MATH 215 C10 - Study Guide: Unit 2
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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 2-2 2-3 2-4 2-5 2-6 The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete Assignment 2. 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 MATH 215 C10 - Study Guide: Unit 2 1
Transcript of MATH 215 C10 - Study Guide: Unit 2
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
2-2
2-3
2-4
2-5
2-6
The unit also contains a self-test. When you have completed the material for this unit, including the self-test, complete
Assignment 2.
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
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2. identify all possible outcomes of an experiment using a tree diagram, a Venn diagram or a cross-classification table.
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.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
The Probability Song (https://www.youtube.com/watch?v=0xm1SDlnvh4) (jojoluvs)
Probability (https://www.youtube.com/watch?v=I0YlchC7lAw) (Jeremy Haselhorst)
Probability & Statistics: Definition of Sets & Elements (https://www.youtube.com/watch?v=GY86lxlWWbM) (Michel vanBiezen)
Probability & Statistics: Definition of Sample Spaces & Factorials (https://www.youtube.com/watch?v=WelPq6Kgf7s)(Michel van Biezen)
Probability & Statistics: Definition of Events (https://www.youtube.com/watch?v=EOk25Tb-1bM) (Michel van Biezen)
Probability & Statistics: Definition of Intersection, Union, Complement & Venn Diagram (https://www.youtube.com/watch?v=GU_2eQvVlCg) (Michel van Biezen)
Probability – Tree Diagrams 1 (https://www.youtube.com/watch?v=mkDzmI7YOx0) (Ron Barrow)
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.
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
MATH 215 C10 - Study Guide: Unit 2
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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.
Read Section 4.2 in Chapter 4 of the textbook.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 4.2 of
the textbook.
Probability & Statistics: Introduction (https://www.youtube.com/watch?v=f26vt5uQ_uI) (Michel van Biezen)
Probability & Statistics: The Probability Function – a First Look (https://www.youtube.com/watch?v=i6_byEFjKEA)(Michel van Biezen)
Probability & Statistics: The Probability Function – Flipping Three Coins, Example (https://www.youtube.com/watch?v=zReGHNdWvIo) (Michel van Biezen)
Calculating the Probability of Simple Events (https://www.youtube.com/watch?v=BAjOEsU_mpE&nohtml5=False)(patrickJMT)
Proability Models: Two-Way Tables & Venn Diagrams (https://www.youtube.com/watch?v=S32UL06rGn8) (JeremyHaselhorst)
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).
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.
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1. Read Section 4.3 in Chapter 4 of the textbook.
2. Read Additional Topic 2A in this Study Guide, below.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook reading.
Joint, Marginal, and Conditional Probabilities (https://www.youtube.com/watch?v=FWrEaSaW2mc) (Bryan Nelson)
Joint and Marginal Probabilities (https://www.youtube.com/watch?v=DkHWKAy47X0&list=PLIeGtxpvyG-I9m4otjYGCQL_1m0Edm0LA) (Brandon Foltz)
Conditional Probability: Basic Definition (https://www.youtube.com/watch?v=cwADSMeiIoE) (Kevin deLaplante)
How to Calculate Conditional Probability (https://www.youtube.com/watch?v=H02B3aMNKzE&nohtml5=False)(statisticsfun)
Boy Girl Conditional Probability (https://www.youtube.com/watch?v=MDzbD2Ay5b4&nohtml5=False) (statisticsfun)
Tree Diagram Conditional Probability Review (https://www.youtube.com/watch?v=znhjRqvvDWE) (Peter Bianchi)
Conditional Probability and Tree Diagrams (https://www.youtube.com/watch?v=WmcoWd8Uv-0) (youngteacher74)
Probability Tree Diagrams (without replacement) (https://www.youtube.com/watch?v=b1aeMqwd7uE) (Miss Banks)
Practice Exercises: Conditional Probability (https://www.youtube.com/watch?v=PPHvCRHneKs) (lbowen11235)
Probability of Mutually Exclusive and Non-Mutually Exclusive Events (https://www.youtube.com/watch?v=rGekybNs2V8) (HCCMathHelp)
Probability: Independent and Dependent Events (https://www.youtube.com/watch?v=yi97YB8EyDg) (Textbook Tactics)
Probability for Independent and Dependent Events (https://www.youtube.com/watch?v=Tl4stJNY10s) (HCCMathHelp)
Mutually Exclusive versus Independent Events (https://www.youtube.com/watch?v=0Vqmkpr1grA) (Steve Mays)
Probability & Statistics: The Probability of an Event NOT Occurring (https://www.youtube.com/watch?v=HpXAlpnoq1U) (Michel van Biezen)
Calculating Probability – “At Least One” Statements (https://www.youtube.com/watch?v=dwjQaJ5xt1o&list=PL8gnhgRJl1x6Zb5ecu1Jvcrb3aJSRx3oT&index=5) (patrickJMT)
In this video, observe that an “at least one satisfies” statement means “one or more satisfy,” which translates intothe complement event of “none satisfy.”
