Secondary II
Probability
Teacher Edition
Unit 11
Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
NUCC| Secondary II Math i
Table of Contents
11.1 VENN DIAGRAMS (TB OR NOT TB)......................................................................................................1
Teacher Notes ..................................................................................................................................................1
Mathematics Content .......................................................................................................................................4
Mathematics Content .......................................................................................................................................5
Mathematics Content .......................................................................................................................................6
Probability – TB or Not TB, Part I A Develop Understanding Task 1a ..........................................................7
Probability – TB or Not TB, Part II A Develop Understanding Task 1b .........................................................8
Ready, Set, Go! ................................................................................................................................................9
Solutions: .......................................................................................................................................................11
11.2 RELATIONSHIPS .....................................................................................................................................12
Teacher Notes ................................................................................................................................................12
Mathematics Content .....................................................................................................................................16
Mathematics Content .....................................................................................................................................17
Mathematics Content .....................................................................................................................................18
Grandma’s Birthday A Solidify Understanding Task 2a ................................................................................19
Sally’s Error A Solidify Understanding Task 2b ............................................................................................20
Ready, Set, Go! ..............................................................................................................................................21
Solutions: .......................................................................................................................................................24
11.3 PROBABILITY EQUATION....................................................................................................................25
Teacher Notes ................................................................................................................................................25
Mathematics Content .....................................................................................................................................29
Mathematics Content .....................................................................................................................................30
Mathematics Content .....................................................................................................................................31
Titanic Tragedy, Part I A Develop Understanding Task 3a ...........................................................................32
Titanic Tragedy, Part II A Develop Understanding Task 3b ..........................................................................33
Titanic Tragedy, Part III A Develop Understanding Task 3c.........................................................................34
Screaming and Throwing Things A Solidify Understanding Task 3d ............................................................35
Solutions: .......................................................................................................................................................36
11.4 CATEGORICAL DATA ...........................................................................................................................37
Teacher Notes ................................................................................................................................................37
Mathematics Content .....................................................................................................................................40
Mathematics Content .....................................................................................................................................41
NUCC| Secondary II Math ii
Representation of Categorical Data, Part I A Practice Understanding Task 4a ............................................42
Representation of Categorical Data, Part II A Practice Understanding Task 4b ...........................................43
Representation of Categorical Data, Part III A Practice Understanding Task 4c ..........................................44
Representation of Categorical Data, Part IV A Practice Understanding Task 4d .........................................45
Ready, Set, Go! ..............................................................................................................................................46
Solutions: .......................................................................................................................................................48
11.5 DEPENDENT & INDEPENDENT EVENTS ...........................................................................................49
Teacher Notes ................................................................................................................................................49
Mathematics Content .....................................................................................................................................51
Mathematics Content .....................................................................................................................................52
Ready, Set, Go! ..............................................................................................................................................53
Solutions: .......................................................................................................................................................55
11.6H PERMUTATIONS & COMBINATIONS ..............................................................................................56
Teacher Notes ................................................................................................................................................56
Mathematics Content .....................................................................................................................................59
Ready, Set, Go! ..............................................................................................................................................60
Solutions: .......................................................................................................................................................63
NUCC| Secondary II Math iii
Unit 11.1
11.1 VENN DIAGRAMS (TB OR NOT TB)
Teacher Notes
Time Frame:
Materials Needed: TB or Not TB Task Sheet, Calculators
Purpose: Define and organize sample spaces using 2-way tables, Venn diagrams, and tree
diagrams. Students will:
1. Define and organize sample spaces in a variety of ways
2. Use 2-way tables, Venn Diagrams and tree diagrams to show and interpret characteristics
of outcomes, including: “and”, “or”, “not”, “complement”, & “conditional”.
Skills: Students are able to compute probabilities in the form of percentages (including
conditional) when given information in a 2-way table.
Core Standards Focus S.CP.1: Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or”, “and”, “not”).
S.CP.5: Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also
belong to A, and interpret the answer in terms of the model.
Vocabulary:
Two-Way Table – A table with one degree of freedom
Sample Space – The set of all possible outcomes of an event
Complement – The set of all possible outcomes that are not contained in the event.
Launch (Whole Class, Individual): Ask students what they know about TB. Discuss the
prevalence of tuberculosis in countries with inadequate health care. Infection rates in some
countries are as high as 30%, and consequently affect people traveling from country to country
and are deadly to many individuals, especially those with compromised immune systems. All
food handlers are required to take the TB skin test, which involves injecting a bubble of serum
underneath the skin, then, several days later, observing the skin to see if the person has shown a
reaction to the serum, which indicates the presence of TB antibodies.
Explore (Individual, Small Groups): Have students start on Part I of the task. Give them time to
work together to fill out the rest of the data table and answer the questions. Stop at the end of the
first page and discuss as a class how accurate the test is and how to make that determination.
Explore and develop differences among groups in their determination of the test’s accuracy.
Then have students complete Part II in their groups. Remind them to show their calculations and
be able to explain them to the rest of the class.
NUCC | Secondary II Math 1
Unit 11.1
Discuss (Whole Class or Group): Discuss each of the Part II questions as the students complete
them. Work through the questions as a class. Be sure to address the following questions:
1. What does the chance of a false positive mean? Is the probability of a false positive low or
high? How low is low? Compare to something like winning the lottery.
2. In this instance, which is worse, a false positive or a false negative? In what setting would a
false positive be worse?
3. How important is it to have a large sample space?
4. Discuss vocabulary: “sample space”, “set”, “subset”, “conditional probability”, “and”, “not”,
“or”, “complement”, & “given”. Include the notation for these with respect to probability.
NUCC | Secondary II Math 2
Unit 11.1
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 3
Unit 11.1
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
other events (“or,” “and,” “not”).
Concepts and Skills to Master



