Math 105 Workbook Exploring Mathematics

Douglas R. Anderson, Professor

Fall 2018: MWF 11:50-1:00, ISC 101 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 important ways. We also want to thank our departmental colleagues and several Concordia mathematics majors for many fruitful discussions and resources on the content of this course and the makeup of this workbook. Some of the topics, examples, and exercises in this workbook are drawn from other works. Most signiﬁcantly, we thank Samantha Briggs, Ellen Kramer, and Dr. Jessie Lenarz for their work in Exploring Mathematics, as well as other Cobber mathematics professors. 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 suggestions for this course and this workbook are always encouraged, either in person or over e-mail. Both the course and workbook are works in progress that will continue to improve each semester with your help. Let’s have a great semester this fall exploring mathematics together and fulﬁlling Concordia’s math requirement in 2018. Skol Cobbs!

i ii 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 2.6 Chapter Projects ...... 93 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 Possible Project: Directed Graphs ...... 138 Homework ...... 143 3.7 Chapter Projects ...... 146 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

iv 1

Homework ...... 170 4.4 Ordinary Annuities ...... 173 Homework ...... 176 4.5 Mortgages ...... 177 Homework ...... 180 4.6 Chapter Projects ...... 182 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 Possible Project: Antagonists ...... 224 5.7 Chapter Projects ...... 226 5.8 Chapter Review ...... 228

A Projects 235

B Syllabus 237 2 Chapter 1

Taxicab Geometry

1.1 Taxicab Distance

1. Suppose, in the city shown below, that we want to ride in a taxicab along city streets from the corner of 8th Street and 10th Avenue to the corner of 3rd Street 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 10th Avenue to the corner of 3rd Street and 13th Avenue take the same distance?

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

3 4 CHAPTER 1. TAXICAB GEOMETRY

Deﬁnition: The taxicab distance between two points is the shortest possible distance between the two points where we are only allowed to move horizontally or 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 tell you what the taxicab distance is? Why or why not?

(d) If you know the taxicab distance between two points, does that tell you what the Euclidean distance is? Why or why not? 6 CHAPTER 1. TAXICAB GEOMETRY

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

B A

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 to the taxicab distance? If so, explain why. If not, give an example where the 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 accident at X = (−1, 4). There are two police cars in the area, car C at (2, 1) and car D at (−1, −1). Which car should be sent to the scene of the accident to arrive most quickly? (Since the cars must drive on the streets, we use taxicab geometry to measure distances.)

1 3 8 1 5. Find the taxicab distance between A = ( 3 , 2 ) and B = ( 3 , − 2 ). 8 CHAPTER 1. TAXICAB GEOMETRY

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 are not a Euclidean distance of 5 from C. (b) Graph all of the points that are a taxicab distance of 5 from C, including those that are a Euclidean distance of 5 from C. Be sure to include all possible points, not just the ones with integer coordinates. (c) Come up with a mathematically appropriate name for the answer to part 2b.

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 same as the taxicab distance between B and C. (b) Find a diﬀerent point D so that the taxicab distance between A and D equals the taxicab distance between B and D. (c) Graph all of the points P where the taxicab distance between A and P and the taxicab distance between B and P is the same. Be sure to include 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 is a 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). 10 CHAPTER 1. TAXICAB GEOMETRY

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) of radii 1, 2, 3, and 4.

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

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

5. Tyrion Lannister has ﬂed King’s Landing and now works in Taxicab City for the 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 plant to do so. Where in the city are restaurants at which Tyrion can eat? Draw their locations on the graph.

M 12 CHAPTER 1. TAXICAB GEOMETRY

6. A developing company wants to construct an apartment building in Taxicab City within six blocks of the mall at M = (−2, 1) and within four blocks of the tennis courts at T = (3, 3). Shade in the area of the graph that suits the builder’s requirements. 1.2. TAXICAB CIRCLES 13

Homework

1. Graph all of the points that are a taxicab distance of exactly 4 from the point A = (−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. 5 (b) Graph the taxicab circle that is centered at (−1, −3) with a radius of 2 .

3. Bernadette and Howard reside in Taxicab City, which is laid out like a perfect grid centered on (0, 0). North-south and east-west streets join every point with integer coordinates. Bernadette works as an administrative assistant at an art school located at A = (−3, −1). Howard works as a bagel baker in a bakery located at B = (3, 3). Bernadette and Howard just got married and are looking for a house in the city. Bernadette has always dreamed of a cozy little house on a corner lot, so they will only consider houses located at street corners. Because they will walk to their jobs along the city streets, they measure 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 A and exactly 4 blocks from B? If so, find all locations that work and plot them on a graph. If not, why is it impossible? (b) Is it possible for Bernadette and Howard to live exactly 8 blocks from A and exactly 6 blocks from B? If so, find all locations that work and plot them on a graph. If not, why is it impossible? (c) Is it possible for Bernadette and Howard to live 8 or fewer blocks from A and 6 or fewer blocks from B? If so, find all locations that work and plot them on a graph. If not, why is it impossible? 14 CHAPTER 1. TAXICAB GEOMETRY

4. Raj and Lalita Gupta also live in Taxicab City. Raj works at the cupcake shop at C = (−2, 3) and Lalita works at the donut shop at D = (2, −1). Raj and Lalita are looking for places to live, but they do not necessarily have to live 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 within 4 blocks of D? If so, find all locations that work and shade them on a graph. If not, why is it impossible? (b) Raj and Lalita have a daughter, Priya, who goes to the elementary school at 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 locations that 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 blocks of D, and within 2 blocks of E? If so, find all locations that work and plot them on a graph. If not, why is it impossible?

5. Recall that a Euclidean square is a ﬁgure with four right angles and four straight sides of equal length, where we measure the length using Euclidean distance. A taxicab square is a ﬁgure with four right angles and four straight sides, and all four sides have the same length when measured using taxicab distance. We’ve seen in class that every taxicab circle is actually a taxicab square. Is every taxicab square also a taxicab circle? If so, why? If not, give an 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 hospitals in our taxicab city at (−1, 1) and (4, 3). Where should we go, using taxicab geometry? 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 every person living within the 12 × 12 square centered at the origin (0, 0) is within three blocks of a drinking fountain. There are three proposed plans from the city council, shown below. Which plan should the mayor choose? 16 CHAPTER 1. TAXICAB GEOMETRY

3. In the previous problem, you chose the better of the three given choices, but even that one is not optimal in terms of the fewest drinking fountains (for example, 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 place the other 10 strategically. 1.3. TAXICAB APPLICATIONS 17

4. In the same 12 × 12 grid, the city decided to install ﬁre hydrants so that every resident is within 4 blocks of a ﬁre hydrant. What is the fewest number of hydrants needed, and where should they be located? (A few grids are provided for practice below.) 18 CHAPTER 1. TAXICAB GEOMETRY

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

6. A furniture company wants to build a factory F in Taxicab City. They store unﬁnished tables in their warehouse at W = (−3, 2); they want to ship their tables from there to the new factory F , and then from F to their retail store at 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 the graph below. 1.3. TAXICAB APPLICATIONS 19

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 a graph, shade in all the places Leonard can live.

8. Now suppose Leonard wants to look for dark matter at D = (4, −2) and live near Caltech at C = (0, 1). He is looking for an apartment A so that the distance 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 20 CHAPTER 1. TAXICAB GEOMETRY

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

A R 1.3. TAXICAB APPLICATIONS 21

Homework

1. (a) The city council of Taxicab City has decided to build parks on street corners so that every resident of Taxicab City is within 6 blocks of a park. If Taxicab City is currently a 14 × 14 grid, what is the minimum number of parks needed and where should they be located? (b) Suppose the city already has a park located at (1, 2). Does that change the minimum number of parks needed? If it does, what is the new number of parks needed and where are they located? If it doesn’t change, why not? 2. Bernadette and Howard reside in Taxicab City. Bernadette works as an administrative assistant at an art school located at A = (−3, −1). Howard works as a bagel baker in a bakery located at B = (3, 3). Bernadette and Howard just got married and are looking for a house in the city. Bernadette has always dreamed of a cozy little house on a corner lot, so they will only consider houses located at street corners. Because they will walk to their jobs along the city streets, 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 of blocks Bernadette has to walk to work plus the number of blocks Howard has to walk to work is as small as possible. Where should they look for a house? (b) Now Howard decides to be chivalrous and insist that Bernadette should not have to walk any farther than he does, but they still want the total amount of walking to be minimal. Now where should they look for a house? (c) Bernadette decides to be generous in return, and wants both her husband and herself to walk exactly the same distance to work. They still want the total amount of walking to be minimal. Now where should they look for a house? 22 CHAPTER 1. TAXICAB GEOMETRY

(d) They still haven’t found a house! Having decided to widen their search, Bernadette and Howard keep only the requirement that they both walk the same distance to work. (So now the total amount of walking does not 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 plus Howard 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 Pam work at diﬀerent places!) Suppose that Jim and Pam are looking for a house and they want to minimize the total combined distance that they will walk to work. Pam must walk at least as far as Jim but no more than twice as far as Jim. They can only live on street corners. Where should they look for a house? 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 store unﬁnished tables in their warehouse at W = (−3, 2); they want to ship their tables from there to the new factory F , and then from F to their retail store at 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 the graph below.

Deﬁnition: Given a collection of points A, B, C, . . . , the point or points P for which the total taxicab distance from P to A, from P to B, from P to C, . . . , is as small as possible is called the minimizing region of the points A, B, C, . . . . 24 CHAPTER 1. TAXICAB GEOMETRY

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 each minimizing region on a new graph. Be sure to include all possible points in the 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). Throughout this problem, assume that buildings can only be located at corners with integer coordinates, and that warehouses and butter stands can be at the same location.

(a) Butter King wants to build a central supply warehouse so that the sum of the distances from the warehouse to the eight butter stands is as small as 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 that it requires twice as many deliveries from the central warehouse as each of 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 say about its minimizing region? 26 CHAPTER 1. TAXICAB GEOMETRY

1.5 Taxicab Midsets

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

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

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

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

(e) Mark all points on the graph that are equally distant from A and from B. (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 the collection of all points (not just the ones with integer coordinates) that are the same distance from A as from B.

