Mathematics 216 Computer-oriented Approach to Statistics

Study Guide :: Unit 2

Probability

Introduction

Three centuries ago, curious gamblers asked the great Italian scientist Galileo, “Why does a throw of three dice turn up the sum 10 more often than the sum 9?” History does not say whether Galileo was able to answer , but his compatriot Girolamo Cardano wrote the first published work on “practical” mathematics, The Book on Games of Chance (originally published in Latin in 1663), for the benefit of gentlemen of the time whose livelihood depended on games of chance.

The mathematical theory of probability did not get off the ground until the 1600s, when a well known French gambler, Chevalier de Mere, asked philosopher and mathematician Blaise Pascal why it was unprofitable to bet even money that at least one double 6 would come up in 24 throws of two dice. You should be able to answer de Mere's question after completing this unit.

Probability theory has overcome its disreputable origin to become part of everyday life and culture. There is hardly any aspect of western life that is not touched by probability theory. Simply stated, probability theory is concerned with events whose outcome is not certain.

We often make or hear probability oriented statements: “Maybe our team will win tonight,” “I am almost sure I will get the job,” “There is a forty percent chance of snow this evening,” “The car probably has a faulty transmission.” Do such statements mean what they say? Some statements may be based on scientific facts and information, others on personal expectations and beliefs, but all are probabilistic inferences. They are conjectures, not facts.

In this unit, you will learn what probability is, and how to compute the mathematical probabilities of certain events occurring. You will examine three concepts of probability: classical probability, empirical probability, and subjective probability.

Once you have identified probabilities based on one of the three concepts of probability, you will apply probability formulae (rules) to compute other probabilities of interest.

After studying probability in this unit and the next, you will be prepared to examine topics in the field of inferential statistics in the last half of this course.

Basic Concepts of Probability and Counting

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the meaning of the key terms:
    • classical probability; empirical probability; subjective probability
    • complement of an event
    • complement probability formula rule
    • event; simple event
    • fundamental counting principle
    • outcome
    • probability experiment
    • sample space
  2. Given a probability experiment, identify all the simple events that make up a given event; identify all outcomes that make up the sample space.
  3. Given the probability of a specific event, classify this probability as classical, empirical, or subjective.
  4. Using the complement probability formula rule, compute the probability of the complement of an event.
  5. Compute the probabilities of various events based on a given frequency distribution; on a given Venn diagram.
  6. Use a tree diagram and the fundamental counting principle to find probabilities of events.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 3, Section 3.1, Basic Concepts of Probability and Counting (pages 130-139)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 3, Section 3.1 Exercises 3, 37, 41, 43, 45, 47, 51, 53, 57, 59, 61, 63, 83 (pages 140-145). Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 3 Section 3.1 are available on the Pearson MyLab website.

Conditional Probability and the Multiplication Rule

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the meaning of the key terms:
    • conditional probability
    • dependent events; independent events
    • multiplication rule for dependent events; multiplication rule for independent events
  2. Compute conditional probabilities related to a probability experiment.
  3. Distinguish between independent and dependent events.
  4. Determine mathematically whether two events are independent or dependent.
  5. Use the multiplication rule to compute the probability of two events being in sequence—P(A and B)—when the two events are dependent.
  6. Use the multiplication rule to compute the probability of two events being in sequence—P(A and B)—when the two events are independent.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 3, Section 3.2, Conditional Probability and the Multiplication Rule (pages 147-151)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 3, Section 3.2 Exercises 5, 7, 9, 17, 19 (pages 152-153). Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 3 Section 3.2 are available on the Pearson MyLab website.

The Addition Rule

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the meaning of the key terms:
    • addition rule for events that are mutually exclusive events; addition rule for events that are NOT mutually exclusive
    • mutually exclusive events
  2. Determine whether two or more events are mutually exclusive.
  3. Use the addition rule to compute the probability of P(A or B) when the two events are mutually exclusive.
  4. Use the addition rule to compute the probability of P(A or B) when the two events are NOT mutually exclusive.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 3, Section 3.3, The Addition Rule (pages 157-161). Take particular note of the Summary of Probability table on page 161.

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 3, Section 3.3 Exercises 7, 9, 11, 17, 21, 25 (pages 162-165); and Case Study Exercises 1, 3, and 4 (page 167). Write out the step-by-step solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 3 Section 3.3 are available on the Pearson MyLab website.

Additional Topics in Probability and Counting

Learning Objectives

After completing the readings and exercises assigned for this topic, you should be able to:

  1. Explain the meaning of the key terms:
    • combinations
    • permutations; distinguishable permutations
  2. Compute the number of permutations of n distinct objects taken n at a time.
  3. Compute the number of permutations of n distinct objects taken r at a time (nPr).
  4. Compute the number of distinguishable permutations of n objects, where n1 are of one type, n2 are of another type, etc.
  5. Compute the number of combinations of r objects from a group of n objects without regard to order (nCr).
  6. Apply the permutation and combination counting rules to compute probabilities.

Important Note: For help accessing the eText resources referred to below, see Navigating Your eText on the course home page.

Required Reading

Elementary Statistics, Chapter 3, Section 3.4, Additional Topics in Probability and Counting (pages 168-173)

Try It Yourself Examples

Work through each Try It Yourself example in this section of the eText. Check your work against the solutions provided.

Exercises in Your eText

Do the following exercises in your eText:

Chapter 3, Section 3.4 Exercises 15, 17, 23, 35, 45, 49 (pages 174-176). Write out step-by-step-solutions or explanations. Check your work against the solutions provided.

Optional Multimedia Resources

Additional optional multimedia resources related to Chapter 3 Section 3.4 are available on the Pearson MyLab website.

Chapter 3 Review (Extra Online Practice)

For more practice working with the topics in this chapter of the eText, work through this review. Or, if you feel you have mastered this material, you may skip to the computer lab section of this unit.

Review Learning Objectives

Before proceeding to the online exercises, briefly review the Learning Objectives for each of the topics covered in previous sections of this Study Guide:

  • Basics Concepts of Probability and Counting
  • Conditional Probability and the Multiplication Rule
  • The Addition Rule
  • Additional Topics in Probability and Counting
Optional Practice in the MyLab Study Plan

For more practice on the topics/sections of Chapter 3, visit Pearson MyLab, and work interactively through the exercises in the Study Plan. For help accessing this resource, see Accessing Pearson MyLab on the course home page.

Computer Lab 2

In Computer Lab 2, you will use a statistical software called StatCrunch to develop solutions to exercises related to topics in the eText’s Chapter 3.

Computer Lab 2 Detailed Instructions

Your Computer Lab activities and the detailed step-by-step instructions (Guided Solutions) that will guide you in using StatCrunch to complete these are in the Computer Lab 2 file.

Computer Lab 2 Quick Reviews

The Computer Lab Quick Reviews (QRs) summarize a few key steps (but not all steps) needed to complete each Activity in the Computer Labs. These QRs will be useful when you are preparing for the computer components of the assignments, midterm exam, and final exam. To access the QRs, click MATH 216 Computer Lab 2 QRs.

Self-Test 2

To access Self-Test 2, click MATH 216 Self-Test 2.

It is important that you work through all the exercises in the unit self-tests and the eText chapter quizzes. No grades are assigned to the self-tests. They are designed to, along with the unit assignments, help you master the content presented in each unit.

Each unit self-test has two parts: one on theory (A) and one on computer work (B). Working through these will help you review key exercises in the unit, which will help you prepare for assignments and exams.

Assignment 2

After completing Self-Test 2, complete Assignment 2, which you will find on the course home page. Submit your solutions to this assignment for marking using the drop box on the course home page.