTS 41

1.7 Chapter Projects

1. Discuss the definition and history of the number π, including attempts tocalculate the digits of π. Explain what the value of π should be in taxicabgeometry (it’s different!).

2. Taxicab Triangles: Does the Euclidean version of the Pythagorean Theoremhold for taxicab triangles? If not, come up with a replacement theorem in taxi-cab geometry. Then, determine how to circumscribe a taxicab circle arounda triangle. When is more than one circumscribing taxicab circle possible?When are there no taxicab circles to circumscribe a triangle? Ask me for somematerials to help you get started on this project.

3. Describe various methods to show how two triangles are congruent in regularEuclidean geometry, including SSS (side-side-side) and SAS (side-angle-side).Determine whether or not these methods work for taxicab geometry.

4. Present various ways in which the taxicab geometry model can be modified tomore accurately describe the real world. For example, maybe there are one-way streets, or certain streets that are missing. Maybe there is an expresswaythat allows one to travel faster or skip over some blocks. Maybe there isconstruction or other detours. Explore these models with specific examples.


1.8 Chapter Review

Concepts:

• Euclidean (regular) distance

• Taxicab distance

• Taxicab circles

• Taxicab squares (in comparison to taxicab circles)

• Covering a grid with taxicab circles

• Minimizing regions

• Midsets

• Distances from points to lines, taxicab parabolas

Some Review Exercises:

1. In our taxicab city, we decide to install fire hydrants so that every residentliving in the 12×12 grid shown below is within four taxicab blocks of a hydrant.Draw a configuration of seven or fewer hydrants so that every resident in thisgrid is covered.

1.8. CHAPTER REVIEW 43

2. Draw the minimizing region for each of the following sets of points. Be sure toinclude all possible points in the minimizing region, not just the points withinteger coordinates.

(a) A = (−6, 0), B = (2, 4), C = (0, 4), D = (−3,−2)

(b) A = (−4, 0), B = (−3, 3), C = (0, 2), D = (1,−2), E = (−1,−2)


3. Bernadette and Howard live in our taxicab city. Bernadette works as an airtraffic controller at an airport that is located at A = (2, 2). Howard works asa baker at a bakery that is located at B = (−4,−2). They are looking fora house in this city, and they only want to consider houses that are locatedat street corners. They will walk to their jobs, and they walk only along citystreets, so they measure all of their distances using taxicab geometry.

(a) Is it possible for Bernadette and Howard to live exactly 5 blocks from Aand exactly 5 blocks from B? If so, plot all such locations that work onthe graph below. If not, why is it impossible?

(b) Is it possible for Bernadette and Howard to live 8 or fewer blocks from Aand 4 or fewer blocks from B? If so, plot all such locations that work onthe graph below. If not, why is it impossible?

(c) Plot all of the points where a house could be located so that Bernadetteand Howard each walk the same distance to work.


4. Our taxicab city has three high schools, Taxicab Halpert, Taxicab Schrute, andTaxicab Kapoor. Halpert is at (−4,−4), Schrute is at (2,−2), and Kapoor isat (−1, 5). Where should they draw the school district boundary lines so thateach student attends the high school nearest their home? (As always in ourcity, we measure distances using taxicab geometry.) For your lines, be sure toinclude all possible points, not just the points with integer coordinates.

If the owner of Windy’s wants to set up a wind stand that is equally distantfrom all three high schools, where should the owner put the stand?

5. Find (i) the Euclidean distance and (ii) the taxicab distance between the givenpoints.

(a) A = (1,−1), B = (2,−2)

(b) C = (0, 1), D = (0, 3)


6. Draw the set of all points that are exactly taxicab distance 5 from the point(−1,−2).

7. Draw the taxicab circle centered at the given point P with the given radius r.

(a) P = (−1, 2), r = 2

(b) P = (1,−3), r = 3


8. In our taxicab city, a builder wants to construct an apartment building within3 blocks of the mall at M = (2, 3) and within 2 blocks of the health club atH = (5, 4). Find all points where the builder might build.

9. Twinville has decided to set up stands selling Twins memorabilia in such a waythat every resident is within 5 blocks of a stand. Using the grid below, whatis the minimum number of stands needed and where should they be located?


10. Potterville has two high schools, Hogwarts High School, located at H =(−3,−1), and Pigbunions High School, located at P .

(a) If P = (−3, 5), where should the city draw the district line so that everystudent attends the high school closest to their home?

(b) If P = (3, 3), where should the city draw the district line so that everystudent attends the high school closest to their home?

(c) If P = (1, 3), how should the city map the school districts so that everystudent attends the high school closest to their home?

(d) If the city now has three schools at H = (−3,−1), P = (3, 3), and S =(6,−6), where should they draw the new district lines?

11. Find the set of all points P so the sum of the distance from A to P and P toB is exactly k units.

(a) A = (−1, 1), B = (3,−3), k = 10

(b) A = (2, 1), B = (5, 1), k = 5

(c) A = (−2, 2), B = (3, 5), k = 12


12. Find the distance from the point P to the line through the points A and B.

(a) P = (2,−1), A = (2, 3), B = (−1,−3)

(b) P = (4, 2), A = (−1, 1), B = (2, 4)

(c) P = (−3,−3), A = (−2, 2), B = (4,−1)

13. Graph the set of all points exactly d units from the line through the points Aand B.

(a) d = 1, A = (−2, 2), B = (4,−1)

(b) d = 3, A = (−1, 1), B = (2, 4)


14. Draw the minimizing region for the set of points

(a) A = (−1,−1), B = (1, 1), C = (−1, 3), D = (−6, 1).

(b) A = (−4, 2), B = (−1, 5), C = (3, 1), D = (1,−1).

(c) A = (−3,−4), B = (4,−6).


Some Review Answers:

1.

2.

(a)

(b)

3.

(a)

(b)

(c)


4. Windy’s should be at (−2, 0).

5. (a) (i)√

2; (ii) 2. (b) (i) 2; (ii) 2

6.C

7.

C

C

8.

MH


9. Minimum number is 6.

10. (a),(b)

P

H

P

H

(c),(d)

P

H

P

H

S

11.

A

B

A BA

B

12. (a) 2; (b) 4; (c) 5.5


A

B

PA

B

P

A

B

P

13.

A

B

`

A

B

`

14. (a) (−1, 1); (b) red rectangle; (c) green rectangle

A

B

C

DA

B

C

DA

B

Chapter 2

Counting and Probability

2.1 Introduction to Counting

1. We flip a fair coin 3 times. How many different sequences of Heads and Tailsare possible?

55

56 CHAPTER 2. COUNTING AND PROBABILITY

Multiplication Principle of Counting:

Suppose a task can be divided into m consecutive subtasks.

If

Subtask 1 can be completed in n1 ways, and then

Subtask 2 can be completed in n2 ways, and then

...

Subtask m can be completed in nm ways,

then the overall task can be completed in ways.

2. If we flip a fair coin 5 times, how many sequences of Heads and Tails arepossible? What about 10 times?

3. There are 26 letters (A-Z) and 10 digits (0-9). If a license plate must contain 3letters followed by 3 digits, how many license plates are possible if repetitionsare allowed? What if repetitions are not allowed?

2.1. INTRODUCTION TO COUNTING 57

Definition: A permutation of a collection of different objects is an orderedarrangement of the objects.

Definition: For any positive integer n, we define n factorial to be

n! = n · (n− 1) · (n− 2) · · · · · 3 · 2 · 1.

We also define 0! = 1.

Definition: A permutation of r objects from a collection of n different ob-jects is an ordered arrangement of r of the n objects. The number of thesepermutations is denoted by P (n, r).

4. If we have a division of 8 basketball teams and we decide to rank our top 2teams (first and second place), how many choices are possible? What if wewant to rank the top 4 teams? All 8 teams?


Formulas for P (n, r):

5. If we have a popularity contest in this class and choose a first, second, andthird place person, how many choices are possible?


6. Suppose we flip a fair coin 5 times. How many sequences of Heads and Tailsstart with either two Heads or with two Tails?

7. Dwight needs to build a password consisting of five different upper-case letters.How many different passwords are possible if Dwight must use the letter Bsomewhere in his password?

More practice problems on the next page!


1. Multiplication Principal of Counting restated:

If one event can occur in a ways, and for each of those a ways another eventcan occur in b ways, then the total number of events is the multiplication a×b.

2. Suppose 3 students are on a road trip, and all 3 are willing to drive. In howmany ways can they be seated in the car?

3. Four students are returning from break. If all are willing to drive, in how manyways can they be seated in the car?

4. A 6 member board sits around a table. How many different seating arrange-ments are possible?

5. A 6 member board self selects a president and a treasurer. In how many wayscan this be done?

6. A true/false quiz has 5 questions. In how many ways can the quiz be com-pleted?

7. A softball team with 9 players needs a batting order. How many different waysare possible if:

• there are no restrictions;

• the pitcher bats last;

• the catcher bats last and the pitcher bats anywhere but first?

8. An identification tag has 2 letters followed by 4 numbers. How many differenttags are possible if:

• repetition of letters and numbers is allowed?

• repetition of neither letters nor numbers is allowed?

9. The athletic department wants a picture for a brochure that includes 4 of the11 starting offensive players from the football team on the left, 3 of the 6volleyball players in the center, and 2 of the 5 starting basketball players onthe right. How many possible pictures are there?

10. Evaluate P (8, 5), then P (9, 8).

11. The starting 9 players on the school baseball team and the starting 5 playersfrom the basketball team are to line up for a picture, with all members of thebaseball team together on the left. How many ways can this be done?


Section 2.1 Answers to Practice Problems

1. Just read.

2. 3! = 6 ways.

3. 4! = 24 ways.

4. 6! = 720 ways.

5. P (6, 2) = 6× 5 = 30.

6. 25 = 32 ways.

7. A softball team with 9 players needs a batting order. How many different waysare possible if:

• there are no restrictions;

9! = 362, 880

• the pitcher bats last;

8! = 40, 320

• the catcher bats last and the pitcher bats anywhere but first?

7!× 7 = 35, 280

8. An identification tag has 2 letters followed by 4 numbers. How many differenttags are possible if:

• repetition of letters and numbers is allowed?

262 × 104 = 6, 760, 000

• repetition of neither letters nor numbers is allowed?

P (26, 2)× P (10, 4) = 26× 25× 10× 9× 8× 7 = 3, 276, 000

9. P (11, 4)×P (6, 3)×P (5, 2) = (11×10×9×8)×(6×5×4)×(5×4) = 19, 008, 000

10. Evaluate P (8, 5), then P (9, 8).

P (8, 5) = 6720 and P (9, 8) = 362, 880.

11. 9!5! = 43, 545, 600


Homework

1. Simplify the following fractions as much as possible (you will get a wholenumber as the answer in each case):

(a)8!

4!

(b)7!

4! · 3!

(c)100!

98!

(d)9!

5! · 4!

(e)n!

(n− 1)!, where n is any integer greater than or equal to one

2. An urn holds 6 balls: a red ball, an orange ball, a yellow ball, a green ball, ablue ball, and a purple ball. Meredith selects one ball from the urn, and then,without replacing the first ball, she selects a second ball. How many choicesare possible for Meredith?

3. Suppose your debit card PIN must be four digits (0-9).

(a) If there are no restrictions on the digits for your PIN, how many choicesare possible?

(b) If you are not allowed to repeat digits for your PIN, how many choicesare possible?

(c) If your PIN must contain the digits 2 and 5 (in some order, and notnecessarily as consecutive digits) and you are not allowed to repeat digits,how many choices are possible?

4. Suppose your e-mail password must be exactly 8 characters long.

(a) If the password must be 6 upper-case letters (A-Z) followed by 2 digits(0-9), how many choices are possible?

(b) If the password must be 7 upper-case letters followed by a single digit,how many choices are possible?

(c) If the password must contain 7 upper-case letters and one digit, but thedigit can be in any position, how many choices are possible?


5. A large kindergarten class of 51 students lines up single-file to walk to recess.How many different lines are possible?

6. Find the number of ways to rearrange the letters in the following words:

(a) MATH

(b) IVERS

(c) DOOR

7. Suppose Alice, Bob, Connie, Daniel, Erika, and Fred are the six membersof a committee, and they need to elect the following officers: Chairperson,Secretary, and Treasurer. (The officers are members of the committee.)

(a) How many different ways are there to select the officers?

(b) How many selections are there in which Daniel is not an officer?

(c) How many selections are there in which neither Connie nor Daniel is anofficer?

(d) How many selections are there in which Connie and Daniel are bothofficers?

(e) How many selections are there in which Connie or Daniel may or maynot be officers, but Connie and Daniel are definitely not both officers?


2.2 Combinations

1. Suppose we have three people: Angela, Bob, and Chuck.

(a) If we need to select one to be president and one to be vice-president, howmany choices are possible?

(b) If we instead only need to select 2 of the 3 people to be “leaders,” thenhow many choices are possible?

Definition: A combination of r objects from a collection of n different objectsis a selection of r of the objects where the order of the objects selected is notimportant. The number of these combinations is denoted by C(n, r).

2. Suppose we have four people: Angela, Bob, Chuck, and Dani. How many waysare there to select three of these four people:

(a) If order does matter

(b) If order doesn’t matter

2.2. COMBINATIONS 65

Finding a formula for C(n, r):

3. If we pick 2 teams from 8 teams to go to the playoffs, where the order of theteams is irrelevant, how many choices are possible?

4. A small company has 12 employees. They will send 3 to a meeting in Min-neapolis, another 1 to a meeting in Chicago, another 1 to a meeting in Mil-waukee, and another 1 to a meeting in Moorhead. How many different choicesare possible?


5. (a) Domino’s Pizza offers 7 pizza toppings and allows you to put exactly 3toppings on your pizza. How many choices are there?

(b) If you can put up to 3 toppings on your pizza, how many choices are nowpossible?

6. Suppose we flip a coin 10 times.

(a) How many sequences of Heads and Tails are possible?

(b) How many sequences have exactly 7 Heads?

(c) How many sequences have at least 7 Heads?

(d) How many sequences have at least 1 Head?

More practice problems on the next page!


1. A 10-member board self selects a president, a vice president, and a treasurer.In how many ways can this be done? (Order matters)

2. A 10-member board self selects a subcommittee of 3. In how many ways canthis be done? (Order no longer matters).

3. Evaluate C(9, 5)

4. Evaluate C(5, 3)

5. How many different 5-card poker hands can be dealt from a standard deck of52 cards?

6. In how many different ways can the 9-member US Supreme Court reach a 6-3decision?

7. A quiz contains 5 true/false questions.

• In how many ways can the quiz be completed?

• How many of the ways from (a) contain exactly 3 correct answers?

• How many of the ways from (a) contain at least 3 correct answers?

8. A committee of 13 has 7 women and 6 men. In how many ways can a sub-committee of 5 be formed if it consists of:

• all women?

• any 5 people?

• exactly 2 men and 3 women?

• at least 3 men?

• at least 1 woman?

9. A telemarketer makes 15 phone calls in 1 hour. In how many ways can theoutcomes of the calls be 3 sales, 8 no-sales, and 4 answering machines?


Section 2.2 Answers to Practice Problems

1. P (10, 3) = 10!(10−3)! = 10!

7!= 10× 9× 8 = 720.

2. P (10,3)3!

= 10!(10−3)!3! = 10!

7!3!= 10×9×8

3×2×1 = 120.

3. C(9, 5) = 9!4!5!

= 126

4. C(5, 3) = 5!2!3!

= 10

5. C(52, 5) = 2, 598, 960

6. C(9, 6) = 84

7. A quiz contains 5 true/false questions.

• 25 = 32

• C(5, 3) = 10

• C(5, 3) + C(5, 4) + C(5, 5) = 16

8. A committee of 13 has 7 women and 6 men. In how many ways can a sub-committee of 5 be formed if it consists of:

• all women?

C(7, 5) = 21

• any 5 people?

C(13, 5) = 1287

• exactly 2 men and 3 women?

C(6, 2)C(7, 3) = 525

• at least 3 men?

C(6, 3)C(7, 2) + C(6, 4)C(7, 1) + C(6, 5)C(7, 0) = 531

• at least 1 woman?

Total number minus the number with no women: C(13, 5) − C(6, 5) =1281, or C(6, 4)C(7, 1) + C(6, 3)C(7, 2) + · · ·+ C(6, 0)C(7, 5) = 1281

9. C(15, 3)C(12, 8)C(4, 4) = 225, 225


Homework

1. Answer the following counting questions. Answers with P s and Cs are fine, ifyou do not want to calculate the actual numbers.

(a) In a regional track meet, 16 runners are competing in the 10000-meterrun. The top eight runners qualify for the national meet. How manydifferent ways are there for the runners to qualify for the national meet?(The order of the qualification doesn’t matter.)

(b) In how many ways can we choose 17 flowers from a collection of 51 flowersto use in a vase?

(c) A couple is planning a menu for a rehearsal dinner. The restaurant makesthem choose 4 of the 7 possible appetizers, 3 of the possible 6 maincourses, and 2 of the possible 5 desserts. How many different menus arepossible?