Example from Euclidean geometry:

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

A 28 CHAPTER 1. TAXICAB GEOMETRY

2. Consider the points in the following diagram.

B C

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

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

3. Given two points X and Y , describe a rule for ﬁnding the taxicab midset of X and Y . 1.5. TAXICAB MIDSETS 29

4. Find the taxicab midset of G and H.

5. When, if ever, is the taxicab midset the same as the Euclidean midset? 30 CHAPTER 1. TAXICAB GEOMETRY

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 department draw a district boundary line so that all homes within the district are served by the closest fire station? N

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

(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 equally distant from all three ﬁre stations? 1.5. TAXICAB MIDSETS 31

Homework

1. Find the midset of A = (1, 3) and B = (−3, 5). (Recall that midsets can include 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 East at (2, 1). (Put your answers to the following questions on diﬀerent graphs!) (a) Where should they draw the school district boundary line so that each student attends the high school nearest their home? (As always, we measure distances in taxicab geometry.) (b) The city builds a third high school, Taxicab South, at (−1, −6). Now where should the boundary lines be drawn? (c) The owner of Pizza Palace wants to set up a pizzeria P that is equally distant from all three high schools. Where should the owner build the restaurant? (d) Wow! Our city is really growing! When Taxicab North High School is built 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 school A = (−3, −1) and Howard at the bagel bakery at B = (3, 3). The family does not 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 same distance to walk to work or school? (b) Where should they live if they have decided that Raj should have the shortest walk, Bernadette the second shortest walk, and Howard the longest walk? Shade the appropriate region of your graph. 6. Given two points A and B, we know that the minimizing region for these two points 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 say about the midset of A and B? (b) Of course, some rectangles are squares, and others are not squares. If the minimizing region for A and B is a square, what can we say about the midset of A and B? 32 CHAPTER 1. TAXICAB GEOMETRY

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.

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

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

(c) Josephine wants to move to a scenic apartment within three blocks’ walk of 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).

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

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

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 34 CHAPTER 1. TAXICAB GEOMETRY

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

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

(Challenge: How many diﬀerent minimal routes could the cable take?) 1.6. TAXICAB LINES 35

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 in taxicab distance.

A0 A B A A P P P

Comparison to Euclidean geometry:

P P

A A 36 CHAPTER 1. TAXICAB GEOMETRY

6. Sheldon and Amy are moving to town. Sheldon got a job at the Synchrotron at S = (−3, −1), whereas Amy will be working for the city light rail line ` that runs through the city as shown. One of Amy’s fringe beneﬁts is that when she comes 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 not need to live at street corners! (a) Sheldon and Amy want to live where the distance Sheldon has to walk to work plus the distance Amy has to walk to work is a minimum. Where should they look?

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

S 1.6. TAXICAB LINES 37

The shape in part (b) is called a taxicab parabola. 38 CHAPTER 1. TAXICAB GEOMETRY

Homework

1. Find the taxicab distance between the point A = (1, 2) and the line that passes through 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 the line `. (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 the line `. (Points with non-integer coordinates are allowed!)

4. Find a line ` carefully chosen so that the collection of points that are a taxicab distance of 3 from ` and the collection of points that are a Euclidean distance of 3 from ` are the same. In other words, every point that is a taxicab distance of 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 be working for the city light rail line ` that runs through the city as shown. One of Amy’s fringe beneﬁts is that when she comes to work she can just get on the train wherever is closest to her home. Sheldon and Amy do not necessarily have to live at street corners.

(a) Sheldon and Amy want to live where the distance Sheldon has to walk to work is no more than 2 blocks and the distance Amy has to walk to 1.6. TAXICAB LINES 39

work is no more than 3 blocks. Where should they look? Shade in your answers on a graph. (b) They change their minds and decide to live where they both walk exactly three blocks to work. Where should they live? List the coordinates of the points.

6. Alex lives at the point A = (−2, 2) in Taxicab City, and Bonnie lives at the point 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 wants to 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?

B 40 CHAPTER 1. TAXICAB GEOMETRY

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

8. Hazel and Gus are moving to town. Hazel is an engineer for the city light rail line ` that runs through the city as shown. One of Hazel’s fringe beneﬁts is that when she goes to work she can just get on the train at the point closest to 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. Where should they look?

D ` 1.7. CHAPTER PROJECTS 41

1.7 Chapter Projects

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

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

3. Describe various methods to show how two triangles are congruent in regular Euclidean 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 to more accurately describe the real world. For example, maybe there are one- way streets, or certain streets that are missing. Maybe there is an expressway that allows one to travel faster or skip over some blocks. Maybe there is construction or other detours. Explore these models with speciﬁc examples.

5. Given a simple polygon constructed on a taxicab grid such that all the polygon’s vertices are at corner points (grid points), Pick’s theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon’s perimeter. Explore this theorem and the simple area formula. 42 CHAPTER 1. TAXICAB GEOMETRY

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 resident living 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 this grid is covered. 1.8. CHAPTER REVIEW 43

2. Draw the minimizing region for each of the following sets of points. Be sure to include all possible points in the minimizing region, not just the points with integer 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) 44 CHAPTER 1. TAXICAB GEOMETRY

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

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

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

(c) Plot all of the points where a house could be located so that Bernadette and Howard each walk the same distance to work. 1.8. CHAPTER REVIEW 45

4. Our taxicab city has three high schools, Taxicab Halpert, Taxicab Schrute, and Taxicab Kapoor. Halpert is at (−4, −4), Schrute is at (2, −2), and Kapoor is at (−1, 5). Where should they draw the school district boundary lines so that each student attends the high school nearest their home? (As always in our city, we measure distances using taxicab geometry.) For your lines, be sure to include 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 distant from all three high schools, where should the owner put the stand?

5. Find (i) the Euclidean distance and (ii) the taxicab distance between the given points. (a) A = (1, −1),B = (2, −2) (b) C = (0, 1),D = (0, 3) 46 CHAPTER 1. TAXICAB GEOMETRY

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 1.8. CHAPTER REVIEW 47

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

9. Twinville has decided to set up stands selling Twins memorabilia in such a way that every resident is within 5 blocks of a stand. Using the grid below, what is the minimum number of stands needed and where should they be located? 48 CHAPTER 1. TAXICAB GEOMETRY

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 every student attends the high school closest to their home?

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

(c) If P = (1, 3), how should the city map the school districts so that every student 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 to B is exactly k units.

(a) A = (−1, 1),B = (3, −3), k = 10

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

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)

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

(a) d = 1,A = (−2, 2),B = (4, −1)

(b) d = 3,A = (−1, 1),B = (2, 4) 50 CHAPTER 1. TAXICAB GEOMETRY

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). 1.8. CHAPTER REVIEW 51

Some Review Answers:

1. 3.

(a)

2. (b)

(a)

(c)

(b) 52 CHAPTER 1. TAXICAB GEOMETRY

4. Windy’s should be at (−2, 0).

√ 5. (a) (i) 2; (ii) 2. (b) (i) 2; (ii) 2

6. C

7. C

H M 8. 1.8. CHAPTER REVIEW 53

9. Minimum number is 6.

P P

10. (a),(b) H H

P P

(c),(d) H H

B A A B A 11. B

12. (a) 2; (b) 4; (c) 5.5 54 CHAPTER 1. TAXICAB GEOMETRY

A B A A P P B B P

A ` A 13. B `

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

C A D B C

A D A 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 Tails are possible?

55 56 CHAPTER 2. COUNTING AND PROBABILITY

Multiplication Principle of Counting:

Suppose a task can be divided into m consecutive subtasks.

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 are possible? What about 10 times?

3. There are 26 letters (A-Z) and 10 digits (0-9). If a license plate must contain 3 letters followed by 3 digits, how many license plates are possible if repetitions are allowed? What if repetitions are not allowed? 2.1. INTRODUCTION TO COUNTING 57

Deﬁnition: A permutation of a collection of diﬀerent objects is an ordered arrangement 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 objects is an ordered arrangement of r of the n objects. The number of these permutations is denoted by P (n, r).

4. If we have a division of 8 basketball teams and we decide to rank our top 2 teams (ﬁrst and second place), how many choices are possible? What if we want to rank the top 4 teams? All 8 teams? 58 CHAPTER 2. COUNTING AND PROBABILITY

Formulas for P (n, r):

5. If we have a popularity contest in this class and choose a ﬁrst, second, and third place person, how many choices are possible? 2.1. INTRODUCTION TO COUNTING 59

6. Suppose we ﬂip a fair coin 5 times. How many sequences of Heads and Tails start 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 B somewhere in his password?

More practice problems on the next page! 60 CHAPTER 2. COUNTING AND PROBABILITY

1. Multiplication Principal of Counting restated: If one event can occur in a ways, and for each of those a ways another event can 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 how many ways can they be seated in the car?

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

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

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

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

7. A softball team with 9 players needs a batting order. How many diﬀerent ways are 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 diﬀerent tags 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 the 11 starting oﬀensive players from the football team on the left, 3 of the 6 volleyball players in the center, and 2 of the 5 starting basketball players on the 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 players from the basketball team are to line up for a picture, with all members of the baseball team together on the left. How many ways can this be done? 2.1. INTRODUCTION TO COUNTING 61

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 ways are 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 diﬀerent tags 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 62 CHAPTER 2. COUNTING AND PROBABILITY

Homework

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

8! (a) 4! 7! (b) 4! · 3! 100! (c) 98! 9! (d) 5! · 4! n! (e) , where n is any integer greater than or equal to one (n − 1)!

2. An urn holds 6 balls: a red ball, an orange ball, a yellow ball, a green ball, a blue 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 choices are 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 choices are possible? (b) If you are not allowed to repeat digits for your PIN, how many choices are possible? (c) If your PIN must contain the digits 2 and 5 (in some order, and not necessarily 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 the digit can be in any position, how many choices are possible? 2.1. INTRODUCTION TO COUNTING 63

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 members of 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 an officer? (d) How many selections are there in which Connie and Daniel are both officers? (e) How many selections are there in which Connie or Daniel may or may not be officers, but Connie and Daniel are definitely not both officers? 64 CHAPTER 2. COUNTING AND PROBABILITY

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, how many choices are possible?

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

Deﬁnition: A combination of r objects from a collection of n diﬀerent objects is a selection of r of the objects where the order of the objects selected is not important. The number of these combinations is denoted by C(n, r).

2. Suppose we have four people: Angela, Bob, Chuck, and Dani. How many ways are 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 the teams 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 choices are possible? 66 CHAPTER 2. COUNTING AND PROBABILITY

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

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

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?

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

More practice problems on the next page! 2.2. COMBINATIONS 67

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 can this 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 of 52 cards?

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

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 subcommittee 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 the outcomes of the calls be 3 sales, 8 no-sales, and 4 answering machines? 68 CHAPTER 2. COUNTING AND PROBABILITY

Section 2.2 Answers to Practice Problems

10! 10! 1. P (10, 3) = (10−3)! = 7! = 10 × 9 × 8 = 720.

P (10,3) 10! 10! 10×9×8 2. 3! = (10−3)!3! = 7!3! = 3×2×1 = 120.