Use correct set notation, with appropriate symbols and words, to identify sets and subsets
within a sample space.
Identify an event as a subset of a set of outcomes (a sample space).
Draw Venn diagrams that show relationships (unions, intersections, or complements)
between sets within a sample space.
Critical Background Knowledge

Represent sample spaces. (7.SP.8).
Academic Vocabulary
Sample space, subset, outcome, union, intersection, complement, 1, c , Ac , A' ,  A, A (Note:
Various notations are commonly used for complement.)
Suggested Instructional Strategies


Create and use Venn diagrams to illustrate relationships between sample spaces and events.
Perform chance experiments, such as rolling dice or tossing coins, to generate sample spaces
and identify events within the same spaces.
Skills Based Task:
Describe the event that the summing two
rolled dice is larger than 7 and even, and
contrast it with the event that the sum is larger
than 7 or even.
Problem Task:
Create a Venn diagram to display the
information in the table. Describe the set of
students who have a curfew but don’t do chores
as a subset of the group.
Chores: Yes
Chores: No
Total
Curfew:
Yes
13
12
25
Curfew:
No
5
3
8
Total
18
15
Some Useful Websites:

http://www.shodor.org Interactivate Venn Diagram Shape Sorter


http://www.khanacademy.org/math/probability/v/basic-probability
http://www.khanacademy.org/math/probability/v/simple-probability
NUCC | Secondary II Math 4
Unit 11.1
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.5: Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. (For example, compare the chance
of having lung cancer if you are a smoker with the chance of being a smoker if you have lung
cancer .
Concepts and Skills to Master
 Interpret conditional probabilities and independence in context.
Critical Background Knowledge




Summarize categorical data in a variety of ways. (I.4.S.ID.5)
Find probabilities of events using tree diagrams. (7.SP.8)
Understand independence. (II.4.S.CP.2)
Understand and calculate conditional probabilities (II.4.S.CP.3)
Academic Vocabulary
conditional probability, independence
Suggested Instructional Strategies