(d) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have three boys and two girls?

(e) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have two boys and three girls?

(f) From a group of seven boys and eight girls, how many 5-player teams canbe formed that have at most two boys and at least three girls?

(g) In how many ways can 15 teams be divided up so that 5 are in the NorthDivision, 5 are in the Central Division, and 5 are in the South Division?

(h) Given 10 points on a sheet of paper, where no three points lie on the sameline, how many triangles can be drawn that use three of these points asvertices?

(i) In a group of 12 people, we select a committee of 5 members. One memberof the committee is the chairperson, and another is the secretary. Howmany different choices for this committee are possible?

(j) Michael, Dwight, Jim, and Pam decide to play a game of cards during aslow day in the paper business. In how many different ways can the 52cards in a deck be dealt to the four players, where each player gets 13cards? Assume that the order of the cards in each player’s hand does notmatter.

(k) Suppose there are three divisions of five teams each. One team from eachdivision will go to the playoffs, along with one other team that can comefrom any of the divisions. How many different combinations of teams cango to the playoffs, where the “wild-card” team is counted just like any


other team? In other words, in how many ways can we select four teamsto go to the playoffs where the order of the teams is irrelevant, but everydivision sends at least one team?

2.3. INTRODUCTION TO PROBABILITY 71

2.3 Introduction to Probability

Definition: An experiment is a process that can be repeated and has observableresults, which are called outcomes. A sample space for an experiment is a set of allof the possible outcomes.

Definition: A sample space is uniform if all of the outcomes in the sample spaceare equally likely to occur.

Definition: An event is a subset of the sample space.


Experimental Probability:

To find the experimental probability of an event, we perform the experiment for alarge number of times and count the number of times our event occurs. Then wedivide the number of times our event occurred by the total number of experiments.

Theoretical Probability:

Suppose the sample space for our experiment is called S, and that S is a uniformsample space (so every outcome is equally likely, in theory, to occur). Then if E isan event (a subset of S), the theoretical probability of E is denoted by P (E) and isgiven by

P (E) =Number of outcomes in E

Number of outcomes in S.


1. Suppose we roll 2 fair dice (each numbered 1 through 6). What are the theo-retical probabilities of the following events?

(a) A = “Roll doubles (each die is the same)”

(b) B = “Roll a total of exactly 9”

(c) C = “Roll at least one 5”


Cards:

A standard deck of cards contains 52 cards. These cards are divided into 4Suits :

Diamonds, Hearts, Clubs, Spades

Diamonds and Hearts are Red, and Clubs and Spades are Black.

Each Suit is divided into 13 Ranks :

Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King

Jacks, Queens, and Kings are Face Cards.

2. Suppose we select 5 cards from a standard 52-card deck.

(a) What is the probability that we get 5 face cards?

(b) What is the probability that we get exactly 2 Spades?


3. A coin is flipped, and it is noted whether heads or tails show. Then a die isrolled, and the number on the top face is noted. What is the sample space ofthis experiment? Indicate the outcomes of the event “the coin shows tails andthe die shows at least 3”.

4. An urn holds 10 identical balls except that 1 is white, 4 are blue, and 5 arered. An experiment consists of selecting two balls consecutively from the urnwithout replacement and observing their color. What is the sample space ofthis experiment? Indicate the outcomes of the event “neither ball is white”.

5. An executive must form an ad hoc committee of 3 people from a group of5, {A,B,C,D,E}. What is the sample space? Indicate the outcomes of theevent “D is selected”.

6. An experiment requires 3 coins to be flipped 1000 times. The results arerecorded in the following table; fill in the third column.

Outcome Frequency Experimental ProbabilityHHH 112HHT 129HTH 120HTT 118THH 133THT 136TTH 128TTT 124

If E is the event of getting exactly 2 tails, find the experimental probabilityof event E.

7. A bin contains 15 components that look identical, but actually 6 are defective.What is the probability that a component selected at random is defective?


8. A bin contains 15 components that look identical, but actually 6 are defective.Suppose one component is selected from the bin, and then another is selectedwithout replacing the first. What is the probability that both components aredefective?

9. A bin contains 15 components that look identical, but actually 6 are defective.What is the probability of selecting 5 components from the bin with 2 defectiveand 3 non-defective?

10. A fair coin is flipped 6 times. Assuming that any outcome is equally likely,find the probability of obtaining exactly 3 heads.

11. Two fair dice are rolled. Find the probability that a sum shows that is equalto 7.

12. A 2-card hand is drawn from a standard deck of 52 cards. Find the probabilitythat the hand contains 2 kings.

13. A 2-card hand is drawn from a standard deck of 52 cards. Find the probabilitythat the hand contains two spades.

14. You wish to invest in the stock market. Among a group of 20 stocks, supposethat 10 stocks will go up and 10 will go down. If you pick 3 stocks at randomfrom the group of 20, what is the probability that all 3 will go up?


Homework

1. A standard six-sided die (1-6) is rolled 100 times. The results of the rolls are

in the following table.

Outcome Frequency1 142 193 184 165 176 16

Find the experimental probability of:

(a) Rolling a 5

(b) Rolling an odd number

(c) Rolling a number less than 3

(d) Rolling a number at least as big as 3

2. A fair six-sided die is rolled once. Find the probability of:

(a) Rolling a 5

(b) Rolling an odd number

(c) Rolling a number less than 3

(d) Rolling a number at least as big as 3

3. A card is randomly selected from a standard 52-card deck. What is the prob-ability that the card is:

(a) A Heart?

(b) A Queen?

(c) A Club or a Spade?

(d) A face card?

4. Two fair six-sided dice are rolled once.

(a) What is the probability that the sum of the dice will be 5?

(b) What is the probability that the sum of the dice will be 11?

(c) What is the probability that the sum of the dice will be 12?

5. A Math 105 class consists of 14 females and 10 males. The dapper instructorrandomly selects 8 of the students to each earn some bonus points. What isthe probability that all 8 of the students selected are females?


6. The English alphabet has 26 letters, 5 of which are vowels. If Alicia randomlyselects 4 different letters from the alphabet, what is the probability that all4 letters are vowels? What is the probability that none of the 4 letters arevowels?

7. An urn contains 10 identical balls except that 1 is white, 4 are red, and 5 areblack.

(a) An experiment consists of selecting a ball from the urn and observing itscolor. What is a sample space for this experiment?

(b) Find the probability of the event “the ball is not white.”

(c) Suppose we have the same urn with the same 10 balls, but our experimentnow is to select 2 balls without replacement (so we select a ball, remove it,and then select another ball). What is a sample space for this experiment?(Your sample space does not need to be a uniform sample space.)

(d) Write all of the outcomes in the event “the white ball is not selected.”

8. A Math 105 class consists of 19 females and 15 males. The friendly instructorselects 6 of these students to form a Math 105 Dance Planning Committee.

(a) What is the probability that the committee consists of exactly 3 femalesand exactly 3 males?

(b) What is the probability that the committee consists of at least 5 females?

2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 79

2.4 Complements, Unions, and Intersections

Definition: Let E be an event is a sample space S. The complement of E is denotedby E and represents the outcomes that are in S but not in E.

Definition: Let A and B be events. The union of A and B is denoted by A ∪ Band contains the outcomes that are in A or B, or both.

The intersection of A and B is denoted by A ∩ B and contains the outcomes thatare in A and B.


1. Suppose we have an experiment where we select one Math 105 student. Soour sample space S is the set of all Math 105 students. Consider the followingthree events.

• M = “The student likes Michael”

• D = “The student likes Dwight”

• P = “The student likes Pam”

(a) Describe the following events in words:

i. M ∩ P

ii. P ∪D

iii. P ∩D

(b) Describe the events below using the set-theoretic notation we just dis-cussed in class:

i. The student does not like Pam

ii. The student likes Michael but not Dwight

iii. The student likes only Dwight (out of the three options)

iv. The student likes Pam or Dwight (or both), but not Michael


Probability Formulas:

• If E is an event, then P (E) ≥ 0 and P (E) ≤ 1

• If E is impossible, then P (E) = 0

• If E must happen, then P (E) = 1

• P (E) + P (E) = 1 (or, equivalently, P (E) = 1− P (E) )

• If A and B are events, then P (A ∪B) = P (A) + P (B)− P (A ∩B)

• In the special case when A and B do not overlap, we say that A andB are mutually exclusive (they cannot both simultaneously happen). In

this special case, P (A ∪B) = P (A) + P (B)


2. Suppose we draw a card from a standard 52-card deck. Find the followingprobabilities.

(a) The probability that the card is not a 6

(b) The probability that the card is an Ace or a black card (or both)

(c) The probability that the card is a face card or a Spade (or both)

(d) The probability that the card is a red card or a black card (or both)


3. Roll two 6-sided dice. Let

• E1 be the event in which both dice show an even number;

• E2 be the event in which the sum of the numbers showing is 6; and

• E3 be the event in which the sum of the numbers showing is less than 11.

Find P (E1), P (E1), P (E2), P (E2), P (E3), and P (E3).

4. One card is drawn from a standard 52-card deck. Find the probability ofselecting at least a 10 (10,J,Q,K,A) or a heart.

5. A new medication being tested produces headaches in 5% of its users, upsetstomach in 15%, and both side effects in 2%.

(a) Find the probability that at least one of these side effects is produced.

(b) Find the probability that neither of these side effects is produced.

6. An 8-sided die is constructed that has two faces marked with 2s, two facesmarked with 3s, two faces marked with 5s, and two faces marked with 8s. Ifthis die is rolled a single time, find the probability of

(a) Getting a 2.

(b) Not getting a 2.

(c) Getting a 2 or a 3.

(d) Getting neither a 2 nor a 3.

(e) Getting an even number.

(f) Not getting an even number.

7. If P (A) = 0.6, P (B) = 0.4, and P (A ∩B) = 0.3, find P (A ∪B).

8. If P (A ∪B) = 0.8, P (A) = 0.6, and P (B) = 0.4, find P (A ∩B).


Homework

1. 17% of American children have blue eyes. 6.6% of American children haveType O- blood. 1.1% of American children have both blue eyes and Type O-blood. What is the probability that a randomly selected American child willhave blue eyes or Type O- blood (or both)?

2. In 2009, there were roughly 21,500,000 teenagers (ages 15-19) in the UnitedStates. 11,000,000 of these teenagers were males and 10,500,000 of theseteenagers were females. 130,000 of the male teenagers had ever been mar-ried, and 240,000 of the female teenagers had ever been married. What is theprobability that a randomly selected teenager in 2009 was female or had everbeen married (or both)?

3. Suppose we select a Concordia student at random. Let A be the event that thestudent is a math major and let B be the event that the student is a junior.Write what the following probabilities represent in words:

(a) P (A ∪B)

(b) P (A ∩B)

(c) P (A)

(d) P (A ∪B)

4. The probability of winning a Math 105 collector’s edition beanie is 0.49. Whatis the probability of not winning the beanie?

5. Two fair six-sided dice are rolled once.

(a) What is the probability that the sum of the dice is not 12?

(b) What is the probability that the sum of the dice is not 10?

(c) What is the probability that the sum of the dice is not an even number?

6. If P (A) = 0.5, P (B) = 0.6, and P (A ∩B) = 0.3, find P (A ∪B).

7. If P (A ∪B) = 0.9, P (A) = 0.6, and P (A ∩B) = 0.2, find P (B).


8. A card is randomly selected from a standard 52-card deck. What is the prob-ability that the card is:

(a) A Heart or a Spade?

(b) A Heart or a King?

(c) A Heart or a face card?

(d) A Red card or a face card?

9. Suppose A and B are events in a sample space S with P (A) = 12

and P (B) =710

. What is the smallest possible value of P (A ∩ B)? What is the largestpossible value of P (A ∩B)?

10. An experiment consists of selecting a car at random from the Hvidsten parkinglot and observing the color and make of the car. Let R be the event “The caris red,” let F be the event “The car is a Ford,” let G be the event “The car is agreen Saturn,” and let B be the event “The car is blue or a Buick (or both).”

Which of the following pairs of events are mutually exclusive?

(a) R and F

(b) R and G

(c) F and G

(d) R and B

(e) F and B

(f) G and B

(g) R and G

(h) F and B


2.5 Conditional Probability

Key Idea: Sometimes learning new information about a situation can change theprobabilities.

Example:

Definition: If A and B are events, then the probability that A will occur giventhat B has occurred is called the conditional probability and is denoted by P (A|B).

P (A|B) =P (A ∩B)

P (B)

2.5. CONDITIONAL PROBABILITY 87

1. Among the employees at Dunder Mifflin:

• 75% are college graduates

• 80% earn more than $51,000 per year

• 70% are college graduates and earn more than $51,000 per year

If a Dunder Mifflin employee selected at random is a college graduate, what isthe probability that they earn more than $51,000 per year?

2. In a survey of Cobbers:

• 70% read The Forum

• 80% read The Concordian

• 90% read at least one of the two papers

If a Cobber reads The Concordian, what is the probability that they read TheForum?


3. Suppose P (A) = 0.7, P (B) = 0.3, and P (A ∪ B) = 0.8. Draw the Venndiagram, then find P (A|B) and P (B|A).

4. Suppose P (A) = 0.4, P (B) = 0.5, and P (A ∩ B) = 0.3. Draw the Venndiagram, then find P (A|B) and P (B|A).

P (A ∩B) = P (A) · P (B|A) and P (A ∩B) = P (B) · P (A|B)


5. Suppose we draw 2 cards from a standard deck without replacement. What isthe probability that both are Kings?

6. Suppose we flip a fair coin 3 times.

(a) What is the probability that we get HHH?

(b) What is the probability that we get HHH given that at least one of thefirst two flips was Heads?

7. Suppose an urn contains 4 white and 6 red balls. Two balls are randomlyselected. If the first ball is white, it is replaced before we make our seconddraw. If the first ball is red, it is not replaced before we make our second draw.What is the probability of drawing at least one white ball?


Homework

1. Suppose we select a Concordia student at random. Let A be the event that thestudent is a math major and let B be the event that the student is a junior.Write what the following probabilities represent in words:

(a) P (A|B)

(b) P (B|A)

(c) P (A|B)

(d) P (A|B)

2. A Math 105 class contains 26 students. Of these students, 14 are math majors,15 are juniors, and 7 are neither math majors nor juniors. Suppose a studentis selected at random from the class and that the student is a junior. What isthe probability that the student is also a math major?

3. Two cards are selected at random in order from a standard 52-card deck. Findthe probability that the first is a Jack and the second is a Queen:

(a) With replacement

(b) Without replacement

4. Two cards are selected at random in order from a standard 52-card deck. Findthe probability that the first is a Jack or the second is a Queen (or both):

(a) With replacement

(b) Without replacement

5. An urn contains 4 white and 6 red balls.

(a) If three balls are drawn from this urn with replacement, what is theprobability that the last ball is red?

(b) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red?

(c) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that the first two balls were red?

(d) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that the first two balls werewhite?


(e) If three balls are drawn from this urn without replacement, what is theprobability that the last ball is red given that at least one of the first twoballs were white?

6. Suppose we flip a fair coin 20 times. What is the probability that:

(a) We obtain no Heads

(b) We obtain Heads exactly once

(c) We obtain Heads at most once

(d) We obtain Heads at least once

(e) We obtain Heads exactly 10 times

7. From a standard 52-card deck, we deal a random 5 card hand. What is theprobability that:

(a) The hand contains only Black cards

(b) The hand has exactly two Spades

(c) The hand has at most two Spades

(d) The hand has more Black cards than Red cards

(e) The hand has more Diamonds than Hearts, and at most two Diamonds

(f) The hand has no pairs

(g) The hand has exactly one pair

(h) The hand has exactly two pairs

(i) The hand is a straight (5 cards with consecutive ranks, where Aces mustbe low)

(j) The hand is a flush (all cards have the same suit)

(k) The hand contains no cards above a Nine (Aces are allowed)

(l) The hand is a full house (3 cards in one rank, 2 cards in another)

(m) The hand has exactly three in one rank, and no other pairs

(n) The hand has exactly three cards in one suit, and the other cards aredifferent suits

(o) The hand contains all four suits


8. The English alphabet contains 26 letters, 5 of which are vowels. What is theprobability that a random 5 letter word:

(a) Has no repetitions

(b) Begins with J, ends with F, and has no repetitions

(c) Begins with J, ends with F, contains L, and has no repetitions

(d) Contains J, and has no repetitions

(e) Contains J and F, and has no repetitions

(f) Contains exactly two vowels, and has no repetitions

(g) Contains at most two vowels, and has no repetitions

(h) Contains more consonants than vowels, and has no repetitions

(i) Has consonants and vowels alternating, and has no repetitions

(j) Contains exactly two L’s, and has no other repetitions

(k) Has exactly one letter appearing exactly twice, and no other letter repeats

(l) Has at most one letter that repeats, and no letter appears more thantwice

(m) Has at most one letter that repeats

(n) Has J appearing exactly twice and F appearing exactly twice

(o) Has exactly two letters appearing exactly twice each

9. In a recent survey of Cobbers, 60% went to the Homecoming football game,70% went to the bonfire, and 80% went to at least one of these two events. Ifa Cobber went to the football game, what is the probability that that studentwent to the bonfire?