9! 3. C(9, 5) = 4!5! = 126

5! 4. C(5, 3) = 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 subcommittee 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 2.2. COMBINATIONS 69

Homework

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

(a) In a regional track meet, 16 runners are competing in the 10000-meter run. The top eight runners qualify for the national meet. How many different 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 flowers to use in a vase? (c) A couple is planning a menu for a rehearsal dinner. The restaurant makes them choose 4 of the 7 possible appetizers, 3 of the possible 6 main courses, and 2 of the possible 5 desserts. How many different menus are possible? (d) From a group of seven boys and eight girls, how many 5-player teams can be formed that have three boys and two girls? (e) From a group of seven boys and eight girls, how many 5-player teams can be formed that have two boys and three girls? (f) From a group of seven boys and eight girls, how many 5-player teams can be 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 North Division, 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 same line, how many triangles can be drawn that use three of these points as vertices? (i) In a group of 12 people, we select a committee of 5 members. One member of the committee is the chairperson, and another is the secretary. How many different choices for this committee are possible? (j) Michael, Dwight, Jim, and Pam decide to play a game of cards during a slow day in the paper business. In how many different ways can the 52 cards in a deck be dealt to the four players, where each player gets 13 cards? Assume that the order of the cards in each player’s hand does not matter. (k) Suppose there are three divisions of five teams each. One team from each division will go to the playoffs, along with one other team that can come from any of the divisions. How many different combinations of teams can go to the playoffs, where the “wild-card” team is counted just like any 70 CHAPTER 2. COUNTING AND PROBABILITY

other team? In other words, in how many ways can we select four teams to go to the playoﬀs where the order of the teams is irrelevant, but every division 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 observable results, which are called outcomes.A sample space for an experiment is a set of all of the possible outcomes.

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

Deﬁnition: An event is a subset of the sample space. 72 CHAPTER 2. COUNTING AND PROBABILITY

Experimental Probability:

To ﬁnd the experimental probability of an event, we perform the experiment for a large number of times and count the number of times our event occurs. Then we divide 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 uniform sample space (so every outcome is equally likely, in theory, to occur). Then if E is an event (a subset of S), the theoretical probability of E is denoted by P (E) and is given by

Number of outcomes in E P (E) = . Number of outcomes in S 2.3. INTRODUCTION TO PROBABILITY 73

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

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

(b) B = “Roll a total of exactly 9”

Cards:

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

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? 2.3. INTRODUCTION TO PROBABILITY 75

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

4. An urn holds 10 identical balls except that 1 is white, 4 are blue, and 5 are red. An experiment consists of selecting two balls consecutively from the urn without replacement and observing their color. What is the sample space of this 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 of 5, {A, B, C, D, E}. What is the sample space? Indicate the outcomes of the event “D is selected”.

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

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

If E is the event of getting exactly 2 tails, ﬁnd the experimental probability of 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? 76 CHAPTER 2. COUNTING AND PROBABILITY

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 selected without replacing the ﬁrst. What is the probability that both components are defective?

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 defective and 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 equal to 7.

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

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

14. You wish to invest in the stock market. Among a group of 20 stocks, suppose that 10 stocks will go up and 10 will go down. If you pick 3 stocks at random from the group of 20, what is the probability that all 3 will go up? 2.3. INTRODUCTION TO PROBABILITY 77

Homework

1. A standard six-sided die (1-6) is rolled 100 times. The results of the rolls are Outcome Frequency 1 14 2 19 in the following table. 3 18 4 16 5 17 6 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 probability 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? 78 CHAPTER 2. COUNTING AND PROBABILITY

5. A large planning committee consists of 14 experienced members and 10 ﬁrst- time volunteers. If we randomly select 8 from the committee to do some subtask, what is the probability that all 8 of the people selected are experienced members?

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

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

(a) An experiment consists of selecting a ball from the urn and observing its color. 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 experiment now 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 cookie jar consists of 19 chocolate chip cookies and 15 sugar cookies, all of similar size. A hungry cookie monster selects 6 of these cookies for some noshing.

(a) What is the probability that the noshing consists of exactly 3 chocolate chip cookies and exactly 3 sugar cookies? (b) What is the probability that the noshing consists of at least 5 chocolate chip cookies? 2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 79

2.4 Complements, Unions, and Intersections

Deﬁnition: Let E be an event in a sample space S. The complement of E is denoted by 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 ∪ B and 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 that are in A and B. 80 CHAPTER 2. COUNTING AND PROBABILITY

1. Suppose we have an experiment where we select one Math 105 student. So our sample space S is the set of all Math 105 students. Consider the following three 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 discussed 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 2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 81

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 and B are mutually exclusive (they cannot both simultaneously happen). In this special case, P (A ∪ B) = P (A) + P (B) 82 CHAPTER 2. COUNTING AND PROBABILITY

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

(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)

(d) The probability that the card is a red card or a black card (or both) 2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 83

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 of selecting 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, upset stomach 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 faces marked with 3s, two faces marked with 5s, and two faces marked with 8s. If this 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). 84 CHAPTER 2. COUNTING AND PROBABILITY

Homework

1. 17% of American children have blue eyes. 6.6% of American children have Type 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 will have blue eyes or Type O- blood (or both)?

2. In 2015, there were roughly 20 million college students in the United States. 14.6 million of these students attended public colleges and 5.4 million attended private colleges. 66% of the public college students had student loans, and 75% of the private college students had student loans. What is the probability that a randomly selected college student attended public college or had student loans (or both)?

3. Suppose we select a Concordia student at random. Let A be the event that the student 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. What is 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). 2.4. COMPLEMENTS, UNIONS, AND INTERSECTIONS 85

8. A card is randomly selected from a standard 52-card deck. What is the probability 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?

1 9. Suppose A and B are events in a sample space S with P (A) = 2 and P (B) = 7 10 . What is the smallest possible value of P (A ∩ B)? What is the largest possible value of P (A ∩ B)?

10. An experiment consists of selecting a car at random from the Hvidsten parking lot and observing the color and make of the car. Let R be the event “The car is red,” let F be the event “The car is a Ford,” let G be the event “The car is a green 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 86 CHAPTER 2. COUNTING AND PROBABILITY

2.5 Conditional Probability

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

Example:

Deﬁnition: If A and B are events, then the probability that A will occur given that 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 is the 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 The Forum? 88 CHAPTER 2. COUNTING AND PROBABILITY

3. Suppose P (A) = 0.7, P (B) = 0.3, and P (A ∪ B) = 0.8. Draw the Venn diagram, 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 Venn diagram, 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) 2.5. CONDITIONAL PROBABILITY 89

5. Suppose we draw 2 cards from a standard deck without replacement. What is the 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 the ﬁrst two ﬂips was Heads?

7. Suppose an urn contains 4 white and 6 red balls. Two balls are randomly selected. If the ﬁrst ball is white, it is replaced before we make our second draw. 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? 90 CHAPTER 2. COUNTING AND PROBABILITY

Homework

1. Suppose we select a Concordia student at random. Let A be the event that the student 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 student is selected at random from the class and that the student is a junior. What is the probability that the student is also a math major?

3. Two cards are selected at random in order from a standard 52-card deck. Find the 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. Find the 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 the probability that the last ball is red? (b) If three balls are drawn from this urn without replacement, what is the probability that the last ball is red? (c) If three balls are drawn from this urn without replacement, what is the probability 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 the probability that the last ball is red given that the ﬁrst two balls were white? 2.5. CONDITIONAL PROBABILITY 91

(e) If three balls are drawn from this urn without replacement, what is the probability that the last ball is red given that at least one of the ﬁrst two balls 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 the probability 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 must be 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 are diﬀerent suits (o) The hand contains all four suits 92 CHAPTER 2. COUNTING AND PROBABILITY

8. The English alphabet contains 26 letters, 5 of which are vowels. What is the probability 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 than twice (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. If a Cobber went to the football game, what is the probability that that student went to the bonﬁre?

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

2.6 Chapter Projects

1. The French mathematicians Pierre de Fermat and Blaise Pascal are usually given credit for originating the theory of probability. The first problems in probability were posed to Pascal by the famous gambler Chevalier de Méré, and Pascal and Fermat exchanged letters developing the theory of probability in order to answer de Méré’squestions. Provide some historical background about these men and the letters, and present a complete solution to at least one of these first problems in probability.

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

3. Discuss the Monty Hall Problem: Suppose you’re on a game show, and you’re given 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 the doors, opens another door, say #3, which has a goat. He says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors? This problem was posed to Marilyn vos Savant in 1990, and she answered it correctly, but thousands of people, including quite a few Ph.D. mathematicians, wrote to her and told her she was wrong. Explain the problem, present some of the letters, and use probability to give a correct solution to the problem. 94 CHAPTER 2. COUNTING AND PROBABILITY

2.7 Chapter Review

Concepts:

• Permutations are where order matters, and the number of permutations is denoted by P (n, r) n! • P (n, r) = = n · (n − 1) · (n − 2) ····· (n − (r − 1)) (n − r)! • Combinations are where order does not matter, and the number of combinations is denoted by C(n, r) P (n, r) n! • C(n, r) = = r! (n − r)! · r! • Sample space

• Uniform sample space

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

number of outcomes in E P (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 not overlap 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) = (this comes from an outstanding Venn diagram) P (B) • P (A ∩ B) = P (A) · P (B|A) and P (A ∩ B) = P (B) · P (A|B) (in particular, we can multiply along a tree diagram) 2.7. CHAPTER REVIEW 95

• Coin questions

• Card questions

• Word questions 96 CHAPTER 2. COUNTING AND PROBABILITY

Review Exercises:

1. A state makes license plates with three letters followed by three digits. There are 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 allowed to repeat.

Assuming that the state makes license plates randomly according to these rules, what is the probability that a license plate made in this state begins with 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. An experiment consists of selecting two balls in succession from the urn without replacing 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 ﬁrst ball 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 probability that 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 the five following infield positions: catcher, first base, second base, third base, and shortstop. There are four possible pitchers, none of whom can play any other position. And there are five players who can play any of the three outfield positions: left field, center field, or right field. In how many ways can the coach assign these players to positions?

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

7. A chef can prepare 10 different entrées.In how many ways can the chef select 6 entréesfor today’s menu?

8. A license plate has 6 digits with repetitions permitted. How many possible license plates of such type are there? 2.7. CHAPTER REVIEW 97

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

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

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

12. The Pi Mu Epsilon honor society consists of 9 men and 7 women. If the society forms a committee with 3 women and 2 men, how many diﬀerent ways can this 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 times with the following results:

Outcome Frequency 1 8 2 8 3 12 4 7 5 15 6 8 7 8 8 13 9 9 10 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? 98 CHAPTER 2. COUNTING AND PROBABILITY

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

(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, and pumpkin. When a quarter is put into the machine, it dispenses 5 gumballs at random. 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 ways can this be done?

