Mathematics 215: Introduction to Statistics

Study Guide

Unit 2: Probability

Unit 1 introduced the two fields of statistics—descriptive and inferential—but focused on descriptive statistics, which consists of the set of methods used to organize, display, and describe data.

The field of inferential statistics, you may recall, consists of methods that use sample results to make conclusions about a population of interest. Since we are relying on sample data, these conclusions are made with a level of uncertainty.

Probability and probability distributions, the topics in the next two units of this course, help us determine the degree of certainty (or uncertainty) with which we can make conclusions about a population based on observed sample results. After you complete Units 2 and 3, you will be prepared to study many of the common methods that are used in the field of inferential statistics.

Unit 2 examines the concepts and rules that allow us to compute the probabilities related to events that occur when conditions are uncertain.

Knowing these probabilities can help you make decisions in your daily life that you can feel comfortable with, regardless of the actual outcome. For example, if you know, based on the best information you have, that the probability that it will rain tomorrow is seventy percent, then you can feel comfortable with your decision to take an umbrella to work the next day. Or, if you know that there is an eighty-percent chance that interest rates will increase over the next week, then you can justify a strategy of negotiating your home mortgage loan right away. If your doctor can tell you the likelihood that a new experimental drug will help ease the pain relating to a condition you have, along with the likely side effects of this drug, then you can make a more reasoned decision about your use of the drug.

Unit 2 of MATH 215 consists of the following sections:.

2-1 Experiments, Outcomes, and Sample Spaces 2-2 Determining Probabilities: Three Conceptual Approaches 2-3 Marginal and Conditional Probabilities 2-4 Intersection of Events and the Multiplication Rule 2-5 Union of Events and the Addition Rule 2-6 Counting Rules, Factorials, and Combinations

The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 2.

Section 2-1: Experiments, Outcomes, and Sample Spaces

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • experiment
   • outcome
   • sample space
   • simple event and compound event
2. identify all possible outcomes of an experiment using a tree diagram, a Venn diagram or a cross-classification table.

Reading

Read the following sections in Chapter 4 of the textbook:

• Chapter 4 Introduction
• Section 4.1

Be prepared to read the material in Chapter 4 twice—the first time for a general overview of topics, and the second time to concentrate on the terms and examples presented. Return to these sections when you need to review these topics.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.

Videos Related to Chapter 4

• The Probability Song (jojoluvs)
• Probability (Jeremy Haselhorst)

Videos Related to Section 4.1

• Probability & Statistics: Definition of Sets & Elements (Michel van Biezen)
• Probability & Statistics: Definition of Sample Spaces & Factorials (Michel van Biezen)
• Probability & Statistics: Definition of Events (Michel van Biezen)
• Probability & Statistics: Definition of Intersection, Union, Complement & Venn Diagram (Michel van Biezen)
• Probability – Tree Diagrams 1 (Ron Barrow)

Exercises

Complete Exercises 4.3, 4.5, 4.7, and 4.9 from Chapter 4 of the textbook (page 133 of the downloadable eText).

Remember to show your work as you develop your answers.

Solutions to these exercises are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on page AN7 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Remember, it is very important that you make a concerted effort to answer each question independently before you refer to the solutions. If your answers differ from those provided and you cannot understand why, contact your tutor for assistance.

Section 2-2: Determining Probabilities: Three Conceptual Approaches

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • probability
   • first and second properties of probability
   • classical probability, relative frequency probability and subjective probability
2. compute probabilities using the classical probability rule.
3. approximate probabilities using the concept of relative frequency.

Reading

Read Section 4.2 in Chapter 4 of the textbook.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 4.2 of the textbook.

• Probability & Statistics: Introduction (Michel van Biezen)
• Probability & Statistics: The Probability Function – a First Look (Michel van Biezen)
• Probability & Statistics: The Probability Function – Flipping Three Coins, Example (Michel van Biezen)
• Calculating the Probability of Simple Events (patrickJMT)
• Probability Models: Two-Way Tables & Venn Diagrams (Jeremy Haselhorst)

Exercises

Complete the following exercises from Chapter 4 of the textbook (page numbers are for the downloadable eText):

• Exercises 4.11, 4.13, 4.15, and 4.17 on page 139
• Exercises 4.19, 4.21, and 4.25 on page 140

Solutions are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on page AN7 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 2-3: Marginal and Conditional Probabilities

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • marginal probability and conditional probability
   • mutually exclusive events
   • independent events and dependent events
   • complementary events
2. compute marginal and conditional probabilities.
3. use a mathematical test to determine whether two events are independent or dependent.
4. calculate probabilities for complementary events.

Reading

1. Read Section 4.3 in Chapter 4 of the textbook.
2. Read Additional Topic 2A in this Study Guide, below.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook reading.