10. Suppose P (A) = 0.4, P (B) = 0.6, and P (A ∩ B) = 0.2. Draw the Venndiagram, then find the conditional probabilities.

(a) P (A|B)

(b) P (B|A)

(c) P (A|B)

(d) P (A|B)

(e) P (A|B)

(f) P (B|A)

(g) P (B|A)



1. The French mathematicians Pierre de Fermat and Blaise Pascal are usuallygiven credit for originating the theory of probability. The first problems inprobability were posed to Pascal by the famous gambler Chevalier de Mere,and Pascal and Fermat exchanged letters developing the theory of probabilityin order to answer de Mere’s questions. Provide some historical backgroundabout these men and the letters, and present a complete solution to at leastone of these first problems in probability.

2. Describe Pascal’s Triangle. Include its connections to counting theory andpresent how it relates to the binomial theorem, which allows us to expandexpressions like (x+ y)2, (x+ y)3, . . . . Also explain how it helps us count thenumber of paths from one corner to another corner in a rectangular grid. Feelfree to include other applications of this triangle as well!

3. Discuss the Monty Hall Problem: Suppose you’re on a game show, and you’regiven the choice of three doors. Behind one door is a car, behind the others,goats. You pick a door, say #1, and the host, who knows what’s behind thedoors, opens another door, say #3, which has a goat. He says to you, “Doyou want to pick door #2?” Is it to your advantage to switch your choice ofdoors?

This problem was posed to Marilyn vos Savant in 1990, and she answered itcorrectly, but thousands of people, including quite a few Ph.D. mathemati-cians, wrote to her and told her she was wrong. Explain the problem, presentsome of the letters, and use probability to give a correct solution to the prob-lem.


2.7 Chapter Review

Concepts:

• Permutations are where order matters, and the number of permutations isdenoted by P (n, r)

• P (n, r) =n!

(n− r)!= n · (n− 1) · (n− 2) · · · · · (n− (r − 1))

• Combinations are where order does not matter, and the number of combina-tions is denoted by C(n, r)

• C(n, r) =P (n, r)

r!=

n!

(n− r)! · r!

• Sample space

• Uniform sample space

• If E is an event in a uniform sample space, then

P (E) =number of outcomes in E

number of outcomes in S

• A ∪B denotes the union of A and B (Or)

• A ∩B denotes the intersection of A and B (And)

• P (A ∪B) = P (A) + P (B)− P (A ∩B)

• P (A ∪ B) = P (A) + P (B) if A and B are mutually exclusive (they do notoverlap at all)

• C denotes the complement of C (Not)

• P (C) = 1− P (C)

• The conditional probability of A given B is denoted by P (A|B)

• P (A|B) =P (A ∩B)

P (B)(this comes from an outstanding Venn diagram)

• P (A∩B) = P (A) ·P (B|A) and P (A∩B) = P (B) ·P (A|B) (in particular, wecan multiply along a tree diagram)


• Coin questions

• Card questions

• Word questions


Review Exercises:

1. A state makes license plates with three letters followed by three digits. Thereare 26 possible letters (A through Z) and 10 possible digits (0 through 9).Letters are not allowed to repeat on a license plate, but the digits are allowedto repeat.

Assuming that the state makes license plates randomly according to theserules, what is the probability that a license plate made in this state beginswith the letter H and ends with the digit 8?

2. An urn holds 10 identical balls except that 4 are red and 6 are white. Anexperiment consists of selecting two balls in succession from the urn withoutreplacing the first ball selected.

(a) What is a sample space for this experiment?

(b) What is the probability that we select two red balls?

(c) What is the probability that we select a red ball if we know that the firstball we select is white?

3. Suppose A and B are events in a sample space S with P (A ∪ B) = 0.7,P (B) = 0.2, and P (A ∩B) = 0.1. What is the value of P (A)?

4. A single card is drawn from a standard 52-card deck. What is the probabilitythat the card is a Jack or Red (or both)?

5. On a children’s baseball team, there are six players who can play any of thefive following infield positions: catcher, first base, second base, third base, andshortstop. There are four possible pitchers, none of whom can play any otherposition. And there are five players who can play any of the three outfieldpositions: left field, center field, or right field. In how many ways can thecoach assign these players to positions?

6. A chef can make 10 main courses. Every day a menu is formed by selecting 6of the main courses and listing them in order. How many different such menuscan be made?

7. A chef can prepare 10 different entrees. In how many ways can the chef select6 entrees for today’s menu?

8. A license plate has 6 digits with repetitions permitted. How many possiblelicense plates of such type are there?


9. At an awards ceremony, 3 men and 4 women are to be called one at a timeto receive an award. In how many ways can this be done if women and menmust alternate?

10. An executive is scheduling meetings with 10 people in succession. The first2 meetings must be with 2 directors on the board, the second 3 with 3 vicepresidents, and the last 5 with 5 junior executives. How many ways can thisschedule be made out?

11. In a certain lotto game, 5 numbered ping pong balls are randomly selectedwithout replacement from a set of balls numbered from 1 to 35 to determine awinning set of numbers (without regard to order). Find the number of possibleoutcomes.

12. The Pi Mu Epsilon honor society consists of 9 men and 7 women. If the societyforms a committee with 3 women and 2 men, how many different ways canthis be done?

13. In a new group of 10 employees, 4 are to be assigned to production, 2 to sales,and 1 to advertising. In how many ways can this be done?

14. A fair (each side is equally likely to land up) 10-sided die is rolled 100 timeswith the following results:

Outcome Frequency

1 82 83 124 75 156 87 88 139 910 12

(a) What is the experimental probability of rolling a 3?

(b) What is the theoretical probability of rolling a 3?

(c) What is the experimental probability of rolling a multiple of 3?

(d) What is the theoretical probability of rolling a multiple of 3?


15. One card is drawn from a standard 52 card deck. What is the probability ofdrawing:

(a) the queen of hearts?

(b) a queen?

(c) a heart?

(d) a face card (J, Q, or K)?

16. Two 6-sided dice are rolled. What is the probability of rolling

(a) a total of 6?

(b) not a total of 6?

(c) a total of 6 or 7?

(d) a total of 6 or more?

17. A gumball machine has gumballs of four flavors: apple, berry, cherry, andpumpkin. When a quarter is put into the machine, it dispenses 5 gumballs atrandom. What is the probability that

(a) each gumball is a different flavor?

(b) at least two gumballs are the same flavor?

18. A coin is flipped ten times in a row. Find the probability that

(a) no tails show.

(b) exactly one tail shows.

(c) exactly twice as many heads as tails occur.

19. A group of 15 students is to be split into 3 groups of 5. In how many wayscan this be done?

20. Five cards are drawn from a standard 52 card deck. What is the probabilityof drawing 5 cards of the same color?

21. Suppose a jar has 4 coins: a penny, a nickel, a dime and a quarter. You removetwo coins at random without replacement. Let A be the event you remove thequarter. Let B be the event you remove the dime. Let C be the event youremove less than 12 cents.

(a) List the sample space.

(b) Draw a probability tree diagram to represent the possible scenarios.


(c) Find P (A); P (C); and P (B).

(d) Compute and interpret P (A ∪B) and P (B ∪ C).

22. The probability is 0.6 that a student will study for an exam. If the studentstudies, she has a 0.8 chance of getting an A on the exam. If she does notstudy, she has a 0.3 probability of getting an A. Make a probability tree forthis situation. What is the probability that she gets an A? If she gets an A,what is the conditional probability that she studied?

23. If P (A) = 0.6, P (B) = 0.3, and P (A ∩B) = 0.2, find P (A ∪B) and P (A|B).

24. If P (A) = 0.4, P (B) = 0.5, and P (A ∪B) = 0.7, find P (A ∩B) and P (B|A).

25. A dental assistant randomly sampled 200 patients and classified them accord-ing to whether or not they had a least one cavity in their last checkup andaccording to what type of tooth decay preventative measures they used. Theinformation is as follows

At least one cavity No CavitiesBrush only 69 2

Brush and floss only 34 11Brush and tooth sealants only 22 13Brush, floss and tooth sealants 3 46

If a patient is picked at random from this group, find the probability that

(a) the patient had at least one cavity;

(b) the patient brushes only;

(c) the patient had no cavities, given s/he brushes, flosses and has toothsealants;

(d) the patient brushes only, given that s/he had at least one cavity.



1.1 · 25 · 24 · 10 · 10 · 1

26 · 25 · 24 · 10 · 10 · 10

2. (a) {RR,RW,WR,WW} is one possibility

(b)4

10· 3

9or

C(4, 2)

C(10, 2)

(c)4

9

3. 0.6

4.28

52

5. P (6, 5) · P (4, 1) · P (5, 3)

6. P (10, 6) =10!

4!= 151, 200

7. C(10, 6) =10!

4!6!= 210

8. 106 = 1, 000, 000

9. 4 3 3 2 2 1 1= 4!3! = 144

10. 2!3!5! = 1440

11. C(35, 5) =35!

30!5!= 324, 632

12. C(7, 3) · C(9, 2) = 1260

13. C(10, 4) · C(6, 2) · C(4, 1) = 12, 600

14. (a) 12/100 = 3/25

(b) 1/10

(c)12 + 8 + 9

100=

29

100

(d)1 + 1 + 1

10=

3

10

15. (a) 1/52

(b) 4/52 = 1/13


(c) 13/52 = 1/4

(d) 12/52 = 3/13

16. (a) 5/36

(b) 1− 5/36 = 31/36

(c) 11/36

(d) 26/36

17. (a) 0, since there are only 4 flavors.

(b) 1− 0 = 1

18. (a)1

210

(b)C(10, 1)

210=

10

210= 0.009765

(c) 0, since it is not possible. 6 heads and 3 tails is only 9 flips, and 8 headsand 4 tails is 12 flips, which is too many.

19.C(15, 5) · C(10, 5) · C(5, 5)

3!= 126, 126

20.C(2, 1)C(26, 5)

C(52, 5)=

253

4998= 0.05062

21. (a) The sample space is {PN,PD, PQ,NP,ND,NQ,DP,DN,DQ,QP,QN,QD}.(b) Probability tree

(c) P (A) = 1/2; P (C) = 4/12 = 1/3; and P (B) = 1/2.

(d) P (A∪B) = P (A)+P (B)−P (A∩B) = 1/2+1/2−1/6 = 5/6 representsthe probability that we remove either a quarter or a dime, and P (B∪C) =1/2 + 1/3− 1/6 = 2/3 is the probability that we remove either a dime orless than 12 cents.

22. What is the probability that she gets an A? 0.6× 0.8 + 0.4× 0.3 = 0.6.

If she gets an A, what is the conditional probability that she studied?0.48

0.6=

0.8.

23. If P (A) = 0.6, P (B) = 0.3, and P (A ∩B) = 0.2, then

P (A ∪B) = P (A) + P (B)− P (A ∩B) = 0.6 + 0.3− 0.2 = 0.7, and

P (A|B) =P (A ∩B)

P (B)=

0.2

0.3=

2

3.


24. If P (A) = 0.4, P (B) = 0.5, and P (A ∪B) = 0.7, then

P (A ∩B) = P (A) + P (B)− P (A ∪B) = 0.4 + 0.5− 0.7 = 0.2, and

P (B|A) =P (A ∩B)

P (A)=

0.2

0.4=

1

2.

25. (a) 128/200 = 0.64

(b) 71/200 = 0.355

(c) 46/49 = 0.938

(d) 69/128 = 0.539

Chapter 3

Graph Theory

3.1 Introduction to Graph Theory

Definition: A graph is a collection of vertices (points) connected by edges (lines).

Examples:

Graph Applications:

103

104 CHAPTER 3. GRAPH THEORY

Definition: A loop is an edge from a vertex to itself.

Definition: If more than one edge joins two vertices, these edges are called multipleedges.

Definition: A graph is simple if it has no loops and no multiple edges.

Definition: A graph is connected if you can get from any vertex to any other vertexby following edges in the graph. If a graph is disconnected, the connected pieces ofthe graph are called the components of the graph.

Definition: The degree of a vertex is the number of edges at that vertex. A loopat a vertex counts as 2 for the degree of the vertex.

3.1. INTRODUCTION TO GRAPH THEORY 105

1. For each of the following lists of numbers:

I. If possible, draw a graph that has the list as its list of vertex degrees.

II. If possible, draw a simple graph that has the list as its list of vertex degrees.

(a) 1, 1, 2, 2, 4

(b) 2, 2, 3, 3, 4

(c) 1, 1, 1, 2, 2, 4

(d) 1, 1, 2, 2, 6

If some of the lists are impossible, can you explain why?


Definition: If n is a positive integer, the complete graph on n vertices isdenoted by Kn and is the simple graph with n vertices and all possible edges.

3.1. INTRODUCTION TO GRAPH THEORY 107

Homework

1. For each of the graphs below:

(a) Write the vertex set

(b) Write the edge set

(c) List the degrees of each vertex

2. For each of the given vertex sets and edge sets, draw two different pictures ofa graph with those sets.

(a) V = {A,B,C,D}, E = {AB,AC,AD,BC}(b) V = {M,A, T,H, Y }, E = {HM,HT,HY,HY,MM,MT,MY }

3. (a) Draw a connected graph with five vertices where each vertex has degree 2.

(b) Draw a disconnected graph with five vertices where each vertex has de-gree 2.

(c) Draw a graph with five vertices where four of the vertices have degree 1and the other vertex has degree 0.

4. Give an example of a graph with four vertices, each of degree 3, with:

(a) No loops and no multiple edges

(b) Loops but no multiple edges

(c) Multiple edges but no loops

(d) Loops and multiple edges.


5. Suppose our city has 51 city council members, and that these members serveon 10 city committees. Each council member serves on at least one committee,and some members serve on multiple committees.

(a) If we wanted to know which pairs of members are on the same committee,would it be better to draw a graph where the vertices represent membersor a graph where the vertices represent committees?

(b) If we wanted to know which committees have members in common, wouldit be better to draw a graph where the vertices represent members or agraph where the vertices represent committees?

6. (a) Draw K6 and K7.

(b) Find a formula, which depends on the positive integer n, for the numberof edges in Kn.

7. For each of the following sets of conditions, explain why no graph satisfiesthose conditions:

(a) A graph with exactly five vertices each of degree 3

(b) A graph with exactly four edges, exactly four vertices, and the verticeshave degrees of 1, 2, 3, and 4

(c) A simple graph with exactly four vertices, where the vertices have degreesof 1, 2, 3, and 4

(d) A simple graph with exactly six vertices, where the vertices have degreesof 1, 2, 3, 4, 5, and 5

3.2. PATHS AND CIRCUITS 109

3.2 Paths and Circuits

Definition: A path is a sequence of consecutive vertices that are joined by edgesin a graph. The length of a path is the number of edges used in the path. Verticescan be repeated in a path, but edges cannot be repeated.

Example:

Definition: A circuit is a path that starts and ends at the same vertex.


1. In the following graphs, is there a path that uses every edge exactly once? Isthere a circuit that uses every edge exactly once?

(a)

A B

C D

E F

(b)

A B

C D

E F

(c)

A B

C D

E F

(d)

A B

C D

E F

(e)

A B

C D

E F


Definition: Suppose we have a connected graph. A path that uses every edgeof the graph exactly once is called an Euler path. A circuit that uses everyedge exactly once is called an Euler circuit.

Konigsberg, from Leonhard Euler, Solutio problematis ad geometriam situs perti-nentis, 1736


Main Theorem:


Exploration: In each of the following five graphs, is there an Euler path orEuler circuit? If so, give the path or circuit; if not, explain why.

A B

CD

B

CA

B

CA

A B

CD

A B

CD

Exploration: In each of the following four graphs, determine whether thereis an Euler circuit, an Euler path but no Euler circuit, or neither. Explaineach answer.

(a) (b) (c) (d)


Homework

1. Consider the following graph:

(a) Find a path of length 5 from A to H.

(b) Find a path of length 7 from A to H.

(c) How many different paths are there from A to D?

(d) How many different paths are there from D to H?

(e) How many different paths are there from A to H?

2. For each of the following graphs, determine if the graph has an Euler circuit,an Euler path but no Euler circuit, or no Euler path and no Euler circuit.Explain your answer. You do not need to show an Euler path and/or Eulercircuit if they exist.

(a)

(b)

(c)


(d)

(e)

(f)

(g)

(h)


3. Find an Euler circuit in the following graph.

4. Find an Euler circuit in the following graph.

5. Find an Euler path in the following graph.



7. In the map below, is it possible to travel a route that crosses each bridgeexactly once? If so, give such a route. If not, explain why not.

8. A letter carrier is responsible for delivering mail to all of the houses on bothsides of the streets shown in the figure below. (The streets are white, and thehouses are in the gray regions.) If the letter carrier does not keep crossing astreet back and forth to get to houses on both sides of a street, then she willneed to walk along a street at least twice, once on each side, to deliver themail.

(a) Is it possible for the letter carrier to construct a round trip so that shewalks on each side of every street exactly once?