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

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

(a) List the sample space. (b) Draw a probability tree diagram to represent the possible scenarios. 2.7. CHAPTER REVIEW 99

22. The probability is 0.6 that a student will study for an exam. If the student studies, she has a 0.8 chance of getting an A on the exam. If she does not study, she has a 0.3 probability of getting an A. Make a probability tree for this 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). 100 CHAPTER 2. COUNTING AND PROBABILITY

Some Review Answers:

1 · 25 · 24 · 10 · 10 · 1 1. 26 · 25 · 24 · 10 · 10 · 10 2. (a) {RR,RW,WR,WW } is one possibility 4 3 C(4, 2) (b) · or 10 9 C(10, 2) 4 (c) 9 3.0 .6 28 4. 52 5. P (6, 5) · P (4, 1) · P (5, 3) 10! 6. P (10, 6) = = 151, 200 4! 10! 7. C(10, 6) = = 210 4!6! 8. 106 = 1, 000, 000

9.4 3 3 2 2 1 1= 4!3! = 144

10. 2!3!5! = 1440 35! 11. C(35, 5) = = 324, 632 30!5! 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 12 + 8 + 9 29 (c) = 100 100 1 + 1 + 1 3 (d) = 10 10 15. (a)1 /52 (b)4 /52 = 1/13 2.7. CHAPTER REVIEW 101

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 1 18. (a) 210 C(10, 1) 10 (b) = = 0.009765 210 210 (c) 0, since it is not possible. 6 heads and 3 tails is only 9 flips, and 8 heads and 4 tails is 12 flips, which is too many. C(15, 5) · C(10, 5) · C(5, 5) 19. = 126, 126 3! C(2, 1)C(26, 5) 253 20. = = 0.05062 C(52, 5) 4998 21. (a) The sample space is {P N, P D, P Q, 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 represents the 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 or less than 12 cents.

22. What is the probability that she gets an A? 0.6 × 0.8 + 0.4 × 0.3 = 0.6. 0.48 If she gets an A, what is the conditional probability that she studied? = 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) 0.2 2 P (A|B) = = = . P (B) 0.3 3 102 CHAPTER 2. COUNTING AND PROBABILITY

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 (A ∩ B) 0.2 1 P (B|A) = = = . P (A) 0.4 2 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 multiple edges.

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 vertex by following edges in the graph. If a graph is disconnected, the connected pieces of the graph are called the components of the graph.

Deﬁnition: The degree of a vertex is the number of edges at that vertex. A loop at 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

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

If some of the lists are impossible, can you explain why? 106 CHAPTER 3. GRAPH THEORY

Deﬁnition: If n is a positive integer, the complete graph on n vertices is denoted 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 of a 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 degree 2. (c) Draw a graph with five vertices where four of the vertices have degree 1 and 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. 108 CHAPTER 3. GRAPH THEORY

5. Suppose our city has 51 city council members, and that these members serve on 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 members or a graph where the vertices represent committees? (b) If we wanted to know which committees have members in common, would it be better to draw a graph where the vertices represent members or a graph where the vertices represent committees?

6. (a) Draw K6 and K7. (b) Find a formula, which depends on the positive integer n, for the number of edges in Kn. 7. For each of the following sets of conditions, explain why no graph satisﬁes those conditions:

(a) A graph with exactly ﬁve vertices each of degree 3 (b) A graph with exactly four edges, exactly four vertices, and the vertices have degrees of 1, 2, 3, and 4 (c) A simple graph with exactly four vertices, where the vertices have degrees of 1, 2, 3, and 4 (d) A simple graph with exactly six vertices, where the vertices have degrees of 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 edges in a graph. The length of a path is the number of edges used in the path. Vertices can 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. 110 CHAPTER 3. GRAPH THEORY

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

A B

C D

(a) E F

A B

C D

(b) E F

A B

C D

A B

C D

(d) E F

A B

C D

(e) E F 3.2. PATHS AND CIRCUITS 111

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

Figure 3.1: * K¨onigsberg, from Leonhard Euler, Solutio problematis ad geometriam situs pertinentis, 1736 112 CHAPTER 3. GRAPH THEORY

Main Theorem: 3.2. PATHS AND CIRCUITS 113

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

A B B B A B A B

D C A C A C D C D C

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

(a) (b) (c) (d) 114 CHAPTER 3. GRAPH THEORY

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 Euler circuit if they exist.

(a)

(b)

(d)

(e)

(f)

(g)

(h) 116 CHAPTER 3. GRAPH THEORY

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.

6. Find an Euler path in the following graph. 3.2. PATHS AND CIRCUITS 117

7. In the map below, is it possible to travel a route that crosses each bridge exactly 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 both sides of the streets shown in the ﬁgure below. (The streets are white, and the houses are in the gray regions.) If the letter carrier does not keep crossing a street back and forth to get to houses on both sides of a street, then she will need to walk along a street at least twice, once on each side, to deliver the mail.

(a) Is it possible for the letter carrier to construct a round trip so that she walks on each side of every street exactly once? (b) If the street diagram was diﬀerent, would you arrive at the same conclu- sion? 118 CHAPTER 3. GRAPH THEORY

3.3 Subgraphs and Trees

Deﬁnition: A subgraph of a graph G is a graph contained in G. A subgraph of G is 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 the smallest possible total weight. 120 CHAPTER 3. GRAPH THEORY

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 connected graph with n vertices and n − 1 edges is a tree. 3.3. SUBGRAPHS AND TREES 121

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

Greedy Algorithms: 122 CHAPTER 3. GRAPH THEORY

Homework

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

(a)

(b)

(c)

(d)

2. Draw all of the diﬀerent spanning trees of the following graph. 3.3. SUBGRAPHS AND TREES 123

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. 124 CHAPTER 3. GRAPH THEORY

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

7. Below is a mileage chart between some cities in Minnesota. Draw a weighted graph that reﬂects the mileages and ﬁnd a minimal spanning tree that connects all of these cities in your graph. Bemidji Frazee International Falls Moorhead Two Harbors White Bear Lake 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. If the 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 the weights are different, then different spanning trees of G have different total weights. (b) If G is a connected simple graph with weighted edges, and e is an edge of G with a smaller weight than any other edge of G, then e must be included in every minimal spanning tree of G. 3.3. SUBGRAPHS AND TREES 125

9. Suppose we have a large collection of 1-cent, 8-cent, and 10-cent stamps available. We want to select the minimum number of stamps needed to make a given amount of postage. Consider a greedy algorithm that selects stamps by ﬁrst choosing as many of the 10-cent stamps as possible, then as many of the 8-cent stamps as possible, then as many 1-cent stamps as possible. Does this greedy algorithm always produce the fewest number of stamps needed for each possible amount of postage? Why or why not? 126 CHAPTER 3. GRAPH THEORY

3.4 Graph Colorings

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

Example:

Deﬁnition: For a graph G, the smallest number of colors needed to color G is called the chromatic number of G and is denoted χ(G). Note that χ is the Greek letter 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), or blue (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 128 CHAPTER 3. GRAPH THEORY

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

Pi χ(Pi)

P1 =

P2 =

P3 =

P4 =

P5 =

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

(b) Draw a tree with 11 vertices, and ﬁnd its chromatic number. 3.4. GRAPH COLORINGS 129

4. Corncob College elects 10 students to serve as oﬃcers on 8 committees. The list 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 committees cannot meet at the same time if they have any members in common. How many diﬀerent meeting times will they need? 130 CHAPTER 3. GRAPH THEORY

An example where Greedy Coloring fails : 3.4. GRAPH COLORINGS 131

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 132 CHAPTER 3. GRAPH THEORY

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

3. The mathematics department at Cornucopia College will oﬀer seven courses next 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, Q Bob: N, R, S Emma: O, M Hodor: R, O Kate: P, S Chuck: R, M Fonz: N, R Isabel: N, Q Lara: P, Q

The department needs to schedule class times for each of these courses, but two courses 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) with no edges crossed. If a graph is drawn in the plane with no edges crossed, we say that the drawing is a plane graph.

Examples:

1. Which complete graphs are planar graphs? 134 CHAPTER 3. GRAPH THEORY

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

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

2. For each of the following graphs, count the number of vertices, the number of edges, and the number of faces. Do you see a relationship between these three numbers? 3.5. PLANAR GRAPHS 135

Euler’s Formula:

Explanation:

Fact: If G is a connected simple plane graph with at least three vertices, then 3V − 6 ≥ E. 136 CHAPTER 3. GRAPH THEORY

Homework

1. For the following plane graph, count the number of vertices, the number of edges, 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 plane so that no edges cross. Otherwise, state that the graph is not planar. (a) (b) A C B

B E D A

E F C D (c) (d) A B A C B F E G J

D F H I E

C D

3. Let G be a connected planar graph. Explain why, no matter how we draw G as 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 degrees of 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? 3.5. PLANAR GRAPHS 137

5. Suppose you have a graph G that is not planar. Is there a way to make it planar 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 not planar by adding more vertices and edges? If so, describe your method; if not, why not? 138 CHAPTER 3. GRAPH THEORY

3.6 Possible Project: Directed Graphs

Deﬁnition: A directed graph is a graph where every edge in the graph is assigned a 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. POSSIBLE PROJECT: DIRECTED GRAPHS 139

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

Task Time Required Start: Decide to build house No time A: Clear land 1 day B: Build foundation 3 days C: Build frame and roof 15 days D: Electrical work 9 days E: Plumbing work 5 days F: Complete exterior work 12 days G: Complete interior work 10 days H: Landscaping 6 days Finish: Move in! No time

The activity directed graph is given below. We draw an arrow from task X to task Y if and only if task Y must be completed directly after completing task X. We label 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 a critical path. The length of the critical path is the length of time needed to complete the overall task.

1. Find the critical path and give its length (in days). 140 CHAPTER 3. GRAPH THEORY

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

Deﬁnition: Two tournaments are isomorphic if they can be redrawn to look like each other. 3.6. POSSIBLE PROJECT: DIRECTED GRAPHS 141

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

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

Fact 1: In any tournament, a longest path that does not repeat any vertices will 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. 142 CHAPTER 3. GRAPH THEORY

2. In each of the tournaments below, ﬁnd a longest path that doesn’t repeat vertices, and ﬁnd a king chicken. 3.6. POSSIBLE PROJECT: DIRECTED GRAPHS 143

Homework

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

Task Time Required Start: Obtain script No time A: Acquire funding 4 weeks B: Sign director 2 weeks C: Hire actors 4 weeks D: Choose filming locations 2 weeks E: Build sets 3 weeks F: Film scenes 10 weeks G: Edit film 5 weeks H: Create soundtrack 3 weeks Finish: Release film No time

The activity directed graph is given below. Find the critical path and give its length (in weeks). 144 CHAPTER 3. GRAPH THEORY

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

Task Time Required Start: Finish playing charades No time A: Put tablecloth on table B: Fold napkins and place on table C: Place dishes and silverware on table D: Put water and ice in water glasses E: Pour wine into wine glasses F: Put food on table Finish: Begin eating No time

Now draw a possible activity directed graph for these tasks, and ﬁnd the minimum amount of time needed to complete the project by ﬁnding a critical path in your graph. Assume that only Dwight has volunteered to help Michael and 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 exactly four vertices.