Videos Related to Section 4.3.1

• Joint, Marginal, and Conditional Probabilities (Bryan Nelson)
• Joint and Marginal Probabilities (Brandon Foltz)
• Conditional Probability: Basic Definition (Kevin deLaplante)
• How to Calculate Conditional Probability (statisticsfun)
• Boy Girl Conditional Probability (statisticsfun)
• Tree Diagram Conditional Probability Review (Peter Bianchi)
• Conditional Probability and Tree Diagrams (youngteacher74)
• Probability Tree Diagrams (without replacement) (Miss Banks)
• Practice Exercises: Conditional Probability (lbowen11235)

Videos Related to Section 4.3.2

• Probability of Mutually Exclusive and Non-Mutually Exclusive Events (HCCMathHelp)

Videos Related to Section 4.3.3

• Probability: Independent and Dependent Events (Textbook Tactics)
• Probability for Independent and Dependent Events (HCCMathHelp)
• Mutually Exclusive versus Independent Events (Steve Mays)

Videos Related to Section 4.3.4

• Probability & Statistics: The Probability of an Event NOT Occurring (Michel van Biezen)
• Calculating Probability – “At Least One” Statements (patrickJMT)
  • In this video, observe that an “at least one satisfies” statement means “one or more satisfy,” which translates into the complement event of “none satisfy.”
• At Least One Probabilities (Kilgore College Mathematics)
• Probability & Statistics: The “At Least One or Once” Rule (Michel van Biezen)
• Probability & Statistics: The “At Least One or Once” Rule, Example (Michel van Biezen)

Exercises

1. Complete Exercises 4.35, 4.37, and 4.39 from Chapter 4 of the textbook (page 149 of the downloadable eText).

   Solutions are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on page AN7 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

2. Complete Exercise for Additional Topic 2A, below.

Required Reading: Additional Topic 2A: Tree Diagrams and Marginal and Conditional Probabilities

A portion of the following notes on tree diagrams and conditional probabilities is taken from the previous edition of the textbook. The purpose of these notes is to show how a tree diagram can be used to describe marginal and conditional probabilities. You can expect to have questions involving tree diagrams in the assignments and exams for this course.

Both marginal and conditional probabilities can be displayed in a tree diagram like the one below. Note that the first set of branches depicts marginal probabilities, while the second set of branches depicts conditional probabilities.

Exercise for Additional Topic 2A

The following exercise is reproduced from Application Exercise 4.47 in the previous edition of the textbook. The solution is provided.

Let $M=\text{Male}$; $F=\text{Female}$; $Y=\text{Yes}$, have Internet shopped; $N=\text{No}$, have not Internet shopped. Assume that one adult is selected at random.

1. Draw a tree diagram that depicts the following: $P\left(M\right)$, $P\left(F\right)$, $P\left(Y|M\right)$, $P\left(N|M\right)$, $P\left(Y|F\right)$, $P\left(N|F\right)$.

Solution

Based on the tree diagram:   $P\left(M\right)=1200⁄2000$;
$P\left(F\right)=800⁄2000$;
$P\left(Y|M\right)=500⁄1200$;
$P\left(N|M\right)=700⁄1200$;
$P\left(Y|F\right)=300⁄800$;
$P\left(N|F\right)=500⁄800$

Section 2-4: Intersection of Events and the Multiplication Rule

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • intersection of events
   • joint probabilities
2. use the multiplication rule to compute joint probabilities for any two types of events.
3. use the multiplication rule to compute conditional probabilities for any two types of events.
4. use the multiplication rule to compute joint probabilities for independent events.

Reading

1. Read Section 4.4 in Chapter 4 of the textbook.
2. Read Additional Topic 2B in this Study Guide, below.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 4.4 of the textbook.

• Joint and Marginal Probabilities (Brandon Foltz)
• Probability: Tree Diagrams 1 (two independent events) (Ron Barrow)
• Probability: Tree Diagrams 2 (two events which are not independent) (Ron Barrow)
• Calculating Probability – “And” Statements, independent events (patrickJMT)
• Calculating Probability – “And” Statements, dependent events (patrickJMT)
• Practice Exercises: Product Rule of Probability (Independent Events) (lbowen11235)
• Calculating Conditional Probabilities Using a Tree Diagram (Dan Ozimek)
• How to add and multiply probabilities using marbles (statisticsfun)

Exercises

1. Complete the following exercises from Chapter 4 of the textbook (page numbers are for the downloadable eText):
   • Exercises 4.47, 4.49, 4.51, 4.53, and 4.55 on page 155
   • Exercises 4.57 (note: use tree diagram), 4.59 (note: assume independent events), and 4.63 on page 156

   Solutions are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on page AN7 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

2. Complete Exercise for Additional Topic 2B, below.

Required Reading: Additional Topic 2B: Tree Diagrams and Joint Probabilities

The following notes on tree diagrams and joint probabilities are taken from the previous edition of the textbook. The purpose of these notes is to show how a tree diagram can be used to compute joint probabilities, given marginal and conditional probabilities. You can expect to have questions involving tree diagrams in the assignments and exams for this course.