(b) If the street diagram was different, would you arrive at the same conclu-sion?


3.3 Subgraphs and Trees

Definition: A subgraph of a graph G is a graph contained in G. A subgraph of Gis spanning if it contains all of the vertices of G.

Examples:

Weighted Graphs:

3.3. SUBGRAPHS AND TREES 119

1. For each of the following graphs, find a connected spanning subgraph with thesmallest possible total weight.


Minimal Spanning Trees

Definition: A connected graph with no circuits is called a tree.

Fact: If a tree has n vertices, it must have n− 1 edges. Also, any connectedgraph with n vertices and n− 1 edges is a tree.


2. How many different spanning trees does the following graph contain?

Greedy Algorithms:


Homework

1. For the following graphs, determine whether or not the graph is a tree. Explainyour answers.

(a)

(b)

(c)

(d)

2. Draw all of the different spanning trees of the following graph.


3. How many different spanning trees does the following graph have?

4. How many different spanning trees does the following graph have?

5. Find a minimal spanning tree, and give its total weight.


6. Find a minimal spanning tree, and give its total weight.

7. Below is a mileage chart between some cities in Minnesota. Draw a weightedgraph that reflects the mileages and find a minimal spanning tree that connectsall of these cities in your graph.

Bem

idji

Frazee

International

Falls

Moorhead

TwoHarbors

WhiteBearLake

Bemidji — 82 112 127 180 224

Frazee 82 — 198 55 223 194

International Falls 112 198 — 242 177 280

Moorhead 127 55 242 — 277 239

Two Harbors 180 223 177 277 — 167

White Bear Lake 224 194 280 239 167 —

8. For each of the following statements, if the statement is true, explain why. Ifthe statement is false, give an example of a graph where the statement is false.

(a) If G is a connected simple graph with weighted edges, and all of theweights are different, then different spanning trees of G have differenttotal weights.

(b) If G is a connected simple graph with weighted edges, and e is an edgeof G with a smaller weight than any other edge of G, then e must beincluded in every minimal spanning tree of G.


9. Suppose we have a large collection of 1-cent, 8-cent, and 10-cent stamps avail-able. We want to select the minimum number of stamps needed to make agiven amount of postage. Consider a greedy algorithm that selects stamps byfirst choosing as many of the 10-cent stamps as possible, then as many of the8-cent stamps as possible, then as many 1-cent stamps as possible. Does thisgreedy algorithm always produce the fewest number of stamps needed for eachpossible amount of postage? Why or why not?


3.4 Graph Colorings

Definition: A (vertex) coloring of a graph is an assignment of colors to the verticesof the graph so that vertices that are joined by an edge (adjacent vertices) havedifferent colors.

Example:

Definition: For a graph G, the smallest number of colors needed to color G iscalled the chromatic number of G and is denoted χ(G). Note that χ is the Greekletter chi, for chromatic.

Example: If G is , then χ(G) = 3. It is possible to color G using

just 3 colors, and we need at least 3 colors because G has a triangle.

Example: Color each of the vertices of the following graph red (R), white (W), orblue (B) in such a way that no adjacent vertices have the same color.

Example: If G is find χ(G).

3.4. GRAPH COLORINGS 127

1. What is the coloring number of the following complete graphs?

Ki χ(Ki)

K1 =

K2 =

K3 =

K4 =

K5 =

Kn


2. What is the coloring number of the following paths?

Pi χ(Pi)

P1 =

P2 =

P3 =

P4 =

P5 =

Pn

3. (a) Find the chromatic number of the following tree:

(b) Draw a tree with 11 vertices, and find its chromatic number.


4. Corncob College elects 10 students to serve as officers on 8 committees. Thelist of the members of each of the committees is:

• Corn Feed Committee: Darcie, Barb, Kyler

• Dorm Policy Committee: Barb, Jack, Anya, Kaz

• Extracurricular Committee: Darcie, Jack, Miranda

• Family Weekend Committee: Kyler, Miranda, Jenna, Natalie

• Homecoming Committee: Barb, Jenna, Natalie, Skye

• Off Campus Committee: Kyler, Jenna, Skye

• Parking Committee: Jack, Anya, Miranda

• Student Fees Committee: Kaz, Natalie

They need to schedule meetings for each of these committees, but two com-mittees cannot meet at the same time if they have any members in common.How many different meeting times will they need?


An example where Greedy Coloring fails :


Homework

1. Find the chromatic number for each of the following graphs.

(a) (b) (c)

(d) (e)

2. Consider the following graph classes:

Cycles

· · ·

C2 C3 C4 C5

Wheels

· · ·

W3 W4 W5 W6


Find χ(Cn) and χ(Wn) for every integer n ≥ 3, where Cn denotes the cyclewith n vertices and Wn denotes the wheel with n spokes (n+1 vertices). Youranswers will change as n changes.

3. The mathematics department at Cornucopia College will offer seven coursesnext semester: Math 105 (M), Numerical Analysis (N), Linear Operators (O),Probability (P), Differential Equations (Q), Real Analysis (R), Statistics (S).The department has twelve students, who will take the following classes:

Alice: N, O, Q Dan: N, O Greg: M, P Jill: N, S, QBob: N, R, S Emma: O, M Hodor: R, O Kate: P, SChuck: R, M Fonz: N, R Isabel: N, Q Lara: P, Q

The department needs to schedule class times for each of these courses, but twocourses cannot meet at the same time if they have any students in common.How many different class times will they need?

3.5. PLANAR GRAPHS 133

3.5 Planar Graphs

Definition: A graph is planar if we can draw it in the plane (a flat surface) withno edges crossed. If a graph is drawn in the plane with no edges crossed, we saythat the drawing is a plane graph.

Examples:

1. Which complete graphs are planar graphs?


Definition: A connected plane graph divides the plane into different regionscalled faces.

Faces are only defined when the drawing is a plane graph — when the edgesdon’t cross.

2. For each of the following graphs, count the number of vertices, the number ofedges, and the number of faces. Do you see a relationship between these threenumbers?


Euler’s Formula:

Explanation:

Fact: If G is a connected simple plane graph with at least three vertices, then3V − 6 ≥ E.


Homework

1. For the following plane graph, count the number of vertices, the number ofedges, and the number of faces. Verify that Euler’s formula holds.

2. For each of the following graphs, if the graph is planar, draw it in the planeso that no edges cross. Otherwise, state that the graph is not planar.

(a)A

B

C D

E

(b)

A

BC

D

E F

(c)

AB

C

DE

F

(d)A

B

C D

EFG

H I

J

3. Let G be a connected planar graph. Explain why, no matter how we draw Gas a plane graph, the number of faces of G will always be the same.

4. Suppose G is a connected plane graph with nine vertices, where the degreesof the vertices are 2, 2, 2, 3, 3, 3, 4, 4, and 5. How many edges does G have?How many faces does G have?


5. Suppose you have a graph G that is not planar. Is there a way to make itplanar by adding more vertices and edges? If so, describe your method; if not,why not?

6. Suppose you have a graph G that is planar. Is there a way to make it notplanar by adding more vertices and edges? If so, describe your method; if not,why not?


3.6 Directed Graphs

Definition: A directed graph is a graph where every edge in the graph is assigneda direction between its two vertices.

For an edge , we say that the edge is directed from X to Y . Alternatively,Y is the head of the edge and X is the tail.

Applications:

3.6. DIRECTED GRAPHS 139

Suppose we have the following tasks as we build a house:

Task Time RequiredStart: Decide to build house No time

A: Clear land 1 dayB: Build foundation 3 daysC: Build frame and roof 15 daysD: Electrical work 9 daysE: Plumbing work 5 daysF: Complete exterior work 12 daysG: Complete interior work 10 daysH: Landscaping 6 days

Finish: Move in! No time

The activity directed graph is given below. We draw an arrow from task X to taskY if and only if task Y must be completed directly after completing task X. Welabel each vertex with the amount of time needed to complete that task.

The longest path following the arrows from Start to Finish (in terms of time) is acritical path. The length of the critical path is the length of time needed to completethe overall task.

1. Find the critical path and give its length (in days).


Definition: A tournament is a complete graph where every edge is directed.

Definition: Two tournaments are isomorphic if they can be redrawn to looklike each other.


Definitions: Given a vertex v in a directed graph, the number of arrows goinginto v is called the in-degree of v, and the number of arrows going out of v iscalled the out-degree of v.

In a tournament, a vertex with only arrows going out (in-degree = 0) is calleda source, and a vertex with only arrows coming in (out-degree = 0) is called asink.

Fact 1: In any tournament, a longest path that does not repeat any verticeswill visit every vertex exactly once.

Fact 2: Every tournament has at least one king chicken.

A king chicken is a team v such that, for any other team w:

• v beat w, or

• v beat a team that beat w.


2. In each of the tournaments below, find a longest path that doesn’t repeatvertices, and find a king chicken.


Homework

1. Suppose we have to complete the following tasks as we produce a film. Theamount of time each intermediate task requires is listed in the table.

Task Time RequiredStart: Obtain script No time

A: Acquire funding 4 weeksB: Sign director 2 weeksC: Hire actors 4 weeksD: Choose filming locations 2 weeksE: Build sets 3 weeksF: Film scenes 10 weeksG: Edit film 5 weeksH: Create soundtrack 3 weeks

Finish: Release film No time

The activity directed graph is given below. Find the critical path and give itslength (in weeks).


2. Michael and Jan have invited some friends over for a dinner party and needto set the dinner table. Estimate the time required (in minutes) needed tocomplete the following tasks.

Task Time RequiredStart: Finish playing charades No time

A: Put tablecloth on tableB: Fold napkins and place on tableC: Place dishes and silverware on tableD: Put water and ice in water glassesE: Pour wine into wine glassesF: Put food on table

Finish: Begin eating No time

Now draw a possible activity directed graph for these tasks, and find theminimum amount of time needed to complete the project by finding a criticalpath in your graph. Assume that only Dwight has volunteered to help Michaeland Jan set the table, so that at most three tasks can occur at any given time.

3. Draw all of the different (non-isomorphic) types of tournaments with exactlyfour vertices.

4. If an undirected graph has E edges and we add all of the vertex degrees in thegraph, we know that the sum is 2E (by Euler’s Handshake Theorem).

(a) If we have a directed graph with E edges and we add all of the in-degrees,what will the sum be?

(b) If we have a directed graph with E edges and we add all of the out-degrees,what will the sum be?

5. Explain why a tournament can have at most one source and at most one sink.

6. (a) Give an example where five teams play in a round robin tournament andall five teams tie for first place.

(b) Explain why, if six teams play in a round robin tournament, it is impos-sible for all six teams to tie for first place.


7. In each of the following tournaments:

(a) Find a longest path that doesn’t repeat vertices (one that ranks all of theteams)

(b) Find a king chicken



1. Present the idea of a hamiltonian circuit in a graph (a circuit that uses everyvertex other than the starting and ending vertex exactly once). Give examplesof graphs that do and graphs that do not have hamiltonian circuits. Discusssome of the history of hamiltonian circuits, including some information aboutHamilton himself, and his game “Around the World.” Describe the connectionto the famous Traveling Salesperson Problem, its history and applications, andgive some examples; how difficult of a problem is the Traveling SalespersonProblem? Possible topics: What did Hamilton contribute to mathematics andother fields? Definition of Hamiltonian circuit, examples of graphs that havehamiltonian circuits and those that do not. Be sure to clarify the difference be-tween a hamiltonian circuit and an Euler circuit. Have the class work throughsome examples.

2. Present the five Platonic solids (tetrahedron, cube, octahedron, dodecahe-dron, icosahedron). Use Euler’s formula to show that these five solids arethe only possible Platonic solids. If you want, describe some of the histor-ical connections between the solids and religion, philosophy, and astronomy.Possible topics: Describe (show) the five Platonic solids. Why are they calledthe five Platonic solids? Historical connections between the solids and reli-gion, philosophy, and astronomy. Euler’s formula (V − E + F = 2) appliesto any convex polyhedron; why? Why are there only five Platonic solids?Use Euler’s formula to explain. Some students have found the explanationsat http://www.mathsisfun.com/geometry/platonic-solids-why-five.html to behelpful.

3. Present the idea of coloring maps. The famous Four Color Theorem was provedin 1976 — the proof is extremely complicated and has never been checked byhand, only by computers. So you don’t need to prove the theorem! At leastpresent what the theorem says, and discuss maps that can be colored withfewer than four colors. Use plenty of examples. If you want, present someof the historical background involving map colorings, including the fact thatthere were incorrect proofs that were believed to be correct for years. Youmay discuss whether or not a proof that no person has ever checked is actuallyvalid. Possible topics: Explain what we mean by coloring maps, give someexamples. How is graph coloring related to map coloring? Be careful aboutthe rules for coloring (for examples, countries that only meet at a corner canget the same color). Possibly involve the class in examples. Give some of thehistory of the problem. Discuss different proof attempts and the validity ofproofs done by computer


3.8 Chapter Review

Concepts:

• Graph

• Vertices, Edges

• Loops, Multiple edges

• Simple graphs

• Connected

• Degrees

• Handshake Theorem (the sum of the degrees in a graph must be even)

• Path, Circuit

• Euler paths and circuits, and the connection to vertex degrees being even

• Network

• Tree

• Spanning Tree

• Kruskal’s Algorithm for Minimal Spanning Trees

• Greedy algorithms

• Complete Graphs: K1, K2, K3, K4, . . .

• Coloring, Chromatic number χ(G)

• Planar graphs, Plane graphs

• Faces

• Euler’s Formula (for connected plane graphs): V − E + F = 2

• Directed graphs

• Activity directed graphs and Critical paths

• Tournaments

• Source and Sink

• Longest paths and King chickens in tournaments


Review Exercises:


2. Find the number of different spanning trees in the following graph.

3. In the following tournament:

(a) Find any sources or sinks

(b) Find a longest path that doesn’t repeat vertices

(c) Find all of the king chickens

4. Draw K6.


5. (a) For each of the following graphs, determine whether the graph is planaror not.

(b) Find the chromatic number for each of the graphs above.

6. Explain why we cannot have a simple graph with exactly six vertices, wherethe vertices have degrees of 1, 1, 1, 3, 5, and 5.

7. For each of the following graphs, answer the following questions or do therequested task.

(i) Does the graph have an Euler path? If so, find an Euler path in the graph.If not, explain why not.

(ii) Does the graph have an Euler circuit? If so, find an Euler circuit in thegraph. If not, explain why not.

(iii) What is the chromatic number χ of the graph?

(a)

A B C

D E F (b)

A B C

D

EFG

H

(c)

A B

CD

E

(d)

A B

CD

E F


8. Find the number of spanning trees in the following graphs:

(a) (b)

9. Find a minimal spanning tree for each of the following graphs.

150

200

210

125

50

30075

100

3.5

1.0

1.3 1.6

1.8

2.1

2.51.5

3.6 2.3

1.12.8


10. Are the following graphs planar? If so, draw a plane graph version of the graphand verify Euler’s formula for it; if not, state that it is not planar.

11. Show that the following graph is planar, find its chromatic number, and verifyEuler’s formula for it.

A

B

C

D

E

F

12. A connected simple planar graph has six vertices with degrees 1, 2, 3, 3, 4,and 5.

(a) How many edges does the graph have?

(b) If the graph is drawn in the plane without edges crossing, how many facesare there?

13. For each of the following sets of conditions, either draw a graph that satisfiesthose conditions, or explain why such a graph is impossible:

(a) A graph with exactly 4 vertices, where the vertices have degrees of 1, 1,2, 3

(b) A graph with exactly 4 vertices, where the vertices have degrees of 1, 3,4, 4

(c) A simple graph with exactly 4 vertices, where the vertices have degreesof 2, 2, 3, and 3


14. Draw an example of a tree that has exactly seven vertices, where exactly threeof the vertices have a degree of 2.



1. One example is CDHGKJIEABCGBFGJFE. Any example must use all17 edges exactly once, and start and end at C or E.

2. 3 · 2 · 4 = 24

3. (a) No source, no sink

(b) One example is AFEBCD

(c) A, D, and F

4.

5. (a) The first is planar, the second is not (M,H, I, J, L forms a K5 in thesecond graph)

(b) 4, 5

6. The two vertices of degree 5 must both be connected to all other vertices ofthe graph since the graph must be simple (no loops or multiple edges), whichdoes not allow for any degree 1 vertices.

7. (a) (i) Yes, two odd vertices B and F : BCFBEADEF

(ii) No, not all of the vertices are even

(iii) χ(a) = 3

(b) (i) Yes, all vertices are even: ABCDEFGHBDFHA; same for (ii)

(iii) χ(b) = 3

(c) (i) No, four odd vertices is two too many

(ii) No, see (i)

(iii) χ(c) = 3

(d) (i) Yes, two odd vertices E and F : EADEBAFECDFCBF

(ii) No, not all of the vertices are even

(iii) χ(d) = 4


8. (a) 3× 3 = 9; (b) 3× 4× 3 = 36

9. (a) 425 = 50 + 75 + 100 + 200; (b) 10.4 = 1.0 + 1.3 + 1.6 + 1.8 + 2.1 + 1.1 + 1.5

10. (a) Planar; V = 6, E = 7, F = 3, so V − E + F = 6− 7 + 3 = 2. (b) Planar;V = 6, E = 8, F = 4, so V − E + F = 6− 8 + 4 = 2. (c) Not planar

11. Stretch BD out and FD way out, and stretch AD way out, and move B abovethe AC edge. χ = 4. V = 6, E = 11, F = 7: V − E + F = 6− 11 + 7 = 2, soEuler’s formula checks.