4. If an undirected graph has E edges and we add all of the vertex degrees in the graph, 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 and all five teams tie for first place. (b) Explain why, if six teams play in a round robin tournament, it is impossible for all six teams to tie for first place. 3.6. POSSIBLE PROJECT: DIRECTED GRAPHS 145

7. In each of the following tournaments:

(a) Find a longest path that doesn’t repeat vertices (one that ranks all of the teams) (b) Find a king chicken 146 CHAPTER 3. GRAPH THEORY

3.7 Chapter Projects

1. Present the idea of a hamiltonian circuit in a graph (a circuit that uses every vertex other than the starting and ending vertex exactly once). Give examples of graphs that do and graphs that do not have hamiltonian circuits. Discuss some of the history of hamiltonian circuits, including some information about Hamilton himself, and his game “Around the World.” Describe the connection to the famous Traveling Salesperson Problem, its history and applications, and give some examples; how difficult of a problem is the Traveling Salesperson Problem? Possible topics: What did Hamilton contribute to mathematics and other fields? Definition of Hamiltonian circuit, examples of graphs that have hamiltonian circuits and those that do not. Be sure to clarify the difference between a hamiltonian circuit and an Euler circuit. Have the class work through some examples.

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

3. Present the idea of coloring maps. The famous Four Color Theorem was proved in 1976 — the proof is extremely complicated and has never been checked by hand, only by computers. So you don’t need to prove the theorem! At least present what the theorem says, and discuss maps that can be colored with fewer than four colors. Use plenty of examples. If you want, present some of the historical background involving map colorings, including the fact that there were incorrect proofs that were believed to be correct for years. You may discuss whether or not a proof that no person has ever checked is actually valid. Possible topics: Explain what we mean by coloring maps, give some examples. How is graph coloring related to map coloring? Be careful about the rules for coloring (for examples, countries that only meet at a corner can get the same color). Possibly involve the class in examples. Give some of the history of the problem. Discuss diﬀerent proof attempts and the validity of proofs done by computer. 3.8. CHAPTER REVIEW 147

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 148 CHAPTER 3. GRAPH THEORY

Review Exercises:

1. Find an Euler path in the following graph.

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. 3.8. CHAPTER REVIEW 149

5. (a) For each of the following graphs, determine whether the graph is planar or 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, where the vertices have degrees of 1, 1, 1, 3, 5, and 5. 7. For each of the following graphs, answer the following questions or do the requested 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 the graph. If not, explain why not. (iii) What is the chromatic number χ of the graph?

A B C

A B C H D

(a) D E F (b) G F E

A B A B

E F E

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.

200

75 300

150 210

50 100

125

1.3 1.6

3.6 2.3 1.0 1.8

3.5 2.1 2.8 1.1

1.5 2.5 3.8. CHAPTER REVIEW 151

10. Are the following graphs planar? If so, draw a plane graph version of the graph and 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 verify Euler’s formula for it. E

F D

A C

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 faces are there?

13. For each of the following sets of conditions, either draw a graph that satisﬁes those 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 degrees of 2, 2, 3, and 3 152 CHAPTER 3. GRAPH THEORY

14. Draw an example of a tree that has exactly seven vertices, where exactly three of the vertices have a degree of 2. 3.8. CHAPTER REVIEW 153

Some Review Answers:

1. One example is CDHGKJIEABCGBF GJF E. Any example must use all 17 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 AF EBCD (c) A, D, and F

5. (a) The ﬁrst is planar, the second is not (M, H, I, J, L forms a K5 in the second graph) (b) 4, 5

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

7. (a) (i) Yes, two odd vertices B and F : BCF BEADEF (ii) No, not all of the vertices are even (iii) χ(a) = 3 (b) (i) Yes, all vertices are even: ABCDEF GHBDF HA; 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 : EADEBAF ECDF CBF (ii) No, not all of the vertices are even (iii) χ(d) = 4 154 CHAPTER 3. GRAPH THEORY

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 above the AC edge. χ = 4. V = 6, E = 11, F = 7: V − E + F = 6 − 11 + 7 = 2, so Euler’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 contain an odd number of odd vertices, since every edge counts toward the degree of two vertices). (b) Possible (c) Recall that a simple graph has no loops or multiple edges. Here it is possible.

14. Start with a degree 2 vertex at the top, followed by a degree 3 vertex on the left 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 is called the principal or present value. You then could charge the person an additional amount, called interest, and the total of the two amounts, which is what you collect from 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 percentage of P , called the interest rate, and multiple this amount by the length of time of the 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 interest and you want $5100 in the CD after 3 years, how much should you deposit today? 158 CHAPTER 4. CONSUMER MATHEMATICS

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 we deposit $2000 today, how much time will it take until we have $2680 in the account? 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 2018, 6.2% of taxable earnings (up to $110,100) is deducted from workers’ paychecks for Social Security. If a worker earns $53,000 in 2018, how much money will the worker pay for the Social Security tax?

4. Given the principal P , the annual simple interest rate r, and the time t, ﬁnd the 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 are given 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. You promise to repay the loan in three years at 5% simple interest. How much will you 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 largest amount of money he could borrow now, if he plans to use his inheritance to repay it in ﬁve months? 160 CHAPTER 4. CONSUMER MATHEMATICS

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

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

10. A coat is marked down 10%. During a special doorbuster sale, the customer is 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: 162 CHAPTER 4. CONSUMER MATHEMATICS

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 t years, then

r i = n m = nt

So we get

r nt F = P · 1 + n

1. Suppose we deposit $2000 into an account that earns 8% annual interest. If the interest is compounded quarterly, how much will be in the account after 10 years? 4.2. COMPOUND INTEREST 163

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 changing the value of n, which is how many times we compound during the year. To help 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)?

(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 your calculator 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.) 4.2. COMPOUND INTEREST 165

Compounding Continuously Formula:

F = P ert where e ≈ 2.718281828

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

5. If an account earns 6% annual interest, compounded continuously, and we want $10,000 in the account after 5 years, how much do we need to deposit today? 166 CHAPTER 4. CONSUMER MATHEMATICS

Homework

1. Given the principal P , the annual interest rate r, the time t, and the frequency of 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 account earning 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 bank compounds interest quarterly. If the bank needs to receive $2,300 in interest from 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. How much money did Angela originally deposit?

5. How much money must Meredith deposit today in order to have $50,000 in twenty 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, determine which 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 4.2. COMPOUND INTEREST 167

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

(a) If this interest is simple interest, how much will be in your account after 40 years? (b) If this interest is compounded annually, how much will be in your account after 40 years? (c) What is the difference in the amounts from parts (a) and (b)? Why is there such a significant difference? 168 CHAPTER 4. CONSUMER MATHEMATICS

4.3 Eﬀective Annual Yield

Example:

Deﬁnition: The eﬀective annual yield of a compound interest account at a given named rate, called the nominal rate, is the simple interest rate that gives the same future 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

r n Y = 1 + − 1 n 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 170 CHAPTER 4. CONSUMER MATHEMATICS

Homework

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

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

(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 produces the same effective annual yield as Bank 1 but only compounds interest quarterly. What nominal annual rate does Bank 2 offer?

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

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

• Option A: Deposit $1000 on November 13, 2018 and deposit another $1000 on November 13, 2019. • Option B: Deposit $2000 on November 13, 2018. • Option C: Deposit $1950 on November 13, 2018.

(a) Without doing any calculations, determine whether Option A or Option B will give a higher account balance on November 13, 2020. Explain your answer. (b) Determine whether Option A or Option C will give a higher account balance on November 13, 2020. (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, compounded semiannually (twice each year). • CD 2: This is a 5-year CD that oﬀers 3.5% annual interest, compounded semiannually (twice each year), but charges an initial $50 fee to earn this higher interest rate.

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

(a) If you have $1500 to deposit into one of these CDs, which CD will have the higher balance after 5 years? (b) If you have $3000 to deposit into one of these CDS, which CD will have the 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 deposit that stays in the account for all three years, would you prefer one account to the other? Why or why not? (b) If, instead, the interest earned is compounded annually, and you make a deposit that stays in the account for all three years, would you prefer one account to the other? Why or why not? (c) If the account earns either simple interest or interest compounded annually 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 offers 5.1% annual interest, compounded daily. Account 3 offers 5.0% annual interest, compounded continuously.

Find the eﬀective annual yield for each of the three accounts, and use these yields to determine which account oﬀers the best return on deposits. 172 CHAPTER 4. CONSUMER MATHEMATICS

9. The United States paid about 4 cents an acre for the Louisiana Purchase in 1803.

(a) Suppose the value of this property grew at an annual rate of 5.5% compounded annually. What would an acre be worth in 2018? (b) What would an acre be worth in 2018 if the annual rate was 6% compounded 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 at equal time periods, where the payments are made at the end of each time period. 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 the ﬁrst 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. 174 CHAPTER 4. CONSUMER MATHEMATICS

Example: Suppose we deposit $50 a month into an ordinary annuity that pays 12% annual interest, compounded monthly, for a total of one year. What is the future value of the annuity? (In other words, how much is in the annuity after one year?) 4.4. ORDINARY ANNUITIES 175

F = future value PMT = payment made each period i = interest rate per time period m = number of time periods

(1 + i)m − 1 F = PMT · i

2. Suppose we pay $250 a quarter into an ordinary annuity for 7 years, where the 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? 176 CHAPTER 4. CONSUMER MATHEMATICS

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, where 8% 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 and pay $100 a month, with the last payment to occur on her 18th birthday. The payments would earn 6% annual interest, compounded monthly. The aunt said they should just give the niece a lump sum of money now that would grow to the same amount (at 6% annual interest, compounded monthly) as the annuity would by the 18th birthday. If they go with the aunt’s plan, how much should they give to the niece now?

4. Mr. Smith decides to pay $300 at the end of each month into an ordinary annuity 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 ﬁve years, but he makes a mistake in his calculations. What was his mistake? Is his answer too big or too small?

(1 + i)m − 1 F = PMT · i (1 + .08)60 − 1 F = 300 · .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’s adjusted monthly income. The adjusted monthly income is the gross monthly income minus any unchanging monthly payments that still have more than 10 months remaining.

The maximum that a person can spend on housing expenses each month is 28% of their adjusted monthly income.

Example: 178 CHAPTER 4. CONSUMER MATHEMATICS

Example:

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-year 5.0 10.61 6.60 5.37 5.5 10.85 6.88 5.68 6.0 11.10 7.16 6.00 6.5 11.35 7.46 6.32 7.0 11.61 7.75 6.65 7.5 11.87 8.06 6.99 8.0 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, is the sum of the down payment plus any points the lender charges. A point is 1% of the amount borrowed.

Followup: 180 CHAPTER 4. CONSUMER MATHEMATICS

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 a month remaining on her student loan and 20 payments of $310 a month on her 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 house at a price of $189,900. If she makes a 25% down payment, then she can ﬁnd a mortgage with an interest rate of 5%. Use Table 4.1 to calculate her monthly principal and interest payment for the remaining 75% of the house price. (c) If insurance and taxes sum to $240 each month, calculate Penny’s total monthly 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 bank requires a 20% down payment and a payment of 3 points on the mortgage amount 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 branch of Dunder Miﬄin, has a gross monthly income of $3,150. He has 11 remaining monthly payments of $200 for supplies for his next beet crop and 48 remaining monthly payments of $5 on a loan for his impressive collection of mustard- colored dress shirts.