Exercise for Additional Topic 2B

The following exercise is reproduced from Example 4-21 in the previous edition of the textbook. Complete a tree diagram showing all joint probabilities. The solution is provided.

Solution: The probability that both are defective is 0.0316.

Section 2-5: Union of Events and the Addition Rule

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the key term “union of events.”
2. use the addition rule to compute probabilities for the union of any types of events.
3. use the addition rule to compute probabilities for the union of mutually exclusive events.

Reading

Read Section 4.5 in Chapter 4 of the textbook.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 4.5 of the textbook.

• How to calculate probability, addition and complements (Independent Events) (statisticsfun)
• Practice Exercises: Union Rule, Probabilities and Venn Diagrams (lbowen11235)
• Probability of Mutually Exclusive and Non-Mutually Exclusive Events (HCCMathHelp)
• Intro to probability: Disjoint or Independent? (MrNystrom)

Exercises

Complete the following exercises from Chapter 4 of the textbook (page numbers are for the downloadable eText):

• Exercises 4.71, 4.73, and 4.75 on page 161
• Exercise 4.81(note: use tree diagram) on page 162

Solutions are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on page AN7 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

Section 2-6: Counting Rule, Factorials, and Combinations

Outcomes

After completing the readings and exercises for this section, you should be able to do the following:

1. define, and use in context, the following key terms:
   • counting rule
   • factorial
   • combination
2. compute the number of combinations using the combinations formula.

Reading

Read sections 4.6.1, 4.6.2, and 4.6.3 in Chapter 4 of the textbook.

Supplementary Video Resources

These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned textbook readings.

Videos Related to Section 4.6

• Counting Rules (Jeremy Haselhorst)
• Coin Toss Probability (lbowen 11235)
• Probability & Statistics: The Probability Function – Flipping Coins – General Formula 1 (Michel van Biezen)
• Probability & Statistics: The Probability Function – Flipping Coins – General Formula 2 (Michel van Biezen)
• Probability – Combinations and Permutations (Textbook Tactics)
• Permutations (Brandon Foltz)
• Combinations (Brandon Foltz)
• Combinations – Losing Your Marbles (Brandon Foltz)
• Combinations – Playing Cards (Brandon Foltz)
• (Probability with) Repeated Independent Events (Larry Feldman)
• Probability with Repeated Events (Art of Problem Solving)

Exercises

1. Complete the following exercises from Chapter 4 of the textbook (page numbers are for the downloadable eText):
   • Exercises 4.87 and 4.93 on page 168
   • Supplementary Exercises 4.95 and 4.97 on page 170
   • Supplementary Exercises 4.99 and 4.101 on page 171

2. Complete the Self-Review Test for Chapter 4 (pages 172–173 of the downloadable eText).

   Solutions are provided in the Student Solutions Manual for Chapter 4 (interactive textbook) and on pages AN7 and AN8 in the Answers to Selected Odd-Numbered Exercises section (downloadable eText).

3. Complete the Unit 2 Self-Test, below.

   A solutions document for this self-test is available on the course home page.

Optional Extra Practice

For extra practice with the material presented in this section, you can complete the following questions and exercises, for which the solutions are provided in the textbook:

1. Any odd-numbered chapter-section practice questions and Supplementary Exercises that are not assigned above.
2. The odd-numbered Advanced Exercises found at the end of Chapter 4 (pages 171–172 of the downloadable eText).

Assignment 2

Once you have completed the Unit 2 Self-Test below, complete Assignment 2. You can access the assignment in the Assessment section of the course home page. Once you have completed the assignment, submit it to your tutor for marking using the drop box on the page for Assignment 2.

Unit 2 Self-Test

The self-test questions are shown here for your information. Download the Unit 2 Self-Test document and write out your answers. Show all your work and keep your calculations to four decimal places. You can access the solutions to this self-test on the course home page.

1. The following survey was recently sent to all 500 employees of a large hospital.

   “All hospital employees should get a flu shot each year. Circle ONE of the following:

   Strongly agree Agree Disagree No opinion”

   The responses to this survey are summarized in the following frequency distribution.