Practice representing conditional probabilities using tree diagrams
Find the probability that a randomly selected athlete is an honors student.
Have students generate questions similar to the example in the standard and pursue the
answers.
Skills Based Task:
Problem Task:
Is owning a smart phone independent from
grade level?
Have students find and interpret probability
statements in media.
Own smart
phone
Do not own
smart phone
204
170
192
160
12th grade
198
Some Useful Websites:
165
10th grade
th
11 grade

http://www.shodor.org Interactivate Venn Diagram Shape Sorter


http://www.khanacademy.org/math/probability/v/basic-probability
http://www.khanacademy.org/math/probability/v/simple-probability
NUCC | Secondary II Math 5
Unit 11.1
Mathematics Content
Cluster Title: Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
Standard S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes
that also belong to A, and interpret the answer in terms of the model.
Concepts and Skills to Master


Find and interpret conditional probabilities using a two-way table, Venn diagram, or tree
diagram.
Understand the difference between compound and conditional probabilities.
Critical Background Knowledge


Find probabilities of compound events. (7.SP.8)
Summarize categorical data in two-way frequency tables. (I.4.S.ID.5)
Academic Vocabulary
random variable, probability model
Suggested Instructional Strategies


Make a “human Venn diagram” where the sample space is all the students in the class. Use
lengths of rope to create three overlapping circles. Assign an event to each of the three
circles, such as: ate breakfast, brought a cell phone to school, and got at least 7 hours of
sleep. Have students place themselves in the appropriate locations. Using correct probability
notation, identify each of the spaces in the Venn diagram (don’t forget to include the space
outside the circles). Analyze, explore and record the results in terms of conditional
probabilities.
Connect to probability models from other standards.
Skills Based Task:
Problem Task:
From the table, determine the probability of
getting the flu, and compare that to the
probability of getting the flu given that an
individual takes high doses of vitamin C.
Life is like a box of chocolates. Suppose your
box of 36 chocolates have some dark and some
milk chocolate, divided into cream or nutty
centers. Out of the dark chocolates, 8 have nutty
centers. Out of the milk chocolates, 6 have nutty
centers. One-third of the chocolates are dark
chocolate. What is the probability that you
randomly select a chocolate with a nutty center?
Given that it has a nutty center, what is the
probability you chose a dark chocolate? Show
how you determined your answers.
Cold
No Cold
Total
Placebo
31
109
140
Vitamin C
17
122
139
Total
48
231
279
Some Useful Websites:
NUCC | Secondary II Math 6
Unit 11.1
Probability – TB or Not TB, Part I
A Develop Understanding Task 1a
Name_____________________________________
If you ever had chicken pox, measles, rubella, or other viruses, then for
the rest of your life you will have antibodies for those diseases in your
blood, and a simple blood test will be able to detect the presence of those
antibodies. In a similar way, if someone has tuberculosis, then they will
have antibodies for tuberculosis in their blood.
Hour___________
Blood tests are designed to detect the presence of specific antibodies, but
the tests are not perfect. Sometimes, even if there are no antibodies
present, other factors will trigger the test to come back positive. So, a
positive test result isn’t absolute evidence that the person has the disease.
To test for tuberculosis (TB), patients are administered a skin test which involves injecting a
bubble of serum underneath the skin, then several days later, observing the skin to see if the
person has shown a reaction to the serum. A skin reaction indicates the presence of TB
antibodies. Suppose that a young adult with no prior evidence of tuberculosis has a skin reaction
to the serum. The person wonders “Do I really have tuberculosis?” Below are the test results
from a large sample of people from countries with insufficient health care.
Test is Positive
Has Tuberculosis (TB)
361
Doesn’t Have Tuberculosis (TB)
Total
Test is Negative
Total
380
558
423
Complete the missing parts of the table. Based on this large sample find the following values:
1)
What percent of the population of people from these countries are infected with TB? Or in
other words, has tuberculosis antibodies?
2)
What percent of individuals in those countries will have a positive test result?
How accurate is the tuberculosis (TB) test?
NUCC | Secondary II Math 7
Unit 11.1
Probability – TB or Not TB, Part II
A Develop Understanding Task 1b
Name_____________________________________
Hour___________
Identify the cells in the table that show counts for the number of individuals who got inaccurate
test results. Put a big frowny face on those cells.
To answer the person who asks “Do I really have tuberculosis?” we will explore some
probabilities.
1) Out of all those who tested positive, what percent of individuals actually had tuberculosis
antibodies? (The test accurately detected the disease.)
2) Out of all those who tested negative, what percent of individuals didn’t have TB antibodies?
(The test accurately said they were TB free.)
3) What percent of those with tuberculosis tested negative? (The test said they were TB free, but
they really had the disease. This is called a false negative.)
4) What percent of those who tested positive didn’t have tuberculosis (The
test said they had TB, but they really didn’t. This is called a false
positive.)
5) Which is worse, a false negative or a false positive? Why?