At Least One Probabilities (https://www.youtube.com/watch?v=KFtCj_46TzA) (Kilgore College Mathematics)
Probability & Statistics: The “At Least One or Once” Rule (https://www.youtube.com/watch?v=VMcBkIhzEhU) (Michelvan Biezen)
Probability & Statistics: The “At Least One or Once” Rule, Example (https://www.youtube.com/watch?v=-b6FuQvHEmQ) (Michel van Biezen)
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 theAnswers to Selected Odd-Numbered Exercises section (downloadable eText).
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2. Complete Exercise for Additional Topic 2A, below.
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.
Suppose all 100 employees of a company were asked whether they are in favor of or against paying high
salaries to CEOs of U.S. companies. Table [4.4] gives a two-way classification of the responses of these
100 employees. Assume that every employee responds either in favor or against.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 159–161. This material is
reproduced with the permission of John Wiley & Sons Canada, Ltd.]
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.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 159–161. This material is
reproduced with the permission of John Wiley & Sons Canada, Ltd.]
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The following exercise is reproduced from Application Exercise 4.47 in the previous edition of the textbook. The solution is
provided.
Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet.
The following table gives a two-way classification of the responses.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 167. This material is
reproduced with the permission of John Wiley & Sons Canada, Ltd.]
Let ; ; , have Internet shopped; , have not Internet shopped. Assume that one adult is
selected at random.
a. Draw a tree diagram that depicts the following: , , , , , .
Solution
Based on the tree diagram: ;
;
;
;
;
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.
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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.
1. Read Section 4.4 in Chapter 4 of the textbook.
2. Read Additional Topic 2B in this Study Guide, below.
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 (https://www.youtube.com/watch?v=DkHWKAy47X0&list=PLIeGtxpvyG-I9m4otjYGCQL_1m0Edm0LA) (Brandon Foltz)
Probability: Tree Diagrams 1 (two independent events) (https://www.youtube.com/watch?v=mkDzmI7YOx0) (RonBarrow)
Probability: Tree Diagrams 2 (two events which are not independent) (https://www.youtube.com/watch?v=NOOMC_rc-8Q) (Ron Barrow)
Calculating Probability – “And” Statements, independent events (https://www.youtube.com/watch?v=xgoQeRyvw5I&nohtml5=False) (patrickJMT)
Calculating Probability – “And” Statements, dependent events (https://www.youtube.com/watch?v=iIzJxFzlZOQ&nohtml5=False) (patrickJMT)
Practice Exercises: Product Rule of Probability (Independent Events) (https://www.youtube.com/watch?v=QSD7mhAuSW0) (lbowen11235)
Calculating Conditional Probabilities Using a Tree Diagram (https://www.youtube.com/watch?v=0CV-8i-pRBE) (DanOzimek)
How to add and multiply probabilities using marbles (https://www.youtube.com/watch?v=lA7rprQecV8&nohtml5=False) (statisticsfun)
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 theAnswers to Selected Odd-Numbered Exercises section (downloadable eText).
2. Complete Exercise for Additional Topic 2B, below.
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.
Table 4.7 gives the classification of all employees of a company by gender and college degree.
MATH 215 C10 - Study Guide: Unit 2
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If one of these employees is selected at random for membership on the employee–management committee,
what is the probability that this employee is a female and a college graduate?
The tree diagram in Figure 4.16 shows all four joint probabilities for this example. The joint probability of
and is highlighted. Note that the joint probabilities have been rounded to 4 decimal places.