12. (a) 1 + 2 + 3 + 3 + 4 + 5 = 18, so there are 18/2 = 9 edges: E = 9.

(b) V = 6, E = 9, V −E+F = 2 =⇒ F = 2−V +E =⇒ F = 2−6+9 = 5.F = 5.

13. (a) Impossible, as a graph cannot contain 3 odd vertices (no graph can containan odd number of odd vertices, since every edge counts toward the degree oftwo vertices).

(b) Possible

(c) Recall that a simple graph has no loops or multiple edges. Here it ispossible.

14. Start with a degree 2 vertex at the top, followed by a degree 3 vertex on theleft and another degree 2 vertex on the right...

Chapter 4

Consumer Mathematics

4.1 Percentages and Simple Interest

Percentages:

Definition: If you lend someone a sum of money, the amount you lend them iscalled the principal or present value. You then could charge the person an additionalamount, called interest, and the total of the two amounts, which is what you collectfrom the person after some time, is called the future value.

P = principal or present value

I = interest

F = future value

F = P + I

155

156 CHAPTER 4. CONSUMER MATHEMATICS

Definition: To determine interest using simple interest, we charge a fixed percent-age of P , called the interest rate, and multiple this amount by the length of time ofthe loan.

r = annual interest rate

t = time in years

I = P · r · t

Future Value using Simple Interest:

F = P · (1 + rt)

Examples:

4.1. PERCENTAGES AND SIMPLE INTEREST 157

1. If a bank offers a certificate of deposit (CD) with 2% annual simple interestand you want $5100 in the CD after 3 years, how much should you deposittoday?


2. If a street vendor promises to turn your $600 now into $1080 after 4 years,and he is offering simple annual interest, what interest rate is he promising?

3. Suppose we can earn 8% simple annual interest in a savings account. If wedeposit $2000 today, how much time will it take until we have $2680 in theaccount?

4.1. PERCENTAGES AND SIMPLE INTEREST 159

Homework

1. Convert the following percentages to decimal form:

(a) 73%

(b) 0.5%

(c) 2.25%

2. Convert the following decimals to percents:

(a) 0.03

(b) 0.0015

(c) 1.0008

3. In 2015, 6.2% of taxable earnings (up to $110,100) is deducted from workers’paychecks for Social Security. If a worker earns $53,000 in 2015, how muchmoney will the worker pay for the Social Security tax?

4. Given the principal P , the annual simple interest rate r, and the time t, findthe amount F that must be repaid.

(a) P = $15,000, r = 6%, t = 5 years

(b) P = $5,300, r = 2%, t = 3 months

(c) P = $9,000, r = 4.5%, t = 50 days

5. Of the four values F , P , r, and t in the formula for simple interest, three aregiven to you. Use the simple interest formula to find the fourth one.

(a) F = $12,000, r = 3%, t = 3 years

(b) F = $8,500, t = 6 months, P = $8,200

(c) F = $4,250, r = 7%, P = $3,500

6. You borrow $2,000 from Havelock Bank to pay for sidewalk repairs. Youpromise to repay the loan in three years at 5% simple interest. How much willyou pay the bank then?

7. Rafe knows he will inherit $10,000 from his dying aunt within five months.The bank will lend him money at 4% simple interest. What is the largestamount of money he could borrow now, if he plans to use his inheritance torepay it in five months?


8. Andy’s friend Falco wants to borrow $220 from him for three months. If Andywants to earn $20 in interest on the loan, what percent simple interest shouldhe charge?

9. Ryan has borrowed $800 from his friend Sophia, who is charging him 2% simpleinterest. Eventually Ryan repaid the loan, but it cost him $1000 to do so. Forhow long did Ryan borrow Kelly’s money?

10. A coat is marked down 10%. During a special doorbuster sale, the customeris given an additional 15% off. Is this the same as receiving a 25% markdown?If not, which is a better deal for the customer?

4.2. COMPOUND INTEREST 161

4.2 Compound Interest

Examples:


i = interest rate per time period

m = number of time periods

Compound Interest Formula:

F = P · (1 + i)m

In particular, if r is an annual interest rate and we compound n times a year for tyears, then

i =r

nm = nt

So we get

F = P ·(

1 +r

n

)nt

1. Suppose we deposit $2000 into an account that earns 8% annual interest. Ifthe interest is compounded quarterly, how much will be in the account after10 years?


2. If an account earns 6% annual interest, compounded monthly, and we want$10,000 in the account after 5 years, how much do we need to deposit today?

3. We are going to experiment with the compound interest formula by changingthe value of n, which is how many times we compound during the year. Tohelp us focus, let’s pick really simple values for the other variables.

Suppose we invest $1 in an account that pays 100% annual interest for 1 year.

(a) Write the values of P , r, and t.

(b) What is the value of F if n = 1 (we compound annually)?

(c) What is the value of F if n = 2 (we compound semiannually)?


(d) What is the value of F if n = 4 (we compound quarterly)?

(e) What is the value of F if n = 12 (we compound monthly)?

(f) What is the value of F if n = 365 (we compound daily)?

(g) Now pick a large value of n (bigger than 365, but small enough for yourcalculator to handle), and find the value of F for your n.

(h) What will happen to F as n gets really big? (To try to see the pattern,don’t round your answers to the nearest cent.)


Compounding Continuously Formula:

F = Pert where e ≈ 2.718281828

4. If we invest $51 dollars for 17 years in an account that pays 3% annual interestcompounded continuously, how much will we have after 17 years?

5. If an account earns 6% annual interest, compounded continuously, and wewant $10,000 in the account after 5 years, how much do we need to deposittoday?


Homework

1. Given the principal P , the annual interest rate r, the time t, and the frequencyof compounding, find the future value F .

(a) P = $15,000, r = 6.5% compounded quarterly, t = 2 years

(b) P = $100,000, r = 3% compounded monthly, t = 30 years

(c) P = $1,500, r = 4% compounded daily, t = 60 days

(d) P = $6,000, r = 5% compounded monthly, t = 6 months

2. When Phyllis was born, her parents deposited $4,000 into a bank accountearning 5% annual interest, compounded monthly. When she reached age 18,how much was in her account?

3. Scranton Bank is lending $10,000 to Chuck Bratton for three years. The bankcompounds interest quarterly. If the bank needs to receive $2,300 in interestfrom Chuck to cover its expenses, what annual interest rate should it charge?

4. Angela deposited some money in a bank account earning 3% annual interest,compounded daily. Twelve years later, there was $1,720 in the account. Howmuch money did Angela originally deposit?

5. How much money must Meredith deposit today in order to have $50,000 intwenty years, if her account earns

(a) 8% annual interest, compounded annually?

(b) 8% annual interest, compounded quarterly?

(c) 8% annual interest, compounded monthly?

(d) 8% annual interest, compounded daily?

(e) 8% annual interest, compounded continuously?

6. Suppose Oscar has $5,000 to invest. For each pair of investments, determinewhich will yield the greater return after 3 years:

(a) Investment 1: 6% annual interest, compounded monthly

Investment 2: 5.75% annual interest, compounded continuously

(b) Investment 1: 8% annual interest, compounded quarterly

Investment 2: 7.95% annual interest, compounded continuously


7. Suppose you deposit $10,200 into an account today that pays 10% annualinterest.

(a) If this interest is simple interest, how much will be in your account after40 years?

(b) If this interest is compounded annually, how much will be in your accountafter 40 years?

(c) What is the difference in the amounts from parts (a) and (b)? Why isthere such a significant difference?


4.3 Effective Annual Yield

Example:

Definition: The effective annual yield of a compound interest account at a givennamed rate, called the nominal rate, is the simple interest rate that gives the samefuture value as the nominal interest rate would give after compounding for one year.

r = nominal annual interest rate

n = number of times we compound each year

Y = effective annual yield

Y =(

1 +r

n

)n

− 1

4.3. EFFECTIVE ANNUAL YIELD 169

Formula Explanation:

1. Find the effective annual yield for each of the following 2 certificates of deposit(CDs):

(a) CD 1: 3.06% annual interest compounded monthly

(b) CD 2: 3.15% annual interest compounded quarterly


Homework

1. What is the maximum amount you can borrow today if it must be repaid in6 months with an annual simple interest rate of 5% and you know that, 6months from now, you will be able to repay at most $1,500?

2. Find the effective annual yields for the given nominal rates and compoundingfrequencies:

(a) 4.5% annual interest, compounded quarterly

(b) 3.75% annual interest, compounded monthly

(c) 5.1% annual interest, compounded daily

3. Bank 1 offers a nominal annual rate of 4%, compounded daily. Bank 2 pro-duces the same effective annual yield as Bank 1 but only compounds interestquarterly. What nominal annual rate does Bank 2 offer?

4. A 30-year-old worker inherits $250,000. If the worker deposits this amountinto an account that earns 5.5% annual interest, compounded quarterly, howmuch money will be in the account when the worker turns 65 (which is whenthe worker plans to retire)?

5. State Bank offers a savings account that pays 4% annual interest, compoundedquarterly. You have three options:

• Option A: Deposit $1000 on November 6, 2015 and deposit another$1000 on November 6, 2016.

• Option B: Deposit $2000 on November 6, 2015.

• Option C: Deposit $1950 on November 6, 2015.

(a) Without doing any calculations, determine whether Option A or OptionB will give a higher account balance on November 6, 2017. Explain youranswer.

(b) Determine whether Option A or Option C will give a higher accountbalance on November 6, 2017. (Feel free to do some calculations here.)

4.3. EFFECTIVE ANNUAL YIELD 171

6. State Bank offers two different certificates of deposit (CDs):

• CD 1: This is a 5-year CD that offers 3% annual interest, compoundedsemiannually (twice each year).

• CD 2: This is a 5-year CD that offers 3.5% annual interest, compoundedsemiannually (twice each year), but charges an initial $50 fee to earn thishigher interest rate.

So, for example, if you deposit $500 into CD 2, only $450 is allowed to earninterest for the 5 years.

(a) If you have $1500 to deposit into one of these CDs, which CD will havethe higher balance after 5 years?

(b) If you have $3000 to deposit into one of these CDS, which CD will havethe higher balance after 5 years?

7. State Bank offers two 3-year variable interest rate accounts.

• Account A: This account offers 3% annual interest the first year, 4%annual interest the second year, and 5% annual interest the third year.

• Account B: This account offers 5% annual interest the first year, 4%annual interest the second year, and 3% annual interest the third year.

(a) If the interest earned is simple annual interest, and you make a depositthat stays in the account for all three years, would you prefer one accountto the other? Why or why not?

(b) If, instead, the interest earned is compounded annually, and you make adeposit that stays in the account for all three years, would you prefer oneaccount to the other? Why or why not?

(c) If the account earns either simple interest or interest compounded annu-ally and you had to withdraw your money at the end of the second year,which account would you prefer?

8. Gringotts Wizarding Bank offers goblins three special savings accounts. Ac-count 1 offers 5.2% annual interest, compounded monthly. Account 2 offers5.1% annual interest, compounded daily. Account 3 offers 5.0% annual inter-est, compounded continuously.

Find the effective annual yield for each of the three accounts, and use theseyields to determine which account offers the best return on deposits.


9. The United States paid about 4 cents an acre for the Louisiana Purchase in1803.

(a) Suppose the value of this property grew at an annual rate of 5.5% com-pounded annually. What would an acre be worth in 2015?

(b) What would an acre be worth in 2015 if the annual rate was 6% com-pounded annually?

(c) Do these numbers seem realistic?

4.4. ORDINARY ANNUITIES 173

4.4 Ordinary Annuities

1. What is 1 + 2 + 3 + 4 + · · ·+ 99 + 100?

Definition: An ordinary annuity is a sequence of equal payments made atequal time periods, where the payments are made at the end of each timeperiod. Interest is also compounded at the end of each time period.

The term of an annuity is the total length of time from the beginning of thefirst time period to the end of the last time period.

The future value is the total amount in the annuity at the end of the term.


Example: Suppose we deposit $50 a month into an ordinary annuity thatpays 12% annual interest, compounded monthly, for a total of one year. Whatis the future value of the annuity? (In other words, how much is in the annuityafter one year?)

4.4. ORDINARY ANNUITIES 175

F = future value

PMT = payment made each period



F = PMT ·(

(1 + i)m − 1

i

)

2. Suppose we pay $250 a quarter into an ordinary annuity for 7 years, wherethe annual interest is 8%, compounded quarterly.

(a) Find the future value of the annuity.

(b) How much interest did we earn over the course of these 7 years?


Homework

1. Find the future value of each of the following ordinary annuities.

(a) Payments of $1200 made at the end of each year for 10 years, where 7%annual interest is compounded annually

(b) Payments of $300 made at the end of each quarter for 10 years, where8% annual interest is compounded quarterly

(c) Payments of $50 made at the end of each month for 20 years, where 6%annual interest is compounded monthly

(d) Payments of $100 made at the end of each week for 2 years, where 8%annual interest is compounded weekly

2. How much total interest was earned in the annuity in 1. (b)?

3. An uncle said he would set up an ordinary annuity for a newly born niece andpay $100 a month, with the last payment to occur on her 18th birthday. Thepayments would earn 6% annual interest, compounded monthly. The auntsaid they should just give the niece a lump sum of money now that wouldgrow to the same amount (at 6% annual interest, compounded monthly) asthe annuity would by the 18th birthday. If they go with the aunt’s plan, howmuch should they give to the niece now?

4. Mr. Smith decides to pay $300 at the end of each month into an ordinaryannuity that pays 8% annual interest, compounded monthly, for five years.He decides to calculate the future value of this annuity at the end of these fiveyears, but he makes a mistake in his calculations. What was his mistake? Ishis answer too big or too small?

F = PMT ·(

(1 + i)m − 1

i

)F = 300 ·

((1 + .08)60 − 1

.08

)F = 300 · 1253.213296

F = $375963.99

5. What is 1 + 2 + 3 + · · ·+ 48 + 49?

4.5. MORTGAGES 177

4.5 Mortgages

Definition: A mortgage is a loan for a house.

Qualifying for a mortgage:

In order to determine if a buyer qualifies for a mortgage, we compute the buyer’sadjusted monthly income. The adjusted monthly income is the gross monthly in-come minus any unchanging monthly payments that still have more than 10 monthsremaining.

The maximum that a person can spend on housing expenses each month is 28% oftheir adjusted monthly income.

Example:


Example:

Principal and Interest for Mortgages:

The following table records the Monthly Principal and Interest Payment per $1000of Mortgage.

Rate % 10-year 20-year 30-year5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34

Table 4.1: Monthly principal and interest per $1000 of mortgage.

4.5. MORTGAGES 179

Definition: The amount that a buyer pays at closing, called the closing costs, isthe sum of the down payment plus any points the lender charges. A point is 1% ofthe amount borrowed.

Followup:


Homework

1. Calculate what a 20% down payment would be for the following house costs:

(a) $249,900

(b) $119,900

(c) $154,500

(d) $255,000

2. Penny has a gross monthly income of $5,900. She has 13 payments of $160 amonth remaining on her student loan and 20 payments of $310 a month onher car loan.

(a) What is 28% of her adjusted monthly income?

(b) Penny wants a 20-year fixed-rate mortgage. She wishes to buy a houseat a price of $189,900. If she makes a 25% down payment, then she canfind a mortgage with an interest rate of 5%. Use Table 4.1 to calculateher monthly principal and interest payment for the remaining 75% of thehouse price.

(c) If insurance and taxes sum to $240 each month, calculate Penny’s totalmonthly mortgage payment.

(d) Does Penny earn enough money to qualify for this mortgage?

3. Sheldon wants to purchase a new home with a price of $299,900. His bankrequires a 20% down payment and a payment of 3 points on the mortgageamount to earn a lower interest rate.

(a) What is the amount of Sheldon’s down payment?

(b) What is his mortgage amount?

(c) What will the cost of the 3 points on the mortgage amount be?

4.5. MORTGAGES 181

4. Dwight Schrute, Assistant (to the) Regional Manager at the Scranton branchof Dunder Mifflin, has a gross monthly income of $3,150. He has 11 remainingmonthly payments of $200 for supplies for his next beet crop and 48 remainingmonthly payments of $5 on a loan for his impressive collection of mustard-colored dress shirts.

Dwight is (heretically!) considering abandoning Mose at Schrute Farms andbuying a house that is selling for $129,500. The insurance and taxes on theproperty are $125 and $145 per month, respectively. Dwight’s bank requires a20% down payment, payable to the seller, and a payment of 2 points, payableto the bank, at closing. The bank will approve a loan with a total monthlymortgage payment of principal, interest, property taxes and insurance that isless than 28% of Dwight’s adjusted monthly income.

