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Contemporary Mathematics

Chapter 7

Contemporary MathematicsChapter 7
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Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Your Turn

7.1
1.
4 \times 15 = 60
7.2
1.
4 \times 15 \times 3 \times 2 = 360
7.3
1.
3 \times 10 \times 10 \times 10 \times 10 \times 10 \times 26 \times 26 \times 26 = 5,272,800,000
7.4
1.
120
7.5
1.
720
2.
132
3.
70
7.6
1.
30
2.
24,024
3.
5,814
7.7
1.
_{15}{P_3} = 2{,}730
7.8
1.
combination
2.
combination
7.9
1.
15
2.
45
3.
2,002
7.10
1.
30,856
2.
111,930
7.11
1.
290,004
7.12
1.
\{\heartsuit ,\spadesuit ,\clubsuit ,\diamondsuit\}
2.
{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}
3.
{1, 2, 3, 4}
4.
{3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
7.13
1.
Independent
2.
Dependent
7.14
1.
{H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
7.15
1.
{J Q, J K, Q J, Q K, K J, K Q}
7.16
1.
P(\text{H} < 5) = 1
2.
0 < P(\text{H} < 4) < 1
3.
P(\text{H} ≥ 5) = 0
7.17
1.
\frac{1}{6}
2.
\frac{2}{3}
3.
\frac{1}{2}
7.18
1.
\frac{5}{{16}}
7.19
1.
\frac{1}{3}
2.
\frac{1}{2}
3.
\frac{2}{3}">
7.20
1.
\frac{{13}}{{1,000}} = 1.3\%
7.21
1.
theoretical
2.
subjective
3.
empirical
7.22
1.
\frac{{13}}{{16}}
7.23
1.
\frac{{144}}{{43680}} = \frac{3}{{910}}
7.24
1.
1 - \frac{{_{53}{C_3}}}{{_{58}{C_3}}} \approx 10.7\%
2.
\frac{{_{13}{C_2}{\, \times \,_{13}}{C_3}}}{{_{52}{C_5}}} \approx 0.86\%
7.25
1.
13{:}3
2.
12{:}4 = 3{:}1
7.26
1.
The odds for E are 4{:}1 and the odds against E are 1{:}4.
7.27
1.
P(E) = \frac{1}{{16}}
2.
P(E) = \frac{{2.5}}{{3.5}} \approx 0.714
7.28
1.
Mutually exclusive
2.
Not mutually exclusive
3.
Mutually exclusive
7.29
1.
\frac{4}{{10}} + \frac{2}{{10}} = \frac{3}{5}
2.
Not appropriate; the events are not mutually exclusive.
3.
\frac{3}{{10}} + \frac{2}{{10}} = \frac{1}{2}
7.30
1.
\frac{1}{2}
2.
\frac{1}{2}
3.
\frac{2}{3}
7.31
1.
\frac{1}{3}
2.
1
3.
\frac{2}{3}
7.32
1.
\frac{3}{{28}}
2.
\frac{3}{{56}}
3.
\frac{3}{{28}}
7.33
1.
Roll Probability
1 \frac{1}{3}
2 \frac{1}{4}
3 \frac{1}{6}
4 \frac{1}{{12}}
5 \frac{1}{{12}}
6 \frac{1}{{12}}
7.34
1.
Not binomial (more than two outcomes)
2.
Not binomial (not independent)
3.
Binomial
4.
Not binomial (number of trials isn’t fixed)
7.35
1.
0.146
2.
0.190
3.
0.060
7.36
1.
0.2972
2.
0.6615
3.
0.0919
4.
0.5207
5.
0.3643
7.37
1.
\frac{{10}}{3}
2.
\frac{3}{2}
3.
\frac{{25}}{4} = $6.25
7.38
1.
If you roll the special die many times, the mean of the numbers showing will be around 3.33.
2.
If you repeat the coin-flipping experiment many times, the mean of the number of heads you get will be around 1.5.
3.
If you play this game many times, the mean of your winnings will be around $10.
7.39
1.
If the player bets on 7, the expected value is - \$ 0.17.
2.
If the player bets on 12, the expected value is - \$ 0.14.
3.
If the player bets on any craps, the expected value is - \$ 0.11.
The best bet for the player is any craps; the best bet for the casino is the bet on 7.

Check Your Understanding

1.
540
2.
14
3.
1,024
4.
800
5.
25,920
6.
120
7.
120
8.
1,320
9.
1,680
10.
_{15}{P_4} = 32{,}760
11.
permutations
12.
combinations
13.
66
14.
560
15.
20
16.
560
17.
{0, 1, 2, 3, 4, 5, 6}
18.
Crinkle Fries Curly Fries Onion Rings
8 Nuggets 8 nuggets with crinkle fries 8 nuggets with curly fries 8 nuggets with onion rings
12 Nuggets 12 nuggets with crinkle fries 12 nuggets with curly fries 12 nuggets with onion rings
19.
A tree diagram with three stages. The tree diagram shows a node branching into three nodes labeled H, E, and S. The node, H branches into two nodes labeled A and B. Node, E leads to a node labeled C. Node, S branches into three nodes labeled J, K, and L. The possible outcomes are as follows: H A, H B, E C, S J, S K, and S L.
20.
{8 nuggets with crinkle fries, 8 nuggets with curly fries, 8 nuggets with onion rings, 12 nuggets with crinkle fries, 12 nuggets with curly fries, 12 nuggets with onion rings}
21.
{history with Anderson, history with Burr, English with Carter, sociology with Johnson, sociology with Kirk, sociology with Lambert}
22.
\frac{1}{4}
23.
\frac{1}{2}
24.
\frac{1}{2}
25.
0
26.
theoretically
27.
subjectively
28.
empirically
29.
{\text{number of heads}} > 20
30.
89.9%
31.
\frac{{1\, \times \,1 \,\times\, 2}}{{_{10}{P_3}}} \approx 0.278\%
32.
\frac{{1\, \times \,1 \,\times \,2\, \times\, 2}}{{_{10}{P_4}}} \approx 0.079\%
33.
\frac{{2\, \times\, 2 \,\times \,1\, \times\, 2}}{{_{10}{P_4}}} \approx 0.16\%
34.
\frac{{1\, \times\, 1\, \times\, 2}}{{_{10}{C_3}}} \approx 1.67\%
35.
\frac{{1\, \times \,1 \,\times\, 2 \,\times\, 2}}{{_{10}{C_4}}} \approx 1.9\%
36.
\frac{{2 \,\times\, 1\, \times \,2}}{{_{10}{C_4}}} \approx 1.9\%
37.
1{:}2
38.
5{:}1
39.
1{:}1
40.
3{:}5
41.
11{:}2
42.
\frac{4}{{13}}
43.
\frac{5}{{12}}
44.
\frac{3}{5}
45.
\frac{3}{5}
46.
\frac{3}{5}
47.
\frac{2}{5}
48.
\frac{2}{5}
49.
\frac{3}{{10}}
50.
\frac{1}{4}
51.
\frac{1}{9}
52.
\frac{1}{6}
53.
\frac{2}{{15}}
54.
\frac{4}{{45}}
55.
\frac{8}{{45}}
56.
No (more than two outcomes)
57.
Yes
58.
No (number of trials is not fixed)
59.
0.9453
60.
0.1366
61.
0.8230
62.
4.1
63.
If you roll this die many times, the mean of the numbers rolled will be around 4.1.
64.
$0.417
65.
If you play this game many times, the mean of the amount won/lost each time will be about 42 cents.
66.
Yes; the expected value is positive.
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