[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 171–172. This material is
reproduced with the permission of John Wiley & Sons Canada, Ltd.]
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.
A box contains 20 DVDs, 4 of which are defective. If two DVDs are selected at random (without replacement)
from this box, what is the probability that both are defective? Keep all calculations to 4 decimal places.
Solution: Let us define the following events for this experiment:
Let = event that the first DVD selected is good
Let = event that the first DVD selected is defective
Let = event that the second DVD selected is good
Let = event that the second DVD selected is defective
MATH 215 C10 - Study Guide: Unit 2
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[Source: Prem S. Mann, Introductory Statistics, 8th ed. (Wiley, 2012) [VitalSource], 172–173. This material
is reproduced with the permission of John Wiley & Sons Canada, Ltd.]
Solution: The probability that both are defective is 0.0316.
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.
Read Section 4.5 in Chapter 4 of the textbook.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in Section 4.5 of
the textbook.
Probability & Statistics: The Probability of or (Independent Events) (https://www.youtube.com/watch?v=tjwc_Y34Pms) (Michel van Biezen)
Probability & Statistics: The Probability of or (Dependent Events) (https://www.youtube.com/watch?v=fAQrTvEYB0A) (Michel van Biezen)
Probability & Statistics: The Probability of or (Dependent Events ) Example (https://www.youtube.com/watch?v=fAQrTvEYB0A) (Michel van Biezen)
How to calculate probability, addition and complements (Independent Events) (https://www.youtube.com/watch?v=dqNvtYWmSrY&nohtml5=False) (statisticsfun)
Practice Exercises: Union Rule, Probabilities and Venn Diagrams (https://www.youtube.com/watch?v=gc0jdp9XDv0)
MATH 215 C10 - Study Guide: Unit 2
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(lbowen11235)
Probability of Mutually Exclusive and Non-Mutually Exclusive Events (https://www.youtube.com/watch?v=rGekybNs2V8) (HCCMathHelp)
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).
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.
Read sections 4.6.1, 4.6.2, and 4.6.3 in Chapter 4 of the textbook.
These videos provide alternative explanations and further exploration of the concepts and techniques presented in the assigned
textbook readings.
Counting Rules (https://www.youtube.com/watch?v=SDpW026olj8) (Jeremy Haselhorst)
Coin Toss Probability (https://www.youtube.com/watch?v=CBcUhcueCnM) (lbowen 11235)
Probability & Statistics: The Probability Function – Flipping Coins – General Formula 1 (https://www.youtube.com/watch?v=7utYtIpTDSU) (Michel van Biezen)
Probability & Statistics: The Probability Function – Flipping Coins – General Formula 2 (https://www.youtube.com/watch?v=wd3-i7kwXmk) (Michel van Biezen)
Probability – Combinations and Permutations (https://www.youtube.com/watch?v=G2WfktjDNDk) (Textbook Tactics)
Permutations (https://www.youtube.com/watch?v=T1CjOkEb1ew) (Brandon Foltz)
Combinations (https://www.youtube.com/watch?v=6XWqDezwbaw) (Brandon Foltz)
Combinations – Losing Your Marbles (https://www.youtube.com/watch?v=1oEQWo28w6U) (Brandon Foltz)
Combinations – Playing Cards (https://www.youtube.com/watch?v=4tyzC-MkzOY) (Brandon Foltz)
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(Probability with) Repeated Independent Events (https://www.youtube.com/watch?v=2N7J4wJj-7c) (Larry Feldman)
Probability with Repeated Events (https://www.youtube.com/watch?v=Bdhriaynamg) (Art of Problem Solving)
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 AN8in 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.
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).
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.
The self-test questions are shown here for your information. Download the Unit 2 Self-Test (https://fst-course.athabascau.ca
/science/math/215/r10/self_test/self_test02.html) 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”
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The responses to this survey are summarized in the following frequency distribution.
a. If an employee is randomly selected from this hospital, find the probability that
i. the employee will strongly agree with the flu shot statement.
ii. the employee will either strongly agree or agree with the flu shot statement.
iii. the employee will neither strongly agree nor agree with the flu shot statement. Use your answer to part ii.above to help answer this.
b. Are the two events strongly agree and agree mutually exclusive? Explain.