(a) What is Dwight’s down payment?

(b) What is the mortgage amount?

(c) Determine the closing costs (down payment and points).

(d) What is 28% of Dwight’s adjusted monthly income?

(e) If Dwight wants a 30 year mortgage and the annual interest rate is 5.5%,determine the total monthly payment for the mortgage by first findingthe payment for principal and interest using Table 4.1, and then addingthe amounts for insurance and taxes.

(f) Does Dwight qualify for the mortgage?



1. Discuss how to calculate payments for loans (other than payments for homemortgages). In class, we found a formula for the future value of ordinaryannuities. Explain how to find the formula for the present value of ordinaryannuities. Apply this formula to a variety of situations, which may includelottery payments, comparing investments, retirement accounts, car payments,or comparing business expenses. You could also discuss amortization tables,and how we calculate how payments are divided between principal and interest.

2. Give an introduction to inflation and deflation. Explain the Consumer PriceIndex. Use compound interest to calculate the changing costs of items due toinflation. Compare changes in costs of certain items to actual inflation rates.

3. Present the concept of open-end credit loans, the type of loans used by creditcards. Present different methods credit card companies use to compute interestcharges and minimum monthly payments.


4.7 Chapter Review

Concepts:

This page and the next page will be provided for the test!

P = principal or present value

r = annual interest rate

t = time in years

F = future value

n = number of times we compound each year

Y = effective annual yield

PMT = payment made each period



• Simple InterestF = P · (1 + rt)

• Compound Interest

F = P ·(

1 +r

n

)nt

• Compounding Continuously

F = Pert where e ≈ 2.718281828

• Effective Annual Yield

Y =(

1 +r

n

)n

− 1

• Future Value of an Ordinary Annuity

F = PMT ·(

(1 + i)m − 1

i

)• Adjusted Monthly Income

The adjusted monthly income is the gross monthly income minus any un-changing monthly payments that still have more than 10 months remaining.To qualify for a mortgage, the maximum that a person can spend on housingexpenses is 28% of their adjusted monthly income.


• Closing Costs

The closing costs are the amount a buyer pays at closing on the day they buythe house. Closing costs include the down payment and any points chargedby the lender. A point is 1% of the amount borrowed.

• Principal and Interest for Mortgages

The following table records the Monthly Principal and Interest Payment per$1000 of Mortgage.

Rate % 10-year 20-year 30-year5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34


Review Exercises:

1. Liz has a gross monthly income of $3,500. She has 17 remaining monthlypayments of $220 for a car loan and 3 remaining monthly payments of $95 onher student loans.

Liz is looking at buying a house that is selling for $155,000. The insuranceand taxes on the property are $110 and $135 per month, respectively. Liz’sbank requires a 20% down payment. The bank will approve a loan with a totalmonthly mortgage payment of principal, interest, property taxes and insurancethat is less than 28% of Liz’s adjusted monthly income.

(a) What is the mortgage amount?

(b) If Liz wants a 20 year mortgage and the annual interest rate is 6%, de-termine the total monthly payment for the mortgage by first finding thepayment for principal and interest, and then adding the amounts for in-surance and taxes.

(c) Does Liz qualify for the mortgage?

2. How much money needs to be deposited into a bank account today so that itgrows to $51,000 after 12 years if:

(a) The account earns 2% annual simple interest

(b) The account earns 2% annual interest, compounded monthly

(c) The account earns 2% annual interest, compounded continuously

3. The same amount of money was invested in each of two different accounts onJanuary 1, 2011. Account A increased by 2.5% in 2011, then by 4% in 2012,and then by 1.5% in 2013. Account B increased by the exact same x% for eachof those three years. The interest is compounded annually in both accounts.At the end of 2013, both accounts had the same amount of money. What isthe value of x?

4. Locksafe Bank offers three special savings accounts. Account 1 offers 4.7%annual interest, compounded quarterly. Account 2 offers 4.6% annual interest,compounded monthly. Account 3 offers 4.5% annual interest, compoundedcontinuously.

Find the effective annual yield for each of the three accounts, and use theseyields to determine which account offers the best return on deposits.


5. Suppose we put $51 at the end of each month into an account earning 3%annual interest, compounded monthly.

(a) How much is in the account after 10 years?

(b) How much interest did this account earn during the 10 years?

6. You borrow $1290 on a credit card that charges simple interest at an annualrate of 11%. What is your interest after two months?

7. If $2500 is invested at a rate of 3.2% for 4 years, find the balance if the interestis compounded (a) annually; (b) monthly; (c) continuously.

8. Suppose you deposit $2300 into an account earning simple interest. Whatsimple interest rate is being charged if the amount at the end of 8 months is$2331?

9. Suppose you deposit $1500 into an account earning simple interest of 3.33%.How long do you have to wait until the account doubles in value?

10. First Bank is advertising a savings account with a 5% interest rate, com-pounded daily. National Bank advertises a savings account with a 5.25% in-terest rate, compounded monthly. Which bank would you deposit your moneywith? Justify your answer.

11. A mid-life worker wants to retire in 15 years. If the worker needs to have$125,000 at retirement to live comfortably, how much should be invested nowat 6% interest compounded weekly?

Monthly Principal and Interest Payment per $1000 of MortgageRate % 10-year 20-year 30-year

5.0 10.61 6.60 5.375.5 10.85 6.88 5.686.0 11.10 7.16 6.006.5 11.35 7.46 6.327.0 11.61 7.75 6.657.5 11.87 8.06 6.998.0 12.13 8.36 7.34

12. The Pella family is considering a house priced at $144,500. The taxes on thehouse will be $3200 per year and the homeowners’ insurance will be $450 peryear. They have applied for a mortgage from their bank. The bank is requiringa 15% down payment, and the interest rate on the loan is 7.5%. Their annual


gross income is $86,500. They have more than 10 monthly payments remainingon each of the following: $220 on a car, $170 on new furniture, and $210 on astudent loan. Their bank will approve the mortgage if their monthly housingexpenses (principal and interest for mortgage, insurance and taxes) are lessthan 28% of their adjusted monthly income.

(a) What is their down payment?

(b) What is the mortgage amount?

(c) What is 28% of their adjusted monthly income?

(d) If they want a 30-year mortgage, determine the monthly principal andinterest for the mortgage using the table above.

(e) What are their monthly housing expenses (principal, interest, insuranceand taxes)?

(f) Do they qualify for the mortgage?



1. (a) $124000

(b) $1132.84

(c) No

2. (a) $41129.03

(b) $40126.04

(c) $40118.02

3. 2.66%

4. Account 1: 4.783%

Account 2: 4.698%

Account 3: 4.603%

Account 1 offers the best return.

5. (a) $7126.81

(b) $1006.81

6. I = Prt =⇒ I = $1290(0.11)(

212

)= $23.65

7. (a)A = 2500(1 + .032

1

)1(4)=$2835.69 (b)A = 2500

(1 + .032

12

)12(4)=$2840.90 (c)

A = 2500e.032(4) =$2841.38

8. $2331 = $2300 (1 + r × 8/12) =⇒ r = 0.02 =⇒ 2%.

9. $3000 = $1500 (1 + .0333t) =⇒ t = 30 years.

10. At First Bank the effective annual yield is Y = (1 + 0.05/365)365−1 ≈ 0.0512.At National Bank the effective annual yield is Y = (1 + 0.0525/12)12 − 1 ≈0.0538. Since 0.0538 > 0.0512 we should pick National Bank.

11. $125, 000 = P (1 + 0.06/52)52(15) =⇒ P = $50, 847.58.

12. (a) $144, 500× 0.15 = $21, 675

(b) $144, 500− $21, 675 = $122, 825

(c) $86, 500/12− $220− $170− $210 = $6608.33; 0.28× $6608.33 = $1850.33

(d) $6.99× $122, 825/$1000 = $858.55

(e) $858.55 + $3200/12 + $450/12 = $1162.71

(f) Since total monthly payment = $1162.71 < $1850.33 = 28% of adjustedmonthly income they qualify for the mortgage.

Chapter 5

Voting Theory

5.1 Voting Systems

1. The Sugar Beet Company is preparing to elect its president. Three membersare considering running for president: Alice, Bob, and Charles. Each of the fivevoters has a first, second, and third choice, as listed in the following table:

Voter: Voter 1 Voter 2 Voter 3 Voter 4 Voter 5

1st choice: Alice Bob Bob Charles Bob2nd choice: Charles Alice Charles Alice Alice3rd choice: Bob Charles Alice Bob Charles

Depending on who decides to run, the ballot the voters see in the booth willlook different. For each of the following ballots, how many votes will eachcandidate receive? Who will win?

(a)

Who should bepresident?

� Alice� Bob� Charles

(c)


� Alice� Charles

(b)


� Alice� Bob

(d)


� Bob� Charles

189

190 CHAPTER 5. VOTING THEORY

2. A company is electing a manager. Four candidates are running for the position:Micah, Darba, Jim, and Pam. The voters have the following preferences:

Number of BallotsRanking 3 2 4 3 2

1st choice M M D J P2nd choice P P P P D3rd choice D J J M J4th choice J D M D M

(a) Here is the ballot:

Who should be thenew manager?

� Micah� Darba� Jim� Pam

Who will be elected manager? What percent of the vote will he or shehave?

(b) As you look at the voters’ preferences, who seems like the best choice tomake the most voters happy?

(c) If your answers to parts (a) and (b) are different, can you explain why?

(d) Can you think of a different voting system that would help the voterselect your choice from part (b)?

5.1. VOTING SYSTEMS 191

3. Plurality voting:

4. Borda count:


5. If the company above runs their election as a Borda count, who will win?


1st choice M M D J P2nd choice P P P P D3rd choice D J J M J4th choice J D M D M


6. A company warehouse is voting for its new warehouse manager, and the can-didates are Donatello, Leonardo, Raphael, and Michelangelo.

# of BallotsRanking 3 10 9 8

1st choice D L M R2nd choice R R D M3rd choice L D R D4th choice M M L L

(a) Each member votes for his favorite candidate. What percentage of votesdid the first-place candidate receive?

(b) Rather than elect a candidate who didn’t receive a majority vote, thewarehouse holds a run-off election between the top two candidates. Whowill win the run-off election?

(c) Was anyone eliminated from the run-off who perhaps shouldn’t havebeen?

(d) Can you think of a better runoff system for the warehouse than a runoffbetween just the top two candidates? What result does it give?


7. Plurality with elimination:


8. The Army Corps of Engineers is trying to decide on a flood prevention projectfor the Fargo-Moorhead metro area. The options are

(D) A diversion on the North Dakota side(R) Retention and reservoir system along the Red River valley(F) Building floodwalls to 50′ throughout both cities(C) Relocating the community to higher ground in, say, Colorado(M) A diversion on the Minnesota side

The eighteen members of the Corps have the following preferences:

Number of BallotsRanking 5 4 2 3 1 3

1st choice F M R D C R2nd choice D D F M D F3rd choice R R C R F D4th choice M C D F M M5th choice C F M C R C

The ACE’s voting rules employ plurality with elimination.

(a) What will the vote tallies be after the first round of voting? Is there awinner yet? If not, who should be eliminated?

(b) What will the vote tallies be after the second round of voting? Is there awinner yet? If not, who should be eliminated?

(c) What will the vote tallies be after the third round of voting? Is there awinner yet? If not, who should be eliminated?

9. If there are n candidates in an election conducted by plurality with elimination,what is the most rounds of voting that could be required?


10. A company is voting on its Accountant-of-the-Month. The candidates are:

• Oberyn

• Khaleesi

• Arya

• Theon

The preference table is the following:


1st choice A T K O O2nd choice O O A K A3rd choice K K T A T4th choice T A O T K

(a) Who would win in a head-to-head race between Oberyn and Khaleesi?

(b) Who would win in a head-to-head race between Oberyn and Arya?

(c) Fill in the following table of who would win head-to-head races; if thereare any ties, put both names in the box.

Theon Arya Khaleesi

Oberyn

Khaleesi

Arya

(d) Who won the most head-to-head races?(Count any ties as 1

2a win for each contender.)


11. Pairwise Comparison:

12. A tip for setting up the head-to-head races:


13. Our Math 105 classes are voting on where to take a math field trip. Theoptions are:

(E) Egypt, home of the pyramids

(H) Hogwarts School of Witchcraft and Wizardry

(I) Italy, home of the Renaissance

(M) Middle Earth, home of the hobbits

(W) West Acres in Fargo

The preference table is the following.


1st choice H W W E M H2nd choice E H E M H I3rd choice I I I H E W4th choice M E H W W M5th choice W M M I I E

We decide to use the pairwise comparison method to decide. Where will wego?


Homework

Show all your work. In particular, for plurality with elimination, show the results ofeach round of voting; for pairwise comparison, show the result of each head-to-headvote.

1. Consider the following preference matrix:


1st choice D D A E B A2nd choice C C B C E B3rd choice E E E B D C4th choice B A D A A D5th choice A B C D C E

This race is being decided by plurality vote.

(a) How many voters are voting in this election?

(b) How many votes would be required to win a majority in this election?

(c) Who will win this election? How many votes will that person get? Doesthat candidate have a majority?

(d) Suppose candidate B drops out of the race. Who would win the election?

(e) Suppose everyone drops out of the race except A and C. Who will winthe election?



1st choice B C D B A2nd choice D A C C B3rd choice C D A A C4th choice A B B D D

Find the winner of an election held using each of the following voting schemes.(If it is a tie, say who tied.)

(a) Using plurality vote.

(b) Using Borda Count.


(c) Find the winner using plurality with elimination.

(d) Find the winner using pairwise comparison.



1st choice B B C A C A2nd choice C A B C A B3rd choice A C A B B C







# of BallotsRanking 3 5 4

1st choice C B D2nd choice E E C3rd choice A A E4th choice D C A5th choice B D B






5.2. VOTING PARADOXES AND PROBLEMS 201

5.2 Voting Paradoxes and Problems

1. Of the four voting systems we’ve studied, which is best? Why?

2. The Fargo Running Club is electing a new team captain. There are threecandidates: Ira, Jannika, and Kelsey. The preference table is the following.


1st choice J I K2nd choice I J I3rd choice K K J

(a) If the club uses the plurality method, who will win the election?

(b) If it were a head-to-head race between Ira and Jannika, who would win?

(c) If it were a head-to-head race between Ira and Kelsey, who would win?

(d) If it were a head-to-head race between Jannika and Kelsey, who wouldwin?

(e) Why does the plurality method’s result seem unfair in this election?


3. Head-to-Head Criterion:

4. The school board is voting whether to award the contract for building a newelementary school to Aasgaard Architects, Bob’s Builders, Cobber Construc-tion, or Dora’s Developers.


1st choice B B C A2nd choice C C A D3rd choice D A D C4th choice A D B B

(a) The school board votes using a Borda count.Which company will get the contract?

(b) Does this seem fair? Why or why not?


5. Majority Criterion:

6. The International Olympic Committee has to select one of four cities for theWinter Olympics: O, T, M, F. As of this moment, the voters’ preferences areas follows:


1st choice F M T O2nd choice O F M T3rd choice T T O F4th choice M O F M

The IOC uses the plurality with elimination method.

(a) If the vote is held today, which city will be chosen as the host city?


(b) After receiving a very nice bribe, the single voter in the last column de-cides that F really is the best candidate after all, and so his preferenceschange to F,O,T,M. Thus when the election is held tomorrow, the pref-erence table is


1st choice F M T ×O F2nd choice O F M ×T O3rd choice T T O ×F T4th choice M O F ×M M

Which city will be chosen?

(c) What is odd about your answers to (a) and (b)?

7. Monotonicity Criterion:


8. Voters have 4 choices: A, B, C, D. Here is the preference table.

# of BallotsRanking 10 6 5 4 2

1st choice D C C A B2nd choice A B A D A3rd choice C A D B C4th choice B D B C D

(a) Find the winner of the election under the plurality with eliminationmethod.

(b) Find the winner of the election under the pairwise comparison method.

(c) What is strange about your answers to (a) and (b)?


9. Voters are trying to decide what to do with a $50 million surplus in the statebudget. Their options are:

(T) Give the money back to taxpayers as property tax relief.

(R) Spend it on road construction and other infrastructure.

(S) Build a new stadium for a professional sports team.

Here is the preference table:

Number of BallotsRanking 70,000 50,000 40,000

1st choice T R S2nd choice R S T3rd choice S T R

(a) The election is run by pairwise comparison. What is the result?

(b) What is odd about your results from part (a)?


10. Irrelevant Alternatives Criterion:

11. Arrow’s Impossibility Theorem:


Homework



1st choice C D A C A2nd choice D C C B B3rd choice B A B A D4th choice A B D D C

(a) Does the Borda Count violate the Majority Criterion for this particularpreference matrix?

(b) Does the Borda Count violate the Head-to-Head Criterion for this par-ticular preference matrix?



1st choice C D C D A B2nd choice A A A A D D3rd choice D C B B C C4th choice B B D C B A

(a) Does the Borda Count violate the Majority Criterion for this particularpreference matrix?

