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Contemporary Mathematics

7.7 What Are the Odds?

Contemporary Mathematics7.7 What Are the Odds?

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
Three lottery tickets have been scratched off to reveal the ticket-holder's winnings.
Figure 7.33 Scratch-off lottery tickets, as well as many other games, represent the likelihood of winning using odds. (credit: “My Scratch-off Winnings” by Shoshanah/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

  1. Compute odds.
  2. Determine odds from probabilities.
  3. Determine probabilities from odds.

A particular lottery instant-win game has 2 million tickets available. Of those, 500,000 win a prize. If there are 500,000 winners, then it follows that there are 1,500,000 losing tickets. When we evaluate the risk associated with a game like this, it can be useful to compare the number of ways to win the game to the number of ways to lose. In the case of this game, we would compare the 500,000 wins to the 1,500,000 losses. In other words, there are 3 losing tickets for every winning ticket. Comparisons of this type are the focus of this section.

Computing Odds

The ratio of the number of equally likely outcomes in an event EE to the number of equally likely outcomes not in the event EE is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the event to the number in the event EE to the number in the event EE is called the odds against the event.

Checkpoint

Both odds and probabilities are calculated as ratios. To avoid confusion, we will always use fractions, decimals, or percents for probabilities, and we’ll use colons to indicate odds. The rules for simplifying fractions apply to odds, too. Thus, the odds for winning a prize in the game described in the section opener are 500,000:1,500,000=1:3500,000:1,500,000=1:3 and the odds against winning a prize are 3:13:1. These would often be described in words as “the odds of winning are one to three in favor” or “the odds of winning are three to one against.”.

Checkpoint

Notice that, while probabilities must always be between zero and one inclusive, odds can be any (non-negative) number, as we’ll see in the next example.

Example 7.25

Computing Odds

  1. If you roll a fair 6-sided die, what are the odds for rolling a 5 or higher?
  2. If you roll two fair 6-sided dice, what are the odds against rolling a sum of 7?
  3. If you draw a card at random from a standard deck, what are the odds for drawing a ?
  4. If you draw 2 cards at random from a standard deck, what are the odds against them both being ?

Your Turn 7.25

You roll a pair of 4-sided dice with faces labeled 1 through 4.
1.
What are the odds for rolling a sum greater than 3?
2.
What are the odds against both dice giving the same number?

Odds as a Ratio of Probabilities

We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with 500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is 500,0002,000,000=14500,0002,000,000=14, and the probability of losing is 114=34114=34. Notice that the ratio of the probability of winning to the probability of losing is 14:34=1:314:34=1:3, which matches the odds in favor of winning.

FORMULA

For an event EE,

odds forE=n(E):n(E)=P(E):P(E)=P(E):(1P(E))odds againstE=n(E):n(E)=P(E):P(E)=(1P(E)):P(E)odds forE=n(E):n(E)=P(E):P(E)=P(E):(1P(E))odds againstE=n(E):n(E)=P(E):P(E)=(1P(E)):P(E)

We can use these formulas to convert probabilities to odds, and vice versa.

Example 7.26

Converting Probabilities to Odds

Given the following probabilities of an event, find the corresponding odds for and odds against that event.

  1. P(E)=35P(E)=35
  2. P(E)=17%P(E)=17%

Your Turn 7.26

1.
If the probability of an event E is 80%, find the odds for and the odds against E .

Now, let’s convert odds to probabilities. Let’s say the odds for an event are A:BA:B. Then, using the formula above, we have A:B=P(E):(1P(E))A:B=P(E):(1P(E)). Converting to fractions and solving for P(E)P(E), we get:

AB =P(E)1P(E)A(1P(E)) =B×P(E)AA×P(E) =B×P(E)A=A×P(E)+B×P(E)A=(A+B)×P(E)AA+B=P(E). AB =P(E)1P(E)A(1P(E)) =B×P(E)AA×P(E) =B×P(E)A=A×P(E)+B×P(E)A=(A+B)×P(E)AA+B=P(E).

Let’s put this result in a formula we can use.

FORMULA

If the odds in favor of EE are A:BA:B, then

P(E)=AA+BP(E)=AA+B.

Example 7.27

Converting Odds to Probabilities

Find P(E)P(E) if EE:

  1. The odds of EE are 2:12:1 in favor
  2. The odds of EE are 6:16:1 against

Your Turn 7.27

Find P ( E ) if E :
1.
The odds of E are 15 : 1 against
2.
The odds of E are 2.5 : 1 in favor

Checkpoint

Some places, particularly state lottery websites, will use the words “odds” and “probability” interchangeably. Never assume that the word “odds” is being used correctly! Compute one of the odds/probabilities yourself to make sure you know how the word is being used!

Check Your Understanding

For the following exercises, you are rolling a 6-sided die with 3 orange faces, 2 green faces, and 1 blue face.
37.
What are the odds in favor of rolling a green face?
38.
What are the odds against rolling a blue face?
39.
What are the odds in favor of rolling an orange face?
40.
What are the odds in favor of an event with probability 3 8 ?
41.
What are the odds against an event with probability 2 13 ?
42.
What is the probability of an event with odds 9 : 4 against?
43.
What is the probability of an event with odds 5 : 7 in favor?

Section 7.7 Exercises

For the following exercises, find the probabilities of events with the given odds in favor.
1 .
9 : 4
2 .
2 : 3
3 .
2 : 3
4 .
5 : 4
5 .
1 : 50
6 .
7 : 5
7 .
1 : 7
8 .
10 : 9
For the following exercises, find the probabilities of events with the given odds against.
9 .
1 : 8
10 .
2 : 3
11 .
3 : 2
12 .
5 : 4
13 .
1 : 50
14 .
7 : 5
15 .
1 : 7
16 .
10 : 9
In the following exercises, find the odds in favor of events with the given probabilities. Give your answer as a ratio of whole numbers. If neither of those two numbers is 1, also give an answer as a ratio involving both 1 and a number greater than or equal to 1 (for example, the odds 5 : 2 and 3 : 8 can be reduced to 2.5 : 1 and 1 : 2.67 ).
17 .
2 7
18 .
12 17
19 .
8 9
20 .
3 8
21 .
9 25
22 .
6 7
23 .
10 13
24 .
8 15
In the following exercises, find the odds against events with the given probabilities. Give your answer as a ratio of whole numbers. If neither of those two numbers is 1, also give an answer as a ratio involving both 1 and a number greater than or equal to 1 (for example, the odds 5 : 2 and 3 : 8 can be reduced to 2.5 : 1 and 1 : 2.67 ).
25 .
2 7
26 .
12 17
27 .
8 9
28 .
3 8
29 .
9 25
30 .
6 7
31 .
10 13
32 .
8 15
In the following exercises, you are drawing from a deck containing only these 10 cards:
A , A , A , A , K , K , Q , Q , J , J .
33 .
Let E be the event “draw an ace.”
  1. What is the probability of E ?
  2. What are the odds in favor of E ?
  3. What are the odds against E ?
34 .
Let F be the event “draw a ”.
  1. What is the probability of F ?
  2. What are the odds in favor of F ?
  3. What are the odds against F ?
35 .
Let T be the event “draw two (without replacement).”
  1. What is the probability of T ?
  2. What are the odds in favor of T ?
  3. What are the odds against T ?
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