2. Circle True (T) or False (F) for each of the following statements:
a. T F The probability that the sample space will occur from the generation of a given experiment is equalto 1.
b. T F An event that includes one and only one of the (final) outcomes for the generation of a givenexperiment is called a compound event.
c. T F For any given event based on the generation of an experiment, the probability of the event willtypically exceed 1.
d. T F Assigning probabilities to a compound event by dividing the number of simple outcomes associatedwith the event by the total possible number of simple outcomes in the sample space is consistent withthe classical concept of probability.
e. T F If two events and are mutually exclusive, then we can state that .
f. T F If , we can conclude that and are independent events.
g. T F If and are complementary events, then .
h. T F Depending on the experiment, it is possible to have the numeric probability of an event exceed 1.
i. T F If the joint probability of events and equals zero, then .
j. T F If two events and are independent, then we can compute .
k. T F If two events and are NOT independent, then we can compute as follows:.
3. A car dealership located on the outskirts of a large subdivision surveyed 200 of its regular customers in order to comparethe level of customer satisfaction of its urban customers with that of its rural customers. The survey responses aresummarized in the two-way classification table below.
a. If one customer is selected at random from the 200 customers that were surveyed, find the probability that thiscustomer: (show your answers to 2 decimals)
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i. is an urban customer.
ii. is a rural customer or is very satisfied with the dealership’s service.
iii. is an urban customer and is not satisfied with the dealership’s service.
iv. is very satisfied with the dealership’s service, given that the customer is from an urban area.
v. is from a rural area, given that the customer is not satisfied with the dealership’s service.
b. Are the events Urban (U) and Very Satisfied (VS) mutually exclusive? Explain.
c. 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 ina classroom. In the past, 65% of students have taken the course in the classroom and 35% have taken it online. Given thatthe student takes the course in the classroom, they have an 80% chance of passing the course. Given that the studenttakes the course online for convenience, they have a 75% chance of passing the course.
a. A new student has just registered in Statistics 101. Draw a tree diagram describing the different possible outcomesthat face the student, in terms of whether the student takes the course in class or online, as well as whether thestudent ends up passing the course or not. At the end of the tree, display all possible joint probabilities. Keep yourwork to 4 decimals.
b. Compute the probability that the student will pass the course.
c. Are the events “takes the course in class” and “student will pass” independent events? Explain by making theappropriate math computations.
5. Consider the following experiment. You draw one card at random from a full deck of playing cards and observe whether itis a “hearts” card or not. You then replace the card in the deck, shuffle the deck, and then draw a second card. Youobserve whether this card is a hearts card or not. Note that in a deck of 52 cards there are 13 hearts cards.
a. Draw a tree diagram to describe all possible outcomes. At the end of the tree, display all possible jointprobabilities in one full play of this experiment. Keep your work to 4 decimals.
b. Find the probability that two hearts cards will be drawn in one full play of this experiment.
c. Find the probability that exactly one hearts card will be drawn in one full play of this experiment
d. Find the probability that no hearts card will be drawn in one full play of this experiment
6. For two events and :
a. Find .
b. Find .
c. Are the events and independent? Explain.
7. Consider an experiment in which you draw 4 cards, without replacement, from a standard deck of 52 cards. How manydistinct 4-card hands could you draw? Show your calculations.
Mann, Prem S. Introductory Statistics, 8th ed. Wiley, 2012. [VitalSource].
MATH 215 C10 - Study Guide: Unit 2
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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.
a. If an employee is randomly selected from this hospital, find the probability that
i. the employee will strongly agree with the flu shot statement.
ii. the employee will either strongly agree or agree with the flu shot statement.
iii. the employee will neither strongly agree nor agree with the flu shot statement. Use your answer to part ii.above to help answer this.
b. Are the two events strongly agree and agree mutually exclusive? Explain.
2. Circle True (T) or False (F) for each of the following statements:
a. T F The probability that the sample space will occur from the generation of a given experiment is equalto 1.
b. T F An event that includes one and only one of the (final) outcomes for the generation of a givenexperiment is called a compound event.
c. T F For any given event based on the generation of an experiment, the probability of the event willtypically exceed 1.
d. T F Assigning probabilities to a compound event by dividing the number of simple outcomes associatedwith the event by the total possible number of simple outcomes in the sample space is consistent withthe classical concept of probability.
e. T F If two events and are mutually exclusive, then we can state that .
f. T F If , we can conclude that and are independent events.
g. T F If and are complementary events, then .
h. T F Depending on the experiment, it is possible to have the numeric probability of an event exceed 1.
i. T F If the joint probability of events and equals zero, then .
j. T F If two events and are independent, then we can compute as follows:.