(b) Does the Borda Count violate the Head-to-Head Criterion for this par-ticular preference matrix?



1st choice B B D D2nd choice D D B B3rd choice C A A C4th choice A C C A

(a) How many points will each candidate receive in a Borda count? Who willwin?


(b) The five voters in the last column really do think D is the best candidateand B is the second-best. However, they decide to be sneaky and lie ontheir Borda count ballots, claiming they think B is the worst candidate;in other words, they say they prefer D, C, A, and B in that order. Nowhow many points will each candidate receive in a Borda count? Who willwin?

(c) Explain in a few complete sentences how these voters manipulated theBorda count and why it is unfair.

4. Use a Borda count to determine the preference of the voters.


1st choice A C E2nd choice B D C3rd choice C E D4th choice D B B5th choice E A A

Suppose the last two voters change their minds with respect to B and D:


1st choice A C E2nd choice B B C3rd choice C E B4th choice D D D5th choice E A A

What is the preference now? Explain in a few sentences what you see happen-ing in these results.

5. Use a Borda count and straight plurality to determine the following election.Explain the results.


1st choice A B C2nd choice B C B3rd choice C A A


6. Use plurality with elimination to determine the winner of the following elec-tion.


1st choice A C B2nd choice B B A3rd choice C A C

Now suppose voters in the middle column change their minds about C:


1st choice A B B2nd choice B C A3rd choice C A C

Has the outcome changed? Explain the results in a sentence or two.

5.3. WEIGHTED VOTING SYSTEMS 211

5.3 Weighted Voting Systems

1. Notes on weighted voting systems:

(a) Weight:

(b) Example: Four partners start a new business. Partner A owns 8 shares,Partner B owns 7 shares, Partner C owns 3 shares, and Partner D owns2 shares. If

1 share = 1 vote

represent this weighted voting system.

(c) Motion:

(d) Quota:

(e) Example: If the quota for [8, 7, 3, 2] is a simple majority, find the quota.What is the quota if a two-thirds majority is required?

(f) Weight/quota notation:


2. In a weighted voting system with weights [30, 29, 16, 8, 3, 1], if a two-thirdsmajority of votes is needed to pass a motion, what is the quota?

3. Consider the weighted voting system [14, 9, 8, 5].

(a) What is the largest reasonable quota for this system?

(b) What is the smallest reasonable quota for this system?

4. Consider the weighted voting system [20|7, 5, 4, 4, 2, 2, 2, 1, 1].

(a) How many voters are there?

(b) What is the quota?

(c) What is the weight for voter P2?

(d) If the first 4 voters vote for a motion and the rest vote against, does themotion pass?

(e) If P1 and P2 vote against a motion, will the motion pass?


5. What is peculiar about each of the following weighted voting systems?

(a) [20|10, 10, 9]

(b) [7|4, 2, 1]

(c) [51|50, 49, 1]

(d) [6|6, 2, 1, 1]

(e) [21|10, 8, 5, 3, 2]


6. A dummy is . . .

7. A dictator is . . .

8. A voter has veto power if . . .

9. Is weight the same as power? Consider the weighted voting systems [12|13, 7, 2]and [19|8, 7, 3, 2].

10. In the weighted voting system [12|9, 5, 4, 2], are there any dummies or dicta-tors?

11. In designing a weighted voting system [q|6, 5, 4, 3, 2, 1], what is the largestquota q you could pick without giving veto power to anyone?

12. In the weighted voting system [q|8, 5, 4, 1], if every voter has veto power, whatis the quota q?


Homework

1. Anton, Bella, Chaz, and Debra are the stockholders in Red River Industries.Anton owns 252 shares, Bella owns 741 shares, Chaz inherited 637 shares, and412 shares are in Debra’s hands. As usual, each share corresponds to a votein the stockholder’s meeting.

(a) If a certain type of motion requires a majority vote, what is the smallestnumber of votes needed to pass the motion?

(b) A different type of motion requires a 2/3 vote to pass. What is thesmallest number of votes needed to pass this motion?

(c) Using the quota you found in part (b), express the weighted voting systemin the correct notation (with brackets and quota).

2. Find all dictators, dummies, and voters with veto power in the followingweighted voting systems:

(a) [51|20, 20, 20]

(b) [51|36, 34, 23, 6]

(c) [25|27, 11, 7, 2]

(d) [31|15, 13, 6, 4, 2]

3. Which voters have veto power in the system [51|29, 21, 8, 3, 1]?

4. In 1958, the Treaty of Rome established the European Economic Community(EEC) and instituted a weighted voting system for the EEC’s governance. Themembers at that time were France, Germany, Italy, Belgium, the Netherlands,and Luxembourg. The three largest countries (France, Germany and Italy)were each given a vote with weight 4, Belgium and the Netherlands had votesof weight 2 and Luxembourg’s vote had weight 1. The quota was 12.

What is unusual or interesting about this weighted voting system?

5. Consider the weighted voting system [25|7, 7, 6, 6, w]. Find all positive integersw that make P5 a dummy.

6. In the weighted voting system [q|32, 25, 23, 10], if every voter has veto power,what is the quota q?


5.4 Banzhaf Power Index

1. A coalition is . . .

2. (a) Consider a weighted voting system with three voters P1, P2, and P3. Listall the coalitions. How many are there?

(b) Consider a weighted voting system with four voters P1, P2, P3, and P4.List all the coalitions. How many are there?

(c) If a weighted voting system has n voters P1, P2, . . . , Pn, how manycoalitions are there?

5.4. BANZHAF POWER INDEX 217

3. A winning coalition is . . .

4. List all the winning coalitions in the weighted voting system [10|5, 4, 3, 2, 1].

5. In the weighted voting system [10|6, 4, 3, 2, 1], consider the winning coalition{P1, P2, P3, P4}. Which voter(s) could change their minds and vote “no” with-out changing the outcome of the vote? Which voter(s) need to keep voting“yes” in order for the motion to pass?

6. A critical voter in a winning coalition is . . .


7. Consider the voting system [19|11, 9, 8, 5].

(a) List all the winning coalitions.

(b) In each winning coalition above, circle the critical voters.

(c) Count the number of times each voter is a critical voter. This is calledthat voter’s Banzhaf power.

Voter Banzhaf power

P1

P2

P3

P4

(d) Add up all the voters’ Banzhaf powers; this sum is called the totalBanzhaf power of the voting system.

(e) Finally, divide each voter’s Banzhaf power by the total Banzhaf power.The percentage that results is called the voter’s Banzhaf power index.

Voter Banzhaf power index

P1

P2

P3

P4


8. Calculate the Banzhaf Power Index for each voter in the weighted voting sys-tem [51|32, 22, 12].

9. Make up a weighted voting system with a dummy, and calculate the BanzhafPower Index for the dummy.

10. Make up a weighted voting system with a dictator, and calculate the BanzhafPower Index for the dictator.

11. Make up a weighted voting system in which several voters have veto power.Calculate the Banzhaf Power Index for the voters with veto power. What doyou notice?


12. In a weighted voting system with 6 voters, how many possible coalitions arethere?

13. Given [21∣∣10, 8, 5, 3, 2], answer the following questions.

(a) How many voters are there?

(b) What is the quota?

(c) What is the total number of possible coalitions?

(d) Is the coalition {P2, P3, P4, P5} a winning coalition?

(e) List all of the winning coalitions.

(f) What do you notice about voter P1? P2? P5?

14. Determine the Banzhaf power index of each voter for [20∣∣12, 9, 8, 2].



To Calculate Voters’ Banzhaf Power Indices:

STEP 1: Locate all winning coalitions in the weighted voting system.

STEP 2: Determine which voters are critical in each of the winning coali-tions.

STEP 3: Calculate each voter’s Banzhaf power by adding up the numberof times each voter is critical.

STEP 4: Calculate the total Banzhaf power for the weighted voting systemby adding all voters’ Banzhaf powers.

STEP 5: Determine each voter’s Banzhaf power index by dividing his/herBanzhaf power by the total Banzhaf power.


Homework

1. Calculate the Banzhaf Power Index for each voter in the weighted voting sys-tem [34|12, 10, 7, 6].

2. Consider the voting system [25|24, 20, 1].

(a) Calculate the percentage of the total weight that each voter holds.

(b) Calculate the Banzhaf Power Index for each voter.

(c) Comparing your answers to parts (a) and (b), explain in complete sen-tences why the weight controlled by the voter is not the same thing asthe power held by each voter.



5. Nassau County, New York used to be governed by a Board of Supervisors.The county had six districts, each of which had one delegate to vote on countyissues. The delegates’ votes were weighted proportionately to the districts’population in 1964:

District WeightHempstead #1 31Hempstead #2 31Oyster Bay 28North Hempstead 21Long Beach 2Glen Cove 2

A simple majority was needed to pass a motion.

(a) Express this weighted voting system in our usual notation.

(b) Calculate the Banzhaf power of each district.

(c) What percentage of the county population lived in districts that are dum-mies?

In 1965 John F. Banzhaf III argued in court that even though the weightswere proportionate to population, this system of government was unfair. Hewon!


5.5 Voting Theory Homework Set

The following problems are to be worked on in class, with the answers written upon a separate sheet of paper to be turned in as homework for this section.

1. The Coombs Method: This method is just like the plurality-with-eliminationmethod except that in each round we eliminate the candidate with the largestnumber of last-place votes (instead of the one with the fewest first-place votes).


1st choice W Z Z Y X2nd choice X W X Z Z3rd choice Y Y W X W4th choice Z X Y W Y

(a) Find the winner of the election above using the Coombs method.

(b) Find the winner of the election above using the pairwise comparison(head-to-head) method.

(c) What do you notice when comparing the two methods for this data?Mention any fairness criteria that are violated.

2. The Coombs method continued: Recall that in each round we eliminate thecandidate with the largest number of last-place votes.


1st choice B C C A2nd choice A A B B3rd choice C B A C

(a) Find the winner of the election above using the Coombs method.

(b) Suppose all 8 voters in the second column switch A and B; that is, theyput B in 2nd and A in 3rd place. Use the Coombs method to determineif the overall results of the election are the same.

(c) What do you notice when comparing the outcomes for this method? Men-tion any fairness criteria that are violated.

5.5. VOTING THEORY HOMEWORK SET 223

3. Consider the weighted voting system [8∣∣5, 3, 1, 1, 1].

(a) Make a list of all winning coalitions.

(b) Using (a), find the Banzhaf power distribution of this weighted votingsystem.

(c) Suppose that P1, with 5 votes, sells one of her votes to P2, resultingin the weighted voting system [8

∣∣4, 4, 1, 1, 1]. Find the Banzhaf powerdistribution of this system.

(d) Compare the power index of P1 in (b) and (c). Describe in your ownwords the paradox that occurred.

4. A business firm is owned by 4 partners, P1, P2, P3, P4. When making decisions,each partner has one vote and the majority rules, except in the case of a 2− 2tie; in that case, the coalition that contains P4–the partner with the leastseniority–loses. What is the Banzhaf power distribution in this partnership?

5. A panel of sportswriters and broadcasters every year selects an NBA rookieof the year using a variation of the Borda count method. The following tableshows the results of the balloting for the 2003− 2004 season.

Player 1st Place 2nd Place 3rd Place Total PtsLeBron James 78 39 1 508Carmelo Anthony 40 76 2 430Dwayne Wade 0 3 108 117

Using the information from the table, figure out how many points are awardedfor each 1st-place, 2nd-place, and 3rd-place vote in this election. Assume thatthese numbers are different positive integers.

6. An NBA team has a head coach H and 3 assistant coaches C1, C2, C3. Playerpersonnel decisions require at least 3 of the coaches to vote yes, one of whichmust be H to pass. If we use [q

∣∣h, c, c, c] to describe this weighted votingsystem, find q, h, c.


5.6 Supplement for Voting Theory: Antagonists

Suppose that in a weighted voting system there is a voter A who hates another voterP so much that she/he will always vote the opposite of P , regardless of the motion.We will call A the antagonist of P . This means every coalition, whether winningor not, must contain either A or P , but not neither or both.

1. Suppose that in the weighted voted system [8∣∣5, 4, 3, 2], P is the voter with

two votes and her/his antagonist A is the voter with five votes.

(a) What are the possible coalitions (not necessarily winning) in this case?

(b) What is the Banzhaf power distribution is this case?

2. Recall the weighted voting system from Page 132 #7: [19∣∣11, 9, 8, 5].

(a) If P4 becomes the antagonist of P1, what happens to the Banzhaf powerindex of P4? P1? (Compare with the original Banzhaf power index of eachvoter.)

(b) If P3 becomes the antagonist of P1, calculate the new Banzhaf power indexfor each of the four voters.

(c) What happens to the Banzhaf power index of each voter if P3 becomes theantagonist of P2?

3. Consider the weighted voting system [4∣∣3, 2, 1].

(a) Determine the Banzhaf power index of each voter.

(b) If P3 becomes the antagonist of P1, what happens to the Banzhaf powerindex of P1? P2? P3? (Compare with the original Banzhaf power index ofeach voter.)

(c) If P2 becomes the antagonist of P1, what happens to the Banzhaf powerindex of P1? P2? P3? (Compare with the original Banzhaf power index ofeach voter.)

(d) If P3 becomes the antagonist of P2, what happens to the Banzhaf powerindex of P1? P2? P3? (Compare with the original Banzhaf power index ofeach voter.)

5.6. SUPPLEMENT FOR VOTING THEORY: ANTAGONISTS 225

4. (a) Citing examples above (or coming up with new ones), show that it ispossible for voter A to increase her/his Banzhaf power index by becoming anantagonist of another player.

(b) Citing examples above (or coming up with new ones), show that it ispossible for voter A to decrease her/his Banzhaf power index by becoming anantagonist of another player.

5. If there are only two voters, A and P , how many coalitions (not necessarilywinning) are there if A is the antagonist of P?

If there are three voters, A, P , and P3, how many coalitions are there if A isthe antagonist of P?

If there are four voters, A, P , P3, and P4, how many coalitions are there if Ais the antagonist of P?

Do you see a pattern?

If there are n voters, A, P , P3, and so on out to Pn, how many coalitions arethere if A is the antagonist of P?



1. Present the Shapley-Shubik power index. (Who are Shapley and Shubik?)Compare and contrast this index to the Banzhaf power index that we dis-cussed in Section 1.4 (you might compute the Banzhaf power index for yoursmall example so that people can see how they’re different/similar). Givesome examples, including real examples, such as the Electoral College. Thereare many good introductions to this index online, along with plenty of smallexamples. Some tables including real examples of the index can be found at

http://www.cut-the-knot.org/Curriculum/SocialScience/ShapleyShubikIndex.shtml

You will need to give and explain at least one small, but meaningful example(think something like what we have done in class or on the homework) of thisvoting system. The context in which you present this example is up to you,but I would recommend using 4 voters in a weighted voting system. Threevoters is too simplistic, and five can get messy. Note: people have publishedonline calculators that can compute the Shapley-Shubik power indices for you.This might be easier for larger examples. Possible topics: What motivatedShapley and Shubik to define this power index? Find the Shapley-Shubikpower index of all the states in the Electoral College; interpret the results.Can you find examples where the Shapley-Shubik index tells us which votersare marginalized? Which voters have a lot of power?

2. Introduce the Bucklin voting system, named after James Bucklin of GrandJunction, Colorado. Explain how the Bucklin system works, give examples ofsimple elections using this system. What are the flaws of this method? Whatfairness criteria, if any, might this system violate? You will need to give andexplain at least one small, but meaningful example (think something like whatwe have done in class or on the homework) of this voting system. The contextin which you present this example is up to you. You may want to weave it intoone or all of the following: Does anyone use this voting system today? Didanyone use this voting system in the past? Why did they move away from it?Describe how this voting system is related to other voting systems we havediscussed in class. Can a person vote strategically to game this system? If so,how?

3. Discuss the Single Transferable Voting (STV) system. Explain what it is, howthe votes are counted, what quotas are set, and how winners are determined.Give examples of simple elections using this system (for hints, see wikipedia’sexample on single transferable voting, for starters). Who uses this methodtoday? Mention governments and the Academy Awards. You will need togive and explain at least one small, but meaningful example (think something


like what we have in done class or on the homework) of this voting system.The context in which you present this example is up to you. You may wantto weave it into one or all of the following: Discuss which governments useSTV, and why. You may choose to focus on Minneapolis, MN, which currentlyuses STV for some city matters. Which matters are decided by STV? Whatprompted the switch to STV? Have there been complaints about this new-fangled voting system? Is it possible for someone to game their voting andcheat the system? If so, give an example. Other topics or neat things youfind. Be creative!

4. Background and applications of the Banzhaf power index. John Banhaf wasa lawyer in Nassau County (NY) who studied the Nassau County Board andfound that their weighted voting system essentially consigned the smaller dis-tricts in the county to the role of dummies! Explain their weighted votingsystem in the 1960s vs. the 1990s and explain the weaknesses of the 1960sversion. Include any interesting information on the subsequent legal fight toget the voting system changed. See homework exercise 5 in Section 1.4.