6) Most health tests are designed to be very sensitive, so sensitive that they
often come back with false positive results. Why do you suppose tests are
planned to be that way?
7) What will you answer the young adult who tested positive and asks, “Do I really have TB?”
NUCC | Secondary II Math 8
Unit 11.1
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Use table below to answer Questions 1 - 5.
Adult
Child
Total
Vanilla
52
26
78
Chocolate
41
105
146
Total
93
131
224
1. What percent of the people like Chocolate?
2. What percent of the Children like Vanilla?
3. What percent of those that like Chocolate are Adults?
4. What percent of the people surveyed were Children that liked Chocolate?
5. Is Chocolate more popular among Children or Adults? Explain your reasoning.
NUCC | Secondary II Math 9
Unit 11.1
Set
Complete the table below and put your answers in the spaces provided in questions 6-14.
The following table represents data from a survey of people asking them if they slept better after
eating a big meal. Data indicating whether or not the participants ate a big meal as well as
whether or not they slept well is recorded in the table below.
Big Meal
Not a Big Meal
Total
Slept Well
#6
505
1517
Didn’t Sleep Well
#7
299
#8
Total
#9
#10
2000
6.
7.
9.
10.
8.
Go!
11. Of those that slept well, what percentage ate a big meal?
12. Of those that ate a big meal, what percentage slept well?
13. What is the sample space of this survey? (What are the possible responses?)
14. From the survey data, would you conclude that eating a big meal will help you sleep well?
Why or why not?
NUCC | Secondary II Math 10
Unit 11.1
Solutions:
NUCC | Secondary II Math 11
Unit 11.2
11.2 RELATIONSHIPS
Teacher Notes
Time Frame:
Materials Needed: Grandma’s Birthday Task Sheet, Calculators
Purpose: Students will recognize the relationship between 2-way tables and Venn diagrams,
accurately representing a sample space and accounting for each outcome. Students will also be
able to relate the spaces in a Venn diagram to the language of probability, including “and”, “or”,
“complement”, and “conditional”.
Skills: Students are able to calculate probabilities from both a 2-way table as well as a Venn
diagram and recognize the connection between the two ways to display data.
Core Standards Focus S.CP.1: Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or”, “and”, “not”).
S.CP.5: Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
S.CP.7: Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the
answer in terms of the model.
Related Standards: S.CP.4 Construct and interpret two-way frequency tables of data when two
categories are associated with each object being classified. Use the two-way table as a sample
space and decide if events are independent and to approximate conditional probabilities.
S.CP.6: Find the conditional probability of A given B as the fraction of B’s outcomes that also
belong to A, and interpret the answer in terms of the model.
Vocabulary:
Disjoint Sets – Mutually exclusive sets where P(A or B) is P(A) + P(B)
Probability Notation:
P(A) – Probability of Event A Occurring
P(A B) – Probability of Event A Occurring Given Event B
Complement - - A,  A, orAN
OR – (This is a union of both sets)
AND (This is the intersection of both sets)
Launch (Whole Class): Talk about Grandma Addams. Possibly use Wikipedia or a video clip
explaining who she is. Explain to the class that she is having a birthday party with people that are
friends, relatives, some that are both, and some that are neither. Pass out the "Grandma's
Birthday" Task Sheet.
NUCC | Secondary II Math 12
Unit 11.2
Explore (Small Groups): Have groups fill out their Venn diagrams. There are at least 4 different
Venn Diagrams that can be created from the given data. The lack of labels is deliberate. Look for
different representations to show to the class. Common Errors to look for are:
1.
Labeling the two circles as “related” and “not related” and then not knowing what to do
about the category of friend.
2.
Not subtracting the overlapping amount from both “Friend” and “Relative”.
3.
The total number in the party room may not equal 200.
4.
Students may forget about the outcomes that aren’t included in the circles.
5.
In normal English, “or” is exclusive, but in probability, “or” is inclusive.
Discuss (Whole Class or Group): Use student work to discuss the connections between the 2way table and the Venn diagram. Address the following:
1.
Is there any "overlapping" between "Friend" and "Relative"?
2.
Address notation of each of the areas on the Venn diagram.
3.
Match the 4 areas on the Venn diagram to the interior cells of the 2-way table.
4.
Point out that Venn diagrams show relationships and 2-way tables show totals.
5.
Discuss abbreviations for probability areas, and notation including "and", "or", and
"complement."
6.
What is the probability that a randomly selected person from the party is either a friend or a
relative = P(Friend or Relative)? Talk about using counts vs. percents and the advantages of
each. Find the probability using the 2-way table.
7.
Develop the equation of the addition rule: P(A or B) = P(A) + P(B) - P(A and B).
Use the "Sally's Error" Worksheet after the discussion to solidify understanding and reinforce the
connection between the probability "Notation" and the Venn diagram. Pass it out and have
NUCC | Secondary II Math 13
Unit 11.2
students work on it individually, then in groups or partners. Then use student work to facilitate a
discussion on Sally's Error and their answers to the prompts.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 14
Unit 11.2
NUCC | Secondary II Math 15
Unit 11.2
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
other events (“or,” “and,” “not”).
Concepts and Skills to Master