MATH 215 C10 - Self-Test: Unit 2
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k. T F If two events and are NOT independent, then we can compute as follows:.
3. A car dealership located on the outskirts of a large subdivision surveyed 200 of its regular customers in order to comparethe level of customer satisfaction of its urban customers with that of its rural customers. The survey responses aresummarized in the two-way classification table below.
a. If one customer is selected at random from the 200 customers that were surveyed, find the probability that thiscustomer: (show your answers to 2 decimals)
i. is an urban customer.
ii. is a rural customer or is very satisfied with the dealership’s service.
iii. is an urban customer and is not satisfied with the dealership’s service.
iv. is very satisfied with the dealership’s service, given that the customer is from an urban area.
v. is from a rural area, given that the customer is not satisfied with the dealership’s service.
b. Are the events Urban (U) and Very Satisfied (VS) mutually exclusive? Explain.
c. 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 ina classroom. In the past, 65% of students have taken the course in the classroom and 35% have taken it online. Given thatthe student takes the course in the classroom, they have an 80% chance of passing the course. Given that the studenttakes the course online for convenience, they have a 75% chance of passing the course.
a. A new student has just registered in Statistics 101. Draw a tree diagram describing the different possible outcomesthat face the student, in terms of whether the student takes the course in class or online, as well as whether thestudent ends up passing the course or not. At the end of the tree, display all possible joint probabilities. Keep yourwork to 4 decimals.
b. Compute the probability that the student will pass the course.
c. Are the events “takes the course in class” and “student will pass” independent events? Explain by making theappropriate math computations.
5. Consider the following experiment. You draw one card at random from a full deck of playing cards and observe whether itis a “hearts” card or not. You then replace the card in the deck, shuffle the deck, and then draw a second card. Youobserve whether this card is a hearts card or not. Note that in a deck of 52 cards there are 13 hearts cards.
a. Draw a tree diagram to describe all possible outcomes. At the end of the tree, display all possible jointprobabilities in one full play of this experiment. Keep your work to 4 decimals.
b. Find the probability that two hearts cards will be drawn in one full play of this experiment.
c. Find the probability that exactly one hearts card will be drawn in one full play of this experiment
d. Find the probability that no hearts card will be drawn in one full play of this experiment
6. For two events and :
a. Find .
b. Find .
c. Are the events and independent? Explain.
7. Consider an experiment in which you draw 4 cards, without replacement, from a standard deck of 52 cards. How manydistinct 4-card hands could you draw? Show your calculations.
MATH 215 C10 - Self-Test: Unit 2
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Show all your work and keep your calculations to four decimal places, unless otherwise stated.
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.
a. If an employee is randomly selected from this hospital, find the probability that
i. the employee will strongly agree with the flu shot statement.
Solution:
ii. the employee will either strongly agree or agree with the flu shot statement.
Solution:
Since both events are mutually exclusive:
iii. the employee will neither strongly agree nor agree with the flu shot statement. Use your answer to part ii.above to help answer this.
Solution:
b. Are the two events strongly agree and agree mutually exclusive? Explain.
Answer: Yes, the events are mutually exclusive, as for the same randomly selected employee the events SA and Acannot both happen at the same time. It has to be one or the other.
or
neither or or
MATH 215 C10 - Self-Test Answer Key: Unit 2
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2. Circle True (T) or False (F) for each of the following statements:
a. F The probability that the sample space will occur from the generation of a given experiment is equalto 1.
b. T An event that includes one and only one of the (final) outcomes for the generation of a givenexperiment is called a compound event.
c. T For any given event based on the generation of an experiment, the probability of the event willtypically exceed 1.
d. F Assigning probabilities to a compound event by dividing the number of simple outcomes associatedwith the event by the total possible number of simple outcomes in the sample space is consistent withthe classical concept of probability.
e. T If two events and are mutually exclusive, then we can state that .
f. T If , we can conclude that and are independent events.
g. F If and are complementary events, then .
h. T Depending on the experiment, it is possible to have the numeric probability of an event exceed 1.
i. F If the joint probability of events and equals zero, then .
j. F If two events and are independent, then we can compute as follows:
.
k. T If two events and are NOT independent, then we can compute as follows:
.