Dwight is (heretically!) considering abandoning Mose at Schrute Farms and buying a house that is selling for $129,500. The insurance and taxes on the property are $125 and $145 per month, respectively. Dwight’s bank requires a 20% down payment, payable to the seller, and a payment of 2 points, payable to the bank, at closing. The bank will approve a loan with a total monthly mortgage payment of principal, interest, property taxes and insurance that is less 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 ﬁnding the payment for principal and interest using Table 4.1, and then adding the amounts for insurance and taxes. (f) Does Dwight qualify for the mortgage? 182 CHAPTER 4. CONSUMER MATHEMATICS

4.6 Chapter Projects

1. Discuss how to calculate payments for loans (other than payments for home mortgages). In class, we found a formula for the future value of ordinary annuities. Explain how to ﬁnd the formula for the present value of ordinary annuities. Apply this formula to a variety of situations, which may include lottery 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 Price Index. Use compound interest to calculate the changing costs of items due to inflation. 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 credit cards. Present diﬀerent methods credit card companies use to compute interest charges and minimum monthly payments. 4.7. CHAPTER REVIEW 183

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 i = interest rate per time period m = number of time periods

• Simple Interest F = P · (1 + rt) • Compound Interest r nt F = P · 1 + n • Compounding Continuously F = P ert where e ≈ 2.718281828

• Eﬀective Annual Yield r n Y = 1 + − 1 n • Future Value of an Ordinary Annuity (1 + i)m − 1 F = PMT · i

• Adjusted Monthly Income The adjusted monthly income is the gross monthly income minus any unchanging monthly payments that still have more than 10 months remaining. To qualify for a mortgage, the maximum that a person can spend on housing expenses is 28% of their adjusted monthly income. 184 CHAPTER 4. CONSUMER MATHEMATICS

• Closing Costs The closing costs are the amount a buyer pays at closing on the day they buy the house. Closing costs include the down payment and any points charged by 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-year 5.0 10.61 6.60 5.37 5.5 10.85 6.88 5.68 6.0 11.10 7.16 6.00 6.5 11.35 7.46 6.32 7.0 11.61 7.75 6.65 7.5 11.87 8.06 6.99 8.0 12.13 8.36 7.34 4.7. CHAPTER REVIEW 185

Review Exercises:

1. Liz has a gross monthly income of $3,500. She has 17 remaining monthly payments of $220 for a car loan and 3 remaining monthly payments of $95 on her student loans.

Liz is looking at buying a house that is selling for $155,000. The insurance and taxes on the property are $110 and $135 per month, respectively. Liz’s bank requires a 20% down payment. The bank will approve a loan with a total monthly mortgage payment of principal, interest, property taxes and insurance that 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%, determine the total monthly payment for the mortgage by ﬁrst ﬁnding the payment for principal and interest, and then adding the amounts for insurance and taxes. (c) Does Liz qualify for the mortgage?

2. How much money needs to be deposited into a bank account today so that it grows 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 on January 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 each of 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 is the 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, compounded continuously.

Find the eﬀective annual yield for each of the three accounts, and use these yields to determine which account oﬀers the best return on deposits. 186 CHAPTER 4. CONSUMER MATHEMATICS

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 annual rate 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 interest is compounded (a) annually; (b) monthly; (c) continuously.

8. Suppose you deposit $2300 into an account earning simple interest. What simple 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, compounded daily. National Bank advertises a savings account with a 5.25% interest rate, compounded monthly. Which bank would you deposit your money with? 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 now at 6% interest compounded weekly?

Monthly Principal and Interest Payment per $1000 of Mortgage Rate % 10-year 20-year 30-year 5.0 10.61 6.60 5.37 5.5 10.85 6.88 5.68 6.0 11.10 7.16 6.00 6.5 11.35 7.46 6.32 7.0 11.61 7.75 6.65 7.5 11.87 8.06 6.99 8.0 12.13 8.36 7.34

12. The Pella family is considering a house priced at $144,500. The taxes on the house will be $3200 per year and the homeowners’ insurance will be $450 per year. They have applied for a mortgage from their bank. The bank is requiring a 15% down payment, and the interest rate on the loan is 7.5%. Their annual 4.7. CHAPTER REVIEW 187

gross income is $86,500. They have more than 10 monthly payments remaining on each of the following: $220 on a car, $170 on new furniture, and $210 on a student loan. Their bank will approve the mortgage if their monthly housing expenses (principal and interest for mortgage, insurance and taxes) are less than 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 and interest for the mortgage using the table above. (e) What are their monthly housing expenses (principal, interest, insurance and taxes)? (f) Do they qualify for the mortgage? 188 CHAPTER 4. CONSUMER MATHEMATICS

Some Review Answers:

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

2 6. I = P rt =⇒ I = $1290(0.11) 12 = $23.65

.032 1(4) .032 12(4) 7. (a) A = 2500 1 + 1 =$2835.69 (b) A = 2500 1 + 12 =$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 adjusted monthly 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 members are considering running for president: Alice, Bob, and Charles. Each of the five voters 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 Bob 2nd choice: Charles Alice Charles Alice Alice 3rd choice: Bob Charles Alice Bob Charles Depending on who decides to run, the ballot the voters see in the booth will look different. For each of the following ballots, how many votes will each candidate receive? Who will win?

Who should be Who should be president? president? (a) Alice (c) Alice Bob Charles Charles

Who should be Who should be president? president? (b) (d) Alice Bob 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 Ballots Ranking 3 2 4 3 2 1st choice M M D J P 2nd choice P P P P D 3rd choice D J J M J 4th choice J D M D M

(a) Here is the ballot: Who should be the new manager? Micah Darba Jim Pam Who will be elected manager? What percent of the vote will he or she have?

(b) As you look at the voters’ preferences, who seems like the best choice to make the most voters happy?

(d) Can you think of a diﬀerent voting system that would help the voters elect your choice from part (b)? 5.1. VOTING SYSTEMS 191

3. Plurality voting:

4. Borda count: 192 CHAPTER 5. VOTING THEORY

5. If the company above runs their election as a Borda count, who will win?

Number of Ballots Ranking 3 2 4 3 2 1st choice M M D J P 2nd choice P P P P D 3rd choice D J J M J 4th choice J D M D M 5.1. VOTING SYSTEMS 193

6. A company warehouse is voting for its new warehouse manager, and the candidates are Donatello, Leonardo, Raphael, and Michelangelo.

# of Ballots Ranking 3 10 9 8 1st choice D L M R 2nd choice R R D M 3rd choice L D R D 4th choice M M L L

(a) Each member votes for her/his favorite candidate. What percentage of votes did the ﬁrst-place candidate receive?

(b) Rather than elect a candidate who didn’t receive a majority vote, the warehouse holds a run-oﬀ election between the top two candidates. Who will win the run-oﬀ election?

(d) Can you think of a better runoﬀ system for the warehouse than a runoﬀ between just the top two candidates? What result does it give? 194 CHAPTER 5. VOTING THEORY

7. Plurality with elimination: 5.1. VOTING SYSTEMS 195

8. The Army Corps of Engineers is trying to decide on a ﬂood prevention project for 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 500 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 Ballots Ranking 5 4 2 3 1 3 1st choice F M R D C R 2nd choice D D F M D F 3rd choice R R C R F D 4th choice M C D F M M 5th 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 a winner yet? If not, who should be eliminated?

(b) What will the vote tallies be after the second round of voting? Is there a winner yet? If not, who should be eliminated?

(c) What will the vote tallies be after the third round of voting? Is there a winner 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? 196 CHAPTER 5. VOTING THEORY

10. A company is voting on its Accountant-of-the-Month. The candidates are:

• Oberyn • Khaleesi • Arya • Theon

The preference table is the following:

Number of Ballots Ranking 4 2 3 2 3 1st choice A T K O O 2nd choice O O A K A 3rd choice K K T A T 4th 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 there are any ties, put both names in the box. Theon Arya Khaleesi

Oberyn

Khaleesi

Arya

(d) Who won the most head-to-head races? 1 (Count any ties as 2 a win for each contender.) 5.1. VOTING SYSTEMS 197

11. Pairwise Comparison:

12. A tip for setting up the head-to-head races: 198 CHAPTER 5. VOTING THEORY

13. Our Math 105 classes are voting on where to take a math ﬁeld trip. The options 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.

Number of Ballots Ranking 5 4 4 3 2 1 1st choice H W W E M H 2nd choice E H E M H I 3rd choice I I I H E W 4th choice M E H W W M 5th choice W M M I I E

We decide to use the pairwise comparison method to decide. Where will we go? 5.1. VOTING SYSTEMS 199

Homework

Show all your work. In particular, for plurality with elimination, show the results of each round of voting; for pairwise comparison, show the result of each head-to-head vote.

1. Consider the following preference matrix:

Number of Ballots Ranking 3 4 4 9 3 6 1st choice D D A E B A 2nd choice C C B C E B 3rd choice E E E B D C 4th choice B A D A A D 5th 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? Does that 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 win the election?

2. Consider the following preference matrix:

Number of Ballots Ranking 8 6 8 5 1 1st choice B C D B A 2nd choice D A C C B 3rd choice C D A A C 4th 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. 200 CHAPTER 5. VOTING THEORY

3. Consider the following preference matrix:

Number of Ballots Ranking 3 7 4 1 7 8 1st choice B B C A C A 2nd choice C A B C A B 3rd choice A C A B B C

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.

4. Consider the following preference matrix:

# of Ballots Ranking 3 5 4 1st choice C B D 2nd choice E E C 3rd choice A A E 4th choice D C A 5th choice B D B

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. 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 three candidates: Ira, Jannika, and Kelsey. The preference table is the following.

# of Ballots Ranking 4 3 2 1st choice J I K 2nd choice I J I 3rd 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?

(d) If it were a head-to-head race between Jannika and Kelsey, who would win?

(e) Why does the plurality method’s result seem unfair in this election? 202 CHAPTER 5. VOTING THEORY

3. Head-to-Head Criterion:

4. The school board is voting whether to award the contract for building a new elementary school to Aasgaard Architects, Bob’s Builders, Cobber Construc- tion, or Dora’s Developers.

# of Ballots Ranking 5 3 3 1 1st choice B B C A 2nd choice C C A D 3rd choice D A D C 4th 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? 5.2. VOTING PARADOXES AND PROBLEMS 203

5. Majority Criterion:

6. The International Olympic Committee has to select one of four cities for the Winter Olympics: O, T, M, F. As of this moment, the voters’ preferences are as follows:

# of Ballots Ranking 7 5 4 1 1st choice F M T O 2nd choice O F M T 3rd choice T T O F 4th 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? 204 CHAPTER 5. VOTING THEORY

(b) After receiving a very nice bribe, the single voter in the last column decides that F really is the best candidate after all, and so his preferences change to F,O,T,M. Thus when the election is held tomorrow, the preference table is # of Ballots Ranking 7 5 4 1 1st choice F M T ×OF 2nd choice O F M ×TO 3rd choice T T O ×FT 4th choice M O F MM× Which city will be chosen?