Use correct set notation, with appropriate symbols and words, to identify sets and subsets
within a sample space.
Identify an event as a subset of a set of outcomes (a sample space).
Draw Venn diagrams that show relationships (unions, intersections, or complements)
between sets within a sample space.
Critical Background Knowledge

Represent sample spaces. (7.SP.8).
Academic Vocabulary
Sample space, subset, outcome, union, intersection, complement, 1, c , Ac , A' ,  A, A (Note:
Various notations are commonly used for complement.)
Suggested Instructional Strategies


Create and use Venn diagrams to illustrate relationships between sample spaces and events.
Perform chance experiments, such as rolling dice or tossing coins, to generate sample spaces
and identify events within the same spaces.
Skills Based Task:
Describe the event that the summing two
rolled dice is larger than 7 and even, and
contrast it with the event that the sum is larger
than 7 or even.
Problem Task:
Create a Venn diagram to display the
information in the table. Describe the set of
students who have a curfew but don’t do chores
as a subset of the group.
Chores: Yes
Chores: No
Total
Curfew:
Yes
13
12
25
Curfew:
No
5
3
8
Total
18
15
Some Useful Websites:
 www.youtube.com/watch?v=bLNfsh8Ax38
 www.youtube.com/watch?v=g7CQCa77lzs
 www.youtube.com/watch?v=nJnYhOwDKDk
NUCC | Secondary II Math 16
Unit 11.2
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.5: Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. (For example, compare the chance
of having lung cancer if you are a smoker with the chance of being a smoker if you have lung
cancer .
Concepts and Skills to Master
 Interpret conditional probabilities and independence in context.
Critical Background Knowledge




Summarize categorical data in a variety of ways. (I.4.S.ID.5)
Find probabilities of events using tree diagrams. (7.SP.8)
Understand independence. (II.4.S.CP.2)
Understand and calculate conditional probabilities (II.4.S.CP.3)
Academic Vocabulary
conditional probability, independence
Suggested Instructional Strategies