3. A car dealership located on the outskirts of a large subdivision surveyed 200 of its regular customers in order to comparethe level of customer satisfaction of its urban customers with that of its rural customers. The survey responses aresummarized in the two-way classification table below.
a. If one customer is selected at random from the 200 customers that were surveyed, find the probability that thiscustomer: (show your answers to 2 decimals)
i. is an urban customer.
Solution:
ii. is a rural customer or is very satisfied with the dealership’s service.
Solution:
iii. is an urban customer and is not satisfied with the dealership’s service.
T
F
F
T
F
F
T
F
T or
T and
and
F and
and
or and
MATH 215 C10 - Self-Test Answer Key: Unit 2
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Solution:
iv. is very satisfied with the dealership’s service, given that the customer is from an urban area.
Solution:
v. is from a rural area, given that the customer is not satisfied with the dealership’s service.
Solution:
b. Are the events Urban (U) and Very Satisfied (VS) mutually exclusive? Explain.
Answer: No, because a randomly selected customer can be both urban as well as very satisfied at the same time.
c. Are the events Urban (U) and Very Satisfied (VS) independent? Perform the appropriate math proof.
Solution:
Compare with
Since these two events are not equal, they are NOT independent.
OR
Compare with
Since these two events are not equal, they are NOT independent.
4. When a new student registers in a Statistics 101 course at a local university, they can decide to take the course online or ina classroom. In the past, 65% of students have taken the course in the classroom and 35% have taken it online. Given thatthe student takes the course in the classroom, they have an 80% chance of passing the course. Given that the studenttakes the course online for convenience, they have a 75% chance of passing the course.
a. A new student has just registered in Statistics 101. Draw a tree diagram describing the different possible outcomesthat face the student, in terms of whether the student takes the course in class or online, as well as whether thestudent ends up passing the course or not. At the end of the tree, display all possible joint probabilities. Keep yourwork to 4 decimals.
Solution:
Let and and and
b. Compute the probability that the student will pass the course.
and
In Class Online Passes Course Fails Course
MATH 215 C10 - Self-Test Answer Key: Unit 2
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Solution:
OR
c. Are the events “takes the course in class” and “student will pass” independent events? Explain by making theappropriate math computations.
Solution:
Take from part b. above.
Take from the question.
Since is NOT EQUAL to , the events “ ” and “ ”
are NOT independent.
5. Consider the following experiment. You draw one card at random from a full deck of playing cards and observe whether itis a “hearts” card or not. You then replace the card in the deck, shuffle the deck, and then draw a second card. Youobserve whether this card is a hearts card or not. Note that in a deck of 52 cards there are 13 hearts cards.
a. Draw a tree diagram to describe all possible outcomes. At the end of the tree, display all possible jointprobabilities in one full play of this experiment. Keep your work to 4 decimals.
Solution:
Let and
Note that and .
b. Find the probability that two hearts cards will be drawn in one full play of this experiment.
Solution:
c. Find the probability that exactly one hearts card will be drawn in one full play of this experiment.
Solution:
Passes Course and
and
Passes Course
Passes Given Takes Course in Class
Passes Course Passes Given Takes Course in Class
Hearts Selected Hearts Not Selected
and
MATH 215 C10 - Self-Test Answer Key: Unit 2
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© Athabasca University
d. Find the probability that no hearts card will be drawn in one full play of this experiment
Solution:
6. For two events and :
1. Find .
Solution:
2. Find .
Solution:
3. Are the events and independent? Explain.
Solution:
Compare the two probabilities:
, while .
Since is not equal to , the two events are NOT independent.
7. Consider an experiment in which you draw 4 cards, without replacement, from a standard deck of 52 cards. How manydistinct 4-card hands could you draw? Show your calculations.
Solution:
Since order does not matter, we are dealing with a combinations problem. Find .
and
and
and
and
and
C
distinct hands
MATH 215 C10 - Self-Test Answer Key: Unit 2
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