5.8 Chapter Review

To prepare for the exam over this chapter, you should review the in-class worksheetsand homework. Be ready to do the kind of problems you faced on the homework.

1. Know how to read a table of voters’ preferences

2. Calculate the winner according to

(a) Plurality

(b) Borda count

(c) Plurality with elimination

(d) Pairwise comparison

3. Know the four fairness criteria

(a) Head-to-head criterion

(b) Majority criterion

(c) Monotonicity criterion

(d) Irrelevant alternatives criterion

4. Devise preference tables that satisfy given conditions (e.g., “Come up with apreference table where the pairwise comparison test produces no winner.”)

5. Weighted voting

(a) Know how weighted voting on yes/no motions works.

(b) Understand the notation [q|w1, . . . , wn].

(c) Given a weighted voting system, find any dictators, dummies, or voterswith veto power.

6. Construct weighted voting systems that satisfy given conditions (e.g., “Comeup with a weighted voting system where two people have veto power.”)

7. Calculate the Banzhaf Power Index for the voters in a weighted voting system.


Review Exercises:

1. Four candidates (A, B, C, and D) run in an election with the following results:


1st choice B C A D2nd choice C A C C3rd choice A D B B4th choice D B D A

(a) Determine the winner of the election using the

i. plurality method

ii. Borda count method

iii. plurality-with-elimination method

iv. pairwise comparison method

(b) Do any of the methods violate the majority criterion?

(c) Do any of the methods violate the head-to-head criterion?

2. Three candidates (A, B, and C) run in an election with the following results:


1st choice A B B C2nd choice B C A A3rd choice C A C B


i. plurality method


iii. plurality with elimination method


(b) Do any of the methods violate the majority criterion?

(c) Do any of the methods violate the head-to-head criterion?


3. Three candidates (A, B, and C) run in an election with the following results:


1st choice A B C2nd choice B C B3rd choice C A A


i. plurality method




(b) Suppose candidate C drops out of the election. Determine the winner ofthe election using the

i. plurality method




(c) Do any of the methods violate the irrelevant alternatives criterion?

4. Voters have 4 choices: A, B, C, D. Here is the preference table.


1st choice D A C B A2nd choice B B A C C3rd choice A C B A B4th choice C D D D D


i. plurality method




(b) Do any of the methods violate any of the fairness criteria?


5. Determine the quota if a simple majority is needed to pass a motion. Useproper notation to express the voting system.

(a) [7, 4, 3, 3, 2, 1]

(b) [10, 6, 5, 4, 2]

(c) [6, 4, 3, 3, 2, 2]

6. Determine the quota if the given percentage is needed to pass a motion. Useproper notation to express the voting system.

(a) [7, 4, 3, 3, 2, 1], 67%

(b) [10, 6, 5, 4, 2], 75%

(c) [6, 4, 3, 3, 2, 2], 80%

7. Consider the weighted voting system [q∣∣5, 3, 1]. Find the Banzhaf power dis-

tribution of this weighted voting system when (a) q = 6 (b) q = 8.

8. In the weighted voting system [q∣∣8, 5, 4, 1], if every voter has veto power what

is q?

9. In the weighted voting system [q∣∣6, 5, 4, 3, 2, 1], what is the largest value of q

so that no voter has veto power?

10. Describe anything you find unusual or interesting about the following weightedvoting systems.

(a) [22∣∣11, 11, 10]

(b) [10∣∣5, 3, 2]

(c) [41∣∣40, 39, 1]

11. Suppose we have the weighted voting system [49∣∣4w, 2w,w,w], a system with

4 voters, one of whom has twice the power of the others, and one of whom hasfour times the power of the others. Find w so that the quota 49 is

(a) a simple majority of the votes;

(b) more than a 23-majority of the votes;

(c) more than a 34-majority of the votes.

12. Consider the weighted voting system [q∣∣8, 4, 1].

(a) What value(s) of q results in a dictator?

(b) What value(s) of q results in exactly one player with veto power?


(c) What value(s) of q results in more than one player with veto power?

(d) What value(s) of q results in one or more dummies?

13. Consider the weighted voting system [21∣∣10, 8, 5, 3, 2].

(a) List all winning coalitions.

(b) Determine the Banzhaf power index of each voter.

(c) Suppose P4 and P5 become antagonists. Recalculate the Banzhaf powerindex of each voter. Is there a shift in power?

(d) Instead, suppose there is a power struggle at the top, so that P1 and P2

become antagonists, while other voters have normal relations. Recalcu-late the Banzhaf power index and write in your own words what results.Why is this?



1. (a) D, C, D, D

(b) Borda count method violates the majority criterion.

(c) Borda count method violates the head-to-head criterion.

2. (a) B, B, A, A

(b) No

(c) Plurality method and the Borda count method violate the head-to-headcriterion.

3. (a) A, B, B, B

(b) B, B, B, B

(c) Plurality method violates the irrelevant alternatives criterion.

4. (a) D, B, C, A

(b) Plurality, plurality with elimination, and Borda count all violate the head-to-head criterion.

5. (a) [11∣∣7, 4, 3, 3, 2, 1]

(b) [14∣∣10, 6, 5, 4, 2]

(c) [11∣∣6, 4, 3, 3, 2, 2]

6. (a) [14∣∣7, 4, 3, 3, 2, 1]

(b) [21∣∣10, 6, 5, 4, 2]

(c) [16∣∣6, 4, 3, 3, 2, 2]

7. (a) P1 has 3/5=60%; P2 and P3 have 1/5=20%. (b) P1 and P2 have 1/2=50%and P3 has none=0%.

8. q = 18, because the only way to guarantee P4 has veto power is to set q =w1 + w2 + w3 + w4 = 8 + 5 + 4 + 1 = 18.

9. q = 15, because in order to prevent P1 from having veto power, we must setthe quota equal to the sum of the other weights: q = w2 +w3 +w4 +w5 +w6 =5 + 4 + 3 + 2 + 1 = 15.

10. (a) [22∣∣11, 11, 10]: Even though P3 has 10 votes, P3 is a dummy and has no

power. P1 and P2 both have veto power.


(b) [10∣∣5, 3, 2]: Even though the voters have different weights, all voters have

the same power and veto power; the system is equivalent to [3∣∣1, 1, 1].

(c) [41∣∣40, 39, 1]: P2 and P3 have the same power despite their different

weights, and P1 has veto power; the system is equivalent to [3∣∣2, 1, 1].

11. (a) For a simple majority, the quota is 50% plus 1 vote: 49 = 12(8w) + 1, so

w = 12.

(b) For a more than 23-majority, the quota is more than two-thirds the total

vote: 49 ≥ 23(8w), so w = 9.

(c) For a more than 34-majority, the quota is more than three-fourths the

total vote: 49 ≥ 34(8w), so w = 8.

12. Consider the weighted voting system [q∣∣8, 4, 1].

(a) If q = 6, 7, 8, then P1 is a dictator.

(b) If q = 9, then P1 is the one player with veto power.

(c) If q = 10, 11, 12, then P1 and P2 both have veto power. If q = 13, thenall 3 have veto power.

(d) If q = 6, 7, 8, then both P2 and P3 are dummies. If q = 10, 11, 12, thenP3 is a dummy.

13. (a) {10, 8, 5, 3, 2}, {10, 8, 5, 3}, {10, 8, 5, 2}, {10, 8, 3, 2}, {10, 8, 5}, {10, 8, 3}.(b) P1 = P2 = 6/16 = 37.5%, P3 = P4 = 2/16 = 12.5%, P5 = 0%.

(c) {10, 8, 5, 3}, {10, 8, 5, 2}, {10, 8, 3}. P1 = P2 = 3/8 = 37.5%, P3 = P4 =1/8 = 12.5%, P5 = 0%. No, there is no shift in overall power!

(d) There are no winning coalitions! If P1 and P2 become antagonists nomotion passes since they both have veto power.

Appendix A

Projects

• Team Presentation: In teams of two, students will present a topic from alist of possible subjects that correspond to chapters covered during the course.We will finish each chapter with two or three presentations during one classperiod. The talks will be 15-20 minutes in length. Ideas for the talks are foundin the workbook at the end of each chapter. You may select your teammateand presentation topic on a first-come, first-served basis. Absolutely feel freeto see me for help in preparing your paper and presentation.

• Follow-up paper: Each individual team member will also submit an indi-vidual project paper after their presentation. Your paper should include 1-2pages summarizing the content of the team presentation, and a short section(roughly half of a page) describing what you yourself learned in completingyour project. Papers will be due 9 days after your presentation – if yourpresentation is on a Monday, then it is due the following week on Wednesday;presentation on Wednesday =⇒ due the following week on Friday; presen-tation on Friday =⇒ due the Monday after next. Groups in Chapter 5 willhave to turn in their paper by the end of finals week.

• Audience: The students not speaking during a presentation day will evaluatethe talks that are presented on that day, and submit these evaluations at theend of the class period as the homework assignment for that day. Speakersearn a perfect homework score for their presentation day. There will be noother homework assigned on presentation days.

• Points and Grading: These projects will be worth 7.7% of your final grade(50 points out of 650 semester points). Presentations and papers will be gradedon mathematical content, organization, appropriate length of the talk andpaper, visual aids (PowerPoint is strongly encouraged), quality of examples,

235

236 APPENDIX A. PROJECTS

enthusiasm and effectiveness at communicating, and if there are equal con-tributions to the presentation by both team members. Papers will also bechecked for correct spelling and grammar. A grading rubric is below:

Presentation Grade (36 pts):

• Presentation is well organized (4 pts)

• Presentation is an appropriate length (4 pts)

• Information is presented clearly/Easy to follow (4 pts)

• Speakers make proper use of visual aids (4 pts)

• Speakers give appropriate examples (4 pts)

• Presentation contains significant mathematical content (4 pts)

• Speakers are knowledgeable about the subject matter (4 pts)

• Speakers are poised and enthusiastic (4 pts)

• Speakers contribute equally to the presentation (4 pts)

Paper Grade (14 pts):

• Paper was submitted on time (1 pt)

• Paper effectively conveys the team presentation content and what you learnedin completing your project (6 pts)

• Paper is well written and clear, with correct spelling and grammar (5 pts)

• Paper is an appropriate length (2 pts)

Appendix B

Syllabus

Math 105K Exploring Mathematics, CRN 18130, Fall 2015

MWF 10:30-11:40, Old Main 102

Activity Points Date

Homework 175 Every classExam I 85 September 16Exam II 85 October 7Exam III 85 October 30Exam IV 85 November 18Exam V 85 Friday December 18, 8:30-10:30Project 50 @ end of your chosen unit

Professor: Dr. Douglas Anderson, Fjelstad 132, 299-4453, [email protected]

Office Hours: Feel free to stop by my office, Fjelstad 132, Tuesday and Thursdaynoon-4, or MWF by discovery. If these times are inconvenient, please set up anappointment.

Free Tutoring: The Mathematics Department supports a Math 105 tutor Sunday,Tuesday, and Thursday nights in Old Main 202 from 7:15 PM to 9:15 PM. TheAcademic Enhancement and Writing Center (AEWC) in Lower Level Fjelstad, roomB06, also has Math tutors: Sunday through Thursday 7 PM to 9 PM. For moreinformation, visit their website.

Text: Math 105 Workbook: Exploring Mathematics, Fall 2015 Yellow Edi-tion, available from the Cobber Bookstore.

Academic Integrity: All homework, tests, and other graded assignments will besubject to Concordia’s policy on Academic Integrity. In particular, you may not

237

238 APPENDIX B. SYLLABUS

obtain assistance from any source on exams.

Homework: There will be daily graded homework assignments from the workbookto complete on separate paper prior to the next class. I encourage you to work withother students to complete the homework, but the assignment you submit mustreflect your own work. Homework is due at the beginning of class. Late homeworkwill not be accepted unless you have missed class due to an excused absence andyou have made arrangements with me in advance. Your three lowest homeworkscores will not be included in your final homework grade, which will be scaled out of175 points. Again, all students must abide by the college’s expectations regardingacademic quality, integrity, and honesty.

Exams: Attendance is required for all exams. If you should miss an exam for anemergency you will be allowed to make it up only if you have notified me beforethe exam and it must be made up in a timely manner (to be discussed with meindividually). The dates for exams are given on this syllabus; I will give you at leastone week notice if the exam date is to be changed. There will be five 70-minute unitexams, each worth 85 points. I do not scale the 70-minute exams. Partial creditmay be given for incorrect answers but correct reasoning; partial discredit may begiven for correct answers but incorrect reasoning. To study for the exams, pleaselook over examples from class and the homework, and work the review problemsfrom the book.

Project: (Presentation and Paper) Each of you will write and present a 15-20minute in-class presentation (as a part of a team of two) following one of the units wecover in class. The topics for these presentations will be extensions and applicationsof the course material. Ideas for the projects are in the workbook at the end of eachunit. More details are included in the Appendix of the workbook.

Calculator/Calculator App: Calculators or phones with calculator apps are al-lowed and will be used occasionally for homework and exams. Any scientific calcu-lator would be appropriate for this class; cell phones may also be used as calculatorsin class and on exams. Please see me if you need help selecting a calculator, or ifyou have questions about your calculator app on your phone.

Grading: With a total of 650 points, the course grades will be as follows:

A 92-100% C 72-76%A- 90-91% C- 70-71%B+ 87-89% D+ 67-69%B 82-86% D 62-66%B- 80-81% D- 60-61%C+ 77-79% F 0-59%

239

Core Criteria and Outcomes: As a Core Exploration Course in Mathematics,Math 105 Exploring Mathematics meets the following criteria relative to Concordia’sGoals for Liberal Learning (GLL) in our core curriculum, Becoming ResponsiblyEngaged in the World :

• Emphasize arithmetical, algebraic, geometric, algorithmic, or statistical meth-ods (GLL 1,2,3);

• Develop an understanding of mathematical models such as formulas, graphs,tables, schematics, or algorithms (GLL 2,3,5);

• Incorporate mathematical models and methods to solve problems (GLL 1,2,3,5).

Learning Outcomes:

• Students will demonstrate an understanding of mathematical methods andmodels.

• Students will represent mathematical information symbolically, visually, andnumerically.

• Students will apply mathematical methods and models to solve multi-stepproblems.

This course uses real-world problems and situations to improve your problem-solvingskills, to improve your ability to apply mathematics, and to enhance your apprecia-tion of the importance of mathematics in our modern world. Topics will be chosenfrom taxicab geometry, counting and probability, graph theory, consumer mathe-matics, and voting theory.

240 APPENDIX B. SYLLABUS

Date Section Homework from Workbook on Separate PaperAug 28 1.1 Taxicab Geometry (8) 1-3, please use graph paper

Aug 31 1.2 Taxicab Circles (13) 1-5, on graph paperSep 2 1.3 Taxicab Applications (21) 1-3, on graph paper

4 1.4 Minimizing Regions (25) 1-3, on graph paper

Sep 7 1.5 Midsets (31) 1-6, on graph paper9 1.6 Lines (38) 1-8, on graph paper11 Presentations

Sep 14 Review (42-50) 1-1416 Test 118 2.1 Permutations (62) 1-7

Sep 21 2.2 Combinations (69) 123 Fall Symposium China Rising (Memorial Auditorium)25 2.3 Probability Basics (77) 1-8

Sep 28 2.4 Probability of Events (84) 1-1030 2.5 Conditional Probability (91) 6,7

Oct 2 Presentations

Oct 5 Review (94-99) 1-257 Test 29 3.1 Intro to Graphs (107) 1-7

Oct 12 3.2 Paths and Circuits (114) 1-814 3.3 Subgraphs & Trees (122) 1-916 3.4 Graph Colorings (131) 1-3

Oct 19 3.5 Planar Graphs (136) 1-621 3.6 Directed Graphs (143) 1-723 Presentations

Oct 26 Fall Break No Class28 Review (147-152) 1-1430 Test 3

Nov 2 4.1 Interest (159) 1-104 4.2 Compound Interest (166) 1-76 4.3 Effective Annual Yield (170) 1-9

241

Date Section Homework from Workbook on Separate PaperNov 9 4.4 Annuities (176) 1-5

11 4.5 Mortgages (180) 1-413 Presentations

Nov 16 Review (183-187) 1-1218 Test 420 5.1 Voting Systems (199) 1-4

Nov 23 5.2 Flaws in Voting Systems (208) 1-625 Thanksgiving Break No Class27 Thanksgiving Break No Class

Nov 30 5.3 Weighted Voting Systems (215) 1-6Dec 2 5.4 Banzhaf Power Index (221) 1-4

4 5.5 Voting Theory Homework Set (222) 1-6

Dec 7 Supplement: Antagonists (224) 1-59 Presentations11 Review (228-232) 1-13

Dec 14Dec 18 Test 5 Friday, 8:30-10:30 AM

Last modified: August 11, 2015

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Douglas R. Anderson, Professor Fall 2015: MWF …faculty.cord.edu/andersod/105bookFall2015.pdf · Douglas R. Anderson, Professor Fall 2015: MWF 10:30-11 ... Excursions in Modern Mathematics,