Practice representing conditional probabilities using tree diagrams
Find the probability that a randomly selected athlete is an honors student.
Have students generate questions similar to the example in the standard and pursue the
answers.
Skills Based Task:
Problem Task:
Is owning a smart phone independent from
grade level?
Have students find and interpret probability
statements in media.
Own smart
phone
Do not own
smart phone
204
170
192
160
12th grade
198
Some Useful Websites:
165
10th grade
th
11 grade

http://www.shodor.org Interactivate Venn Diagram Shape Sorter


http://www.khanacademy.org/math/probability/v/basic-probability
http://www.khanacademy.org/math/probability/v/simple-probability
NUCC | Secondary II Math 17
Unit 11.2
Mathematics Content
Cluster Title: Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
Standard S.CP.7: Apply the Addition rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret
the answer in terms of the model.
Concepts and Skills to Master


Define the probability of event (A or B) as the probability of their union.
Understand and use the formula (P(A or B) = P(A) + P(B) – P(A and B).
Critical Background Knowledge

Find probabilities of compound events. (7.SP.8)
Academic Vocabulary
Or, and, P(A)l, ^ , _
Suggested Instructional Strategies

Make a connection between the formula for the addition rule and a probability model.
Skills Based Task:
Problem Task:
Given the following table, which includes
data regarding boating preferences of boys
and girls, use the Addition Rule to find P(L ^
G).
Sally shaded the following Venn diagram to
illustrate the Addition Rule. What was wrong
with her reasoning? How could you represent
the addition rule
pictorially?
Lake (L)
River (R)
21
29
Boys (B)
32
Some Useful Websites:
18
Girls (G)
NUCC | Secondary II Math 18
Unit 11.2
Grandma’s Birthday
A Solidify Understanding Task 2a
Name_____________________________________
Hour___________
You’ve been invited to Grandma Addam’s birthday party at the haunted mansion! All your crazy
relatives and friends will be there. When you arrive, this is what you discover:




200 people are at the party
24 are relatives
43 are neither a friend or a relative
20 are both a friend and a relative
How many of your friends came to the party?
Note: a friend is anyone you’ve met. You are that kind of guy or gal.
Once you’ve completed the Venn diagram, create a two-way table that displays the same data.
What information is more obvious from the Venn diagram?
What information is more obvious from the two-way table?
What is the probability that a randomly selected individual is a friend or a relative, that is:
P(Friend or Relative).
NUCC | Secondary II Math 19
Unit 11.2
Sally’s Error
A Solidify Understanding Task 2b
Name_____________________________________
Hour___________
Sally was assigned to create a Venn diagram to represent 𝑃(𝐴 or 𝐵). Sally remembers that
𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 and 𝐵), so she creates the following diagram.
What was Sally’s error?
Make a Venn diagram that correctly represents 𝑃(𝐴 or 𝐵).
Create a Venn Diagram for 𝑃(𝐴 or 𝐵) such that A and B are disjoint1.
1
Disjoint sets are also known as mutually exclusive sets.
NUCC | Secondary II Math 20
Unit 11.2
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Use the information from Task 2 to answer the following questions. Use F to represent “Friend”
and R to represent “Relative.”
1.
Find P(F)
2.
Find P(R)
3.
Find P(R’)
4.
Find P(F’)
5.
Find P(R|F)
6.
Find P(F|R)
7.
Find P(F|R’)
8.
Find P(R’|F’)
NUCC | Secondary II Math 21
Unit 11.2
Set
For exercises 9-24, use the following information.
Let U = 0,1,2,3,4,5,6,7,8,9,10,11,12
A = 1,2,6,9,10,12 , B = 2,9,10 , C = 0,1,6,9,11 ,
D = 4,5,10, E = 2,3,6 , and F = 2,9 .
Determine whether each statement is true or false. Explain your reasoning
9. 3 D
10.
8 A
11. B  A
12.
UA
13. 5 ó D
14.
2 E
15. 0  F
16.
6ó F
17. CN
18.
UN
19. AN
20.
D
E
E
22.
E
F
23. A B
24.
A B
Find each of the following.
21. C
NUCC | Secondary II Math 22
Taken from the Glencoe McGraw-Hill Precalculus Text book page P5
Unit 11.2
Go!
Use the Venn diagram to find each of the following.
25. A
B
26.
A D
27. C
D
28.
AN
30.
( A B) C
29. A B
D
NUCC | Secondary II Math 23
Taken from the Glencoe McGraw-Hill Precalculus Text book page P5
Unit 11.2
Solutions:
NUCC | Secondary II Math 24
Unit 11.3
11.3 PROBABILITY EQUATION
Teacher Notes
Time Frame:
Materials Needed: Titanic Tragedy Task Sheet, Calculators
Purpose: Students will understand how the probability equation relates to the proportionality of
independent events and know how to apply the probability equation to determine if two events
are independent of one another or not.
Skills: Students will be able to compute conditional probabilities and joint probabilities, and
understand how to use the proportionality equation of independent events to determine if events
are independent or dependent.
Core Standards Focus S.CP.2 Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
the independence of A and B as saying that the conditional probability of A given B is the same
as the probability of A, and the conditional probability of B given A is the same as the
probability of B.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space and decide
if events are independent and to approximate conditional probabilities.
S.CP.5 Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
Related Standards Focus S.CP.6 Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms of the model.
Vocabulary:
Proportionality Equation of Independent Events - P(A)=P(A|B)
Conditional Probability P(A|B) = P(A and B) / P(B)
Joint Probability P(A and B) = P(A)P(B)
General Multiplication Rule P(A and B) = P(A)P(B|A) = P(B)P(A|B)
NUCC | Secondary II Math 25
Unit 11.3
Launch (Whole Class): Give some background on the tragedy of the Titanic. Talk about what
students know.
NUCC | Secondary II Math 26
Unit 11.3
Explore (Individual, small group or pairs): Distribute the “Titanic Tragedy Part I” task sheet
and allow student to explore the data. Choose some individuals or groups to share their graphical
representations, numerical calculations, and conclusions about independence.
 Compare and contrast different representations and ask students to find similarities. Look
for common mistakes and things that are misleading because students used counts instead
of percentages.
 Expect to see Venn diagrams and 2-way tables. Students may also incorporate graphs.
Point out how graphs don’t show the relationship between survival and gender.
 It is important to bring out that survival on the Titanic was dependent on various
categories.
Distribute the “Titanic Tragedy Part II” task sheet. Have students analyze the survival rate and
the class of the passenger. Share and discuss similar to Part I.
Distribute the “Titanic Tragedy Part III” task sheet.
 Ask students what the probability of survival is. As a class, look at P(Survival),
P(Survive|Female), P(Survive|Male), and compare them.
 As students work through Part III, guide them to arrive at the solution that survival is
independent of gender when the overall survival rate is equal to the survival rates of both
men and women.
Discuss (Whole Class or Group): Ask students to summarize the characteristics of data that
would indicate independence between categories, i.e., that proportionality between categories of
subgroups will equal each other and that of the overall population.