   Response	Strongly agree (SA)	Agree (A)	Disagree (D)	No opinion (NO)
   Frequency	310	120	40	30

   1. If an employee is randomly selected from this hospital, find the probability that
      1. the employee will strongly agree with the flu shot statement.
      2. the employee will either strongly agree or agree with the flu shot statement.
      3. the employee will neither strongly agree nor agree with the flu shot statement. Use your answer to part ii. above to help answer this.
   2. Are the two events strongly agree and agree mutually exclusive? Explain.
2. Circle True (T) or False (F) for each of the following statements:
   1. TF

      The probability that the sample space will occur from the generation of a given experiment is equal to 1.
   2. TF

      An event that includes one and only one of the (final) outcomes for the generation of a given experiment is called a compound event.
   3. TF

      For any given event based on the generation of an experiment, the probability of the event will typically exceed 1.
   4. TF

      Assigning probabilities to a compound event by dividing the number of simple outcomes associated with the event by the total possible number of simple outcomes in the sample space is consistent with the classical concept of probability.
   5. TF

      If two events $A$ and $B$ are mutually exclusive, then we can state that $P\left(A⁄B\right)=P\left(A\right)$.
   6. TF

      If $P\left(A⁄B\right)=P\left(B\right)$, we can conclude that $A$ and $B$ are independent events.
   7. TF

      If $A$ and $B$ are complementary events, then $P\left(A\right)+P\left(B\right)=1$.
   8. TF

      Depending on the experiment, it is possible to have the numeric probability of an event exceed 1.
   9. TF

      If the joint probability of events $A$ and $B$ equals zero, then $P\left(A\text{or}B\right)=P\left(A\right)+P\left(B\right)$.
   10. TF

       If two events $A$ and $B$ are independent, then we can compute $P\left(A\text{and}B\right)$ $P\left(A\text{and}B\right)=P\left(A\right)\times P\left(B\right)$.
   11. TF

       If two events $A$ and $B$ are NOT independent, then we can compute $P\left(A\text{and}B\right)$ as follows: $P\left(A\text{and}B\right)=P\left(A\right)\times P\left(A⁄B\right)$.

3. A car dealership located on the outskirts of a large subdivision surveyed 200 of its regular customers in order to compare the level of customer satisfaction of its urban customers with that of its rural customers. The survey responses are summarized in the two-way classification table below.

   	Very satisfied (VS)	Satisfied (S)	Not satisfied (NS)
   Rural (R)	10	35	65
   Urban (U)	60	25	5

   1. If one customer is selected at random from the 200 customers that were surveyed, find the probability that this customer: (show your answers to 2 decimals)
      1. is an urban customer.
      2. is a rural customer or is very satisfied with the dealership’s service.
      3. is an urban customer and is not satisfied with the dealership’s service.
      4. is very satisfied with the dealership’s service, given that the customer is from an urban area.
      5. is from a rural area, given that the customer is not satisfied with the dealership’s service.
   2. Are the events Urban (U) and Very Satisfied (VS) mutually exclusive? Explain.
   3. Are the events Urban (U) and Very Satisfied (VS<) independent? Perform the appropriate math proof.
4. When a new student registers in a Statistics 101 course at a local university, they can decide to take the course online or in a classroom. In the past, 65% of students have taken the course in the classroom and 35% have taken it online. Given that the student takes the course in the classroom, they have an 80% chance of passing the course. Given that the student takes the course online for convenience, they have a 75% chance of passing the course.
   1. A new student has just registered in Statistics 101. Draw a tree diagram describing the different possible outcomes that face the student, in terms of whether the student takes the course in class or online, as well as whether the student ends up passing the course or not. At the end of the tree, display all possible joint probabilities. Keep your work to 4 decimals.
   2. Compute the probability that the student will pass the course.
   3. Are the events “takes the course in class” and “student will pass” independent events? Explain by making the appropriate math computations.
5. Consider the following experiment. You draw one card at random from a full deck of playing cards and observe whether it is a “hearts” card or not. You then replace the card in the deck, shuffle the deck, and then draw a second card. You observe whether this card is a hearts card or not. Note that in a deck of 52 cards there are 13 hearts cards.
   1. Draw a tree diagram to describe all possible outcomes. At the end of the tree, display all possible joint probabilities in one full play of this experiment. Keep your work to 4 decimals.
   2. Find the probability that two hearts cards will be drawn in one full play of this experiment.
   3. Find the probability that exactly one hearts card will be drawn in one full play of this experiment
   4. Find the probability that no hearts card will be drawn in one full play of this experiment
6. For two events $A$ and $B$:  

   $\begin{array}{l}P\left(A\right)=0.30\\ P(B)=0.40\\ P(B|A)=0.50\end{array}$
   1. Find $P\left(A\text{and}B\right)$.
   2. Find $P\left(A|B\right)$.
   3. Are the events $A$ and $B$ independent? Explain.
7. Consider an experiment in which you draw 4 cards, without replacement, from a standard deck of 52 cards. How many distinct 4-card hands could you draw? Show your calculations.

References

Mann, Prem S. Introductory Statistics, 8th ed. Wiley, 2012. [VitalSource].