7. Monotonicity Criterion: 5.2. VOTING PARADOXES AND PROBLEMS 205

8. Voters have 4 choices: A, B, C, D. Here is the preference table.

# of Ballots Ranking 10 6 5 4 2 1st choice D C C A B 2nd choice A B A D A 3rd choice C A D B C 4th choice B D B C D

(a) Find the winner of the election under the plurality with elimination method.

(b) Find the winner of the election under the pairwise comparison method.

9. Voters are trying to decide what to do with a $50 million surplus in the state budget. 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 Ballots Ranking 70,000 50,000 40,000 1st choice T R S 2nd choice R S T 3rd 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)? 5.2. VOTING PARADOXES AND PROBLEMS 207

10. Irrelevant Alternatives Criterion:

11. Arrow’s Impossibility Theorem: 208 CHAPTER 5. VOTING THEORY

Homework

1. Consider the following preference matrix:

Number of Ballots Ranking 4 10 2 3 5 1st choice C D A C A 2nd choice D C C B B 3rd choice B A B A D 4th choice A B D D C

(a) Does the Borda Count violate the Majority Criterion for this particular preference matrix? (b) Does the Borda Count violate the Head-to-Head Criterion for this particular preference matrix?

2. Consider the following preference matrix:

Number of Ballots Ranking 11 2 5 1 8 4 1st choice C D C D A B 2nd choice A A A A D D 3rd choice D C B B C C 4th choice B B D C B A

(a) Does the Borda Count violate the Majority Criterion for this particular preference matrix? (b) Does the Borda Count violate the Head-to-Head Criterion for this particular preference matrix?

3. Consider the following preference matrix:

# of Ballots Ranking 9 6 8 5 1st choice B B D D 2nd choice D D B B 3rd choice C A A C 4th choice A C C A

(a) How many points will each candidate receive in a Borda count? Who will win? 5.2. VOTING PARADOXES AND PROBLEMS 209

(b) The ﬁve voters in the last column really do think D is the best candidate and B is the second-best. However, they decide to be sneaky and lie on their 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. Now how many points will each candidate receive in a Borda count? Who will win? (c) Explain in a few complete sentences how these voters manipulated the Borda count and why it is unfair.

4. Use a Borda count to determine the preference of the voters.

# of Ballots Ranking 3 1 1 1st choice A C E 2nd choice B D C 3rd choice C E D 4th choice D B B 5th choice E A A

Suppose the last two voters change their minds with respect to B and D:

# of Ballots Ranking 3 1 1 1st choice A C E 2nd choice B B C 3rd choice C E B 4th choice D D D 5th 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.

# of Ballots Ranking 55 35 10 1st choice A B C 2nd choice B C B 3rd choice C A A 210 CHAPTER 5. VOTING THEORY

6. Use plurality with elimination to determine the winner of the following election.

# of Ballots Ranking 2 2 1 1st choice A C B 2nd choice B B A 3rd choice C A C

Now suppose voters in the middle column change their minds about C:

# of Ballots Ranking 2 2 1 1st choice A B B 2nd choice B C A 3rd 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 owns 2 shares. If 1 share = 1 vote represent this weighted voting system.

(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: 212 CHAPTER 5. VOTING THEORY

2. In a weighted voting system with weights [30, 29, 16, 8, 3, 1], if a two-thirds majority 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?

(d) If the ﬁrst 4 voters vote for a motion and the rest vote against, does the motion pass?

(e) If P1 and P2 vote against a motion, will the motion pass? 5.3. WEIGHTED VOTING SYSTEMS 213

5. What is peculiar about each of the following weighted voting systems?

(a) [20|10, 10, 9]

(b) [7|4, 2, 1]

(d) [6|6, 2, 1, 1]

(e) [21|10, 8, 5, 3, 2] 214 CHAPTER 5. VOTING THEORY

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 dictators?

11. In designing a weighted voting system [q|6, 5, 4, 3, 2, 1], what is the largest quota 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, what is the quota q? 5.3. WEIGHTED VOTING SYSTEMS 215

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, and 412 shares are in Debra’s hands. As usual, each share corresponds to a vote in the stockholder’s meeting.

(a) If a certain type of motion requires a majority vote, what is the smallest number of votes needed to pass the motion? (b) A diﬀerent type of motion requires a 2/3 vote to pass. What is the smallest number of votes needed to pass this motion? (c) Using the quota you found in part (b), express the weighted voting system in the correct notation (with brackets and quota).

2. Find all dictators, dummies, and voters with veto power in the following weighted 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. The members 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 votes of 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 integers w 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? 216 CHAPTER 5. VOTING THEORY

5.4 Banzhaf Power Index

1.A coalition is . . .

2. (a) Consider a weighted voting system with three voters P1, P2, and P3. List all 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?

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” without 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 . . . 218 CHAPTER 5. VOTING THEORY

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 called that voter’s Banzhaf power. Voter Banzhaf power

P4 (d) Add up all the voters’ Banzhaf powers; this sum is called the total Banzhaf 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

P4 5.4. BANZHAF POWER INDEX 219

8. Calculate the Banzhaf Power Index for each voter in the weighted voting system [51|32, 22, 12].

9. Make up a weighted voting system with a dummy, and calculate the Banzhaf Power Index for the dummy.

10. Make up a weighted voting system with a dictator, and calculate the Banzhaf Power 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 do you notice? 220 CHAPTER 5. VOTING THEORY

12. In a weighted voting system with 6 voters, how many possible coalitions are there?

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].

15. Determine the Banzhaf power index of each voter for [12 5, 5, 2, 2].

16. Determine the Banzhaf power index of each voter for [10 6, 5, 4, 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 coalitions.

STEP 3: Calculate each voter’s Banzhaf power by adding up the number of times each voter is critical.

STEP 4: Calculate the total Banzhaf power for the weighted voting system by adding all voters’ Banzhaf powers.

STEP 5: Determine each voter’s Banzhaf power index by dividing his/her Banzhaf power by the total Banzhaf power. 5.4. BANZHAF POWER INDEX 221

Homework

1. Calculate the Banzhaf Power Index for each voter in the weighted voting system [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 sentences why the weight controlled by the voter is not the same thing as the power held by each voter.

3. Calculate the Banzhaf Power Index for each voter in the weighted voting system [27|15, 7, 5].

4. Calculate the Banzhaf Power Index for each voter in the weighted voting system [26|15, 13, 7].

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 county issues. The delegates’ votes were weighted proportionately to the districts’ population in 1964:

District Weight Hempstead #1 31 Hempstead #2 31 Oyster Bay 28 North Hempstead 21 Long Beach 2 Glen 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 dummies?

In 1965 John F. Banzhaf III argued in court that even though the weights were proportionate to population, this system of government was unfair. He won! 222 CHAPTER 5. VOTING THEORY

5.5 Voting Theory Homework Set

The following problems are to be worked on in class, with the answers written up on 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-elimination method except that in each round we eliminate the candidate with the largest number of last-place votes (instead of the one with the fewest ﬁrst-place votes).

# of Ballots Ranking 10 6 6 3 3 1st choice W Z Z Y X 2nd choice X W X Z Z 3rd choice Y Y W X W 4th 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 the candidate with the largest number of last-place votes.

# of Ballots Ranking 10 8 7 4 1st choice B C C A 2nd choice A A B B 3rd 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, they put B in 2nd and A in 3rd place. Use the Coombs method to determine if 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 voting system.

in the weighted voting system [8 4, 4, 1, 1, 1]. Find the Banzhaf power distribution of this system.

(d) Compare the power index of P1 in (b) and (c). Describe in your own words 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 − 2 tie; in that case, the coalition that contains P4–the partner with the least seniority–loses. What is the Banzhaf power distribution in this partnership?

5. A panel of sportswriters and broadcasters every year selects an NBA rookie of the year using a variation of the Borda count method. The following table shows the results of the balloting for the 2003 − 2004 season.

Player 1st Place 2nd Place 3rd Place Total Pts LeBron James 78 39 1 508 Carmelo Anthony 40 76 2 430 Dwayne Wade 0 3 108 117

Using the information from the table, ﬁgure out how many points are awarded for each 1st-place, 2nd-place, and 3rd-place vote in this election. Assume that these numbers are diﬀerent positive integers.

6. An NBA team has a head coach H and 3 assistant coaches C1, C2, C3. Player personnel decisions require at least 3 of the coaches to vote yes, one of which

must be H to pass. If we use [q h, c, c, c] to describe this weighted voting system, ﬁnd q, h, c. 224 CHAPTER 5. VOTING THEORY

5.6 Possible Project: Antagonists

Suppose that in a weighted voting system there is a voter A who hates another voter P 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 winning or 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 218 #7: [19 11, 9, 8, 5].

(a) If P4 becomes the antagonist of P1, what happens to the Banzhaf power index of P4? P1? (Compare with the original Banzhaf power index of each voter.)

(b) If P3 becomes the antagonist of P1, calculate the new Banzhaf power index for each of the four voters.

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 power index of P1? P2? P3? (Compare with the original Banzhaf power index of each voter.)

(c) If P2 becomes the antagonist of P1, what happens to the Banzhaf power index of P1? P2? P3? (Compare with the original Banzhaf power index of each voter.)

(d) If P3 becomes the antagonist of P2, what happens to the Banzhaf power index of P1? P2? P3? (Compare with the original Banzhaf power index of each voter.) 5.6. POSSIBLE PROJECT: ANTAGONISTS 225

4. (a) Citing examples above (or coming up with new ones), show that it is possible for voter A to increase her/his Banzhaf power index by becoming an antagonist of another player. (b) Citing examples above (or coming up with new ones), show that it is possible for voter A to decrease her/his Banzhaf power index by becoming an antagonist of another player.

5. If there are only two voters, A and P , how many coalitions (not necessarily winning) 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 is the antagonist of P ?

If there are four voters, A, P , P3, and P4, how many coalitions are there if A is 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 are there if A is the antagonist of P ? 226 CHAPTER 5. VOTING THEORY

5.7 Chapter Projects

1. Present the Shapley-Shubik power index. (Who are Shapley and Shubik?) Compare and contrast this index to the Banzhaf power index that we discussed in Section 5.4 (you might compute the Banzhaf power index for your small example so that people can see how they’re different/similar). Give some examples, including real examples, such as the Electoral College. There are many good introductions to this index online, along with plenty of small examples. 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 this voting 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. Three voters is too simplistic, and five can get messy. Note: people have published online calculators that can compute the Shapley-Shubik power indices for you. This might be easier for larger examples. Possible topics: What motivated Shapley and Shubik to define this power index? Find the Shapley-Shubik power index of all the states in the Electoral College; interpret the results. Can you find examples where the Shapley-Shubik index tells us which voters are marginalized? Which voters have a lot of power?

2. Introduce the Bucklin voting system, named after James Bucklin of Grand Junction, Colorado. Explain how the Bucklin system works, give examples of simple elections using this system. What are the ﬂaws of this method? What fairness criteria, if any, might this system violate? 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 this voting system. The context in which you present this example is up to you. You may want to weave it into one or all of the following: Does anyone use this voting system today? Did anyone 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 have discussed 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, how the votes are counted, what quotas are set, and how winners are determined. Give examples of simple elections using this system (for hints, see wikipedia’s example on single transferable voting, for starters). Who uses this method today? Mention governments and the Academy Awards. You will need to give and explain at least one small, but meaningful example (think something 5.7. CHAPTER PROJECTS 227

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 want to weave it into one or all of the following: Discuss which governments use STV, and why. You may choose to focus on Minneapolis, MN, which currently uses STV for some city matters. Which matters are decided by STV? What prompted the switch to STV? Have there been complaints about this new- fangled voting system? Is it possible for someone to game their voting and cheat the system? If so, give an example. Other topics or neat things you ﬁnd. Be creative!

4. Background and applications of the Banzhaf power index. John Banhaf was a lawyer in Nassau County (NY) who studied the Nassau County Board and found that their weighted voting system essentially consigned the smaller districts in the county to the role of dummies! Explain their weighted voting system in the 1960s vs. the 1990s and explain the weaknesses of the 1960s version. Include any interesting information on the subsequent legal ﬁght to get the voting system changed. See homework exercise 5 in Section 5.4. 228 CHAPTER 5. VOTING THEORY

5.8 Chapter Review

To prepare for the exam over this chapter, you should review the in-class worksheets and 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 a preference 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 voters with veto power.

6. Construct weighted voting systems that satisfy given conditions (e.g., “Come up 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. 5.8. CHAPTER REVIEW 229

Review Exercises:

1. Four candidates (A, B, C, and D) run in an election with the following results:

# of Ballots Ranking 4 9 4 19 1st choice B C A D 2nd choice C A C C 3rd choice A D B B 4th 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:

# of Ballots Ranking 4 2 4 3 1st choice A B B C 2nd choice B C A A 3rd choice C A C B

(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? 230 CHAPTER 5. VOTING THEORY

3. Three candidates (A, B, and C) run in an election with the following results:

# of Ballots Ranking 6 5 2 1st choice A B C 2nd choice B C B 3rd choice C A 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) Suppose candidate C drops out of the election. Determine the winner of the election using the i. plurality method ii. Borda count method iii. plurality-with-elimination method iv. pairwise comparison 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.

# of Ballots Ranking 14 10 8 4 1 1st choice D A C B A 2nd choice B B A C C 3rd choice A C B A B 4th choice C D D D D

(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 any of the fairness criteria? 5.8. CHAPTER REVIEW 231

5. Determine the quota if a simple majority is needed to pass a motion. Use proper 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. Use proper 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 distribution 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 weighted voting systems.

(a) [22 11, 11, 10]

(b) [10 5, 3, 2]

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 has four times the power of the others. Find w so that the quota 49 is

(a) a simple majority of the votes; 2 (b) more than a 3 -majority of the votes; 3 (c) more than a 4 -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? 232 CHAPTER 5. VOTING THEORY

(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.

(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? 5.8. CHAPTER REVIEW 233

Some Review Answers:

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-head criterion.

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]

6. (a) [14 7, 4, 3, 3, 2, 1]

(b) [21 10, 6, 5, 4, 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 set the 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. 234 CHAPTER 5. VOTING THEORY

(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].

weights, and P1 has veto power; the system is equivalent to [3 2, 1, 1].

1 11. (a) For a simple majority, the quota is 50% plus 1 vote: 49 = 2 (8w) + 1, so w = 12. 2 (b) For a more than 3 -majority, the quota is more than two-thirds the total 2 vote: 49 ≥ 3 (8w), so w = 9. 3 (c) For a more than 4 -majority, the quota is more than three-fourths the 3 total vote: 49 ≥ 4 (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.

(d) If q = 6, 7, 8, then both P2 and P3 are dummies. If q = 10, 11, 12, then P3 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 no motion passes since they both have veto power. Appendix A

Projects

• Team Presentation: In teams of two, students will present a topic from a list of possible subjects that correspond to chapters covered during the course. We will finish each chapter with two or three presentations during one class period. The talks will be 15-20 minutes in length. Ideas for the talks are found in the workbook at the end of each chapter. You may select your teammate and presentation topic on a first-come, first-served basis. Absolutely feel free to see me for help in preparing your paper and presentation.

• Follow-up paper: Each individual team member will also submit an individual project paper after their presentation. Your paper should include 1-2 pages summarizing the content of the team presentation, and a short section (roughly half of a page) describing what you yourself learned in completing your project. Papers will be due 9 days after your presentation – if your presentation is on a Monday, then it is due the following week on Wednesday; presentation on Wednesday =⇒ due the following week on Friday; presentation on Friday =⇒ due the Monday after next. Groups in Chapter 5 will have to turn in their paper by the end of ﬁnals week.

• Audience: The students not speaking during a presentation day will evaluate the talks that are presented on that day, and submit these evaluations at the end of the class period as the homework assignment for that day. Speakers earn a perfect homework score for their presentation day. There will be no other 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 graded on mathematical content, organization, appropriate length of the talk and paper, 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 be checked 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 learned in 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 17052, Fall 2018 MWF 11:50-1:00, ISC 101

Activity Points Date Homework 175 Every class Test 1 85 September 21 Test 2 85 October 10 Test 3 85 November 5 Test 4 85 November 28 Test 5 85 December 18, Tuesday 11 am-1 pm Project 50 @ end of your chosen unit

Professor: Dr. Douglas Anderson, ISC 364, 299-4453, [email protected] Oﬃce Hours: Feel free to stop by my oﬃce, ISC 364, Tuesday and Thursday 9:00- 12:30 and 2:30-4:30, or MWF by discovery. If these times are inconvenient, please set up an appointment. Free Tutoring: The Mathematics Department together with the Center for Student Success supports a tutor Sunday, Tuesday, and Thursday nights in ISC 132; Sundays 7-10 pm, Tuesdays & Thursdays 8-10 pm. Text: Math 105 Workbook: Exploring Mathematics, Fall 2018 Yellow Edi- tion, available from the Cobber Bookstore. Academic Integrity: All homework, tests, and other graded assignments will be subject to Concordia’s policy on Academic Integrity. In particular, you may not obtain assistance from any source on exams.

237 238 APPENDIX B. SYLLABUS

Homework: There will be daily graded homework assignments from the workbook to complete on separate paper prior to the next class. I encourage you to work with other students to complete the homework, but the assignment you submit must reflect your own work. Homework is due at the beginning of class. Late homework will not be accepted unless you have missed class due to an excused absence and you have made arrangements with me in advance. Your 3 (three) lowest homework scores will not be included in your final homework grade, which will be scaled out of 175 points. Again, all students must abide by the college’s expectations regarding academic quality, integrity, and honesty. Exams: Attendance is required for all exams. If you should miss an exam for an emergency you will be allowed to make it up only if you have notified me before the exam and it must be made up in a timely manner (to be discussed with me individually). The dates for exams are given on this syllabus; I will give you at least one week notice if the exam date is to be changed. There will be five 70-minute unit exams, each worth 85 points. I do not scale the 70-minute exams. Partial credit may be given for incorrect answers but correct reasoning; partial discredit may be given for correct answers but incorrect reasoning. To study for the exams, please look over examples from class and the homework, and work the review problems from the book. Project: (Presentation and Paper) Each of you will write and present a 15-20 minute in-class presentation (as a part of a team of two) following one of the units we cover in class. The topics for these presentations will be extensions and applications of the course material. Ideas for the projects are in the workbook at the end of each unit. More details are included in the Appendix of the workbook. Calculator/Calculator App: Calculators or phones with calculator apps are allowed and will be used occasionally for homework and exams. Any scientific calculator would be appropriate for this class; cell phones may also be used as calculators in class and on exams. Please see me if you need help selecting a calculator, or if you 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%

Core Criteria and Outcomes: As a Core Exploration Course in Mathematics, Math 105 Exploring Mathematics meets the following criteria relative to Concordia’s 239

Goals for Liberal Learning (GLL) in our core curriculum, Becoming Responsibly Engaged in the World:

• Emphasize arithmetical, algebraic, geometric, algorithmic, or statistical methods (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 and models.

• Students will represent mathematical information symbolically, visually, and numerically.

• Students will apply mathematical methods and models to solve multi-step problems.

This course uses real-world problems and situations to improve your problem-solving skills, 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 chosen from taxicab geometry, counting and probability, graph theory, consumer mathematics, and voting theory. 240 APPENDIX B. SYLLABUS

Date Section Homework from Workbook on Separate Paper

Aug 31 1.1 Taxicab Geometry (8) 1-3, please use graph paper

Sep 3 1.2 Taxicab Circles (13) 1-5, on graph paper 5 1.3 Taxicab Applications (21) 1-3, on graph paper 7 1.4 Minimizing Regions (25) 1-3, on graph paper

Sep 10 1.5 Midsets (31) 1-6, on graph paper 12 1.6 Lines (38) 1-8, on graph paper 14 Presentations

Sep 17 Review (42-50) 1-14 19 Fall Symposium Power Plays (Memorial Auditorium) 21 Test 1

Sep 24 2.1 Permutations (62) 1-7 26 2.2 Combinations (69) 1 28 2.3 Probability Basics (77) 1-8

Oct 1 2.4 Probability of Events (84) 1-10 3 2.5 Conditional Probability (91) 2-6,9,10 5 Presentations

Oct 8 Review (94-99) 1-25 10 Test 2 12 3.1 Intro to Graphs (107) 1-7

Oct 15 3.2 Paths and Circuits (114) 1-8 17 3.3 Subgraphs & Trees (122) 1-9 19 3.4 Graph Colorings (131) 1-3

Oct 29 3.5 Planar Graphs (136) 1-6 31 Presentations Nov 2 Review (147-152) 1-14

Nov 5 Test 3 7 4.1 Interest (159) 1-10 9 4.2 Compound Interest (166) 1-7 241

Date Section Homework from Workbook on Separate Paper Nov 12 4.3 Eﬀective Annual Yield (170) 1-9 14 4.4 Annuities (176) 1-5 16 4.5 Mortgages (180) 1-4

Nov 19 Presentations 21 Thanksgiving Break No Class 23 Thanksgiving Break No Class

Nov 26 Review (183-187) 1-12 28 Test 4 30 5.1 Voting Systems (199) 1-4

Dec 3 5.2 Flaws in Voting Systems (208) 1-6 5 5.3 Weighted Voting Systems (215) 1-6 7 5.4 Banzhaf Power Index (221) 1-4

Dec 10 5.5 Voting Theory Homework Set (222) 1-6 12 Presentations 14 Review (228-232) 1-13

Dec 18 Test 5 Tuesday, 11 am -1 pm

Last modiﬁed: July 20, 2018