Formalize the proportionality relationship of independent events with the probability
equation, P(A)=P(A|B) [i.e., P(Survive) = P(Survive|Female) = P(Survive|Male)]
The focus of this discussion is the equation of independent events. Additional probability
equations such as Conditional Probability [P(A|B) = P(A and B) / P(B)], Joint Probability
[P(A and B) = P(A)P(B)], or the General Multiplication Rule [P(A and B) = P(A)P(B|A) =
P(B)P(A|B)] may come up but they are not the focus of this task. It would be good to link
them as reminders and build connections, if they surface.
Relate the equation back to the Titanic data and verify conclusions.
NUCC | Secondary II Math 27
Unit 11.3
Distribute the “Screaming and Throwing Things” task and have students determine if there is
independence between categories. Verify with calculations using both intuitive estimate and
probability calculations.
NUCC | Secondary II Math 28
Unit 11.3
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.2: Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this characterization to
determine if they are independent.
Concepts and Skills to Master
 Use appropriate probability notation for individual events as well as their intersection (joint
probability).
 Calculate probabilities for events, including joint probabilities, using various methods (e.g.
Venn diagram, frequency table).
 Compare the product of probabilities for individual events ( P ( A) AP ( B )) with their joint
probability ( P( A_ B)).
 Understand that independent events satisfy the relationship P( A) AP( B)  P( A_ B).
Critical Background Knowledge
 Understand basic properties of probability. (7.SP.5)
 Approximate probabilities of chance events through experiment. (7.SP.6)
 Use Venn diagrams (II.4.S.CP.1) and two-way frequency tables. (I.S.ID.5)
 P( A_ B) is the equivalent of the probability of event A and event B occurring together.
(II.4.S.CP.1)
Academic Vocabulary
joint probability, intersection, event independent events, P( A), P( A_ B), P ( A and B )
Suggested Instructional Strategies
 Convert frequencies from a Venn diagram or a two-way frequency table into probabilities
with correct notation.
 Generate a two-way frequency table to describe characteristics of your class (e.g., gender and
eye color) and use the table to determine if eye color and gender are independent.
 Compare experimental results to theoretical (long run) probabilities.
Skills Based Task:
Problem Task:
When rolling two dice:
Roll a pair of dice 100 times and keep track of
1) What is the probability of rolling a sum
the outcomes. Find pairs of events that are
that is greater than 7?
independent and pairs that are not. Justify your
2) What is the probability of rolling a sum
conclusions. (For example, the probability of
that is odd?
rolling doubles and the probability of rolling 7
3) What is the probability of rolling a sum
vs. the probability of rolling doubles and the
that is greater than 7 and is odd?
probability of rolling a sum that is less than 4.)
4) Are the events rolling a sum greater than 7
and rolling a sum that is odd independent?
Justify.
Some Useful Websites:
http://www.mathgoodies.com/lessons/vol6/conditional.html
http://www.stat.yale.edu/Courses/1997-98/101/conprob.htm
http://www.khanacademy.org/math/probability/v/conditional-probability-and-combinations
NUCC | Secondary II Math 29
Unit 11.3
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and
interpret independence of A and B as saying that the conditional probability of A given B is the
same as the probability of A, and the conditional probability of B given A is the same as the
probability of B.
Concepts and Skills to Master
 Understand conditional probability and how it applies to real-life events.
P( A_ B )
 Use P ( A#B ) 
to calculate conditional probabilities.
P( B )

Understand that events A and B are independent if and only if they satisfy P( A-B )  P( A) or

satisfy P( B#A)  P ( B ).
Apply the definition of independence to a variety of change events.
Critical Background Knowledge


Use basic probability notation, particularly PA_ B). (II.4.S.CP.2)
Understand independent events (II.4.S.CP.2)
Academic Vocabulary
conditional, independence, conditional probability, P ( A#B )
Suggested Instructional Strategies

Use Venn diagrams to explore and compute conditional probabilities.
Skills Based Task:
Problem Task:
Given the following
Venn diagram,
determine whether
events A and B are
independent.
Is participation in sports independent of
participation in the arts?
Some Useful Websites:
Cut the Knot – Conditional Probability and Independent Events: http://www.cut-theknot.org/Curriculum/Probability/ConditionalProbability.shtml
Texas A&M – Conditional Probability Applet:
http://www.stat.tamu.edu/~west/applets/Venn1.html
NUCC | Secondary II Math 30
Unit 11.3
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.4: Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to decide if events
are independent and to approximate conditional probabilities. (For example, collect data from a random
sample of students in your school on their favorite subject among math, science and English. Estimate the
probability that a randomly selected student from your school will favor science given that the student is
in tenth grade. Do the same for other subjects and compare the results.)
Concepts and Skills to Master


Model real-life data using two-way frequency tables.
Recognize that the conditional probability, represents the joint probability for A and B divided by the
marginal probability of B.

Use P( A-B ) 

Apply the definition of independence to a variety of chance events as represented by a two-way
frequency table.
P( A_ B )
to calculate conditional probabilities from a two-way frequency table.
P( B )
Critical Background Knowledge

Summarize categorical data in a variety of ways. (I.4.S.ID.5)
Understand what it means for two events to be independent. (II.S.CP.2)

Academic Vocabulary
conditional, independence, joint probability ( P( A_ B)), conditional probability ( P( A_ B)), marginal
probability
Suggested Instructional Strategies

Construct two-way tables based on data from news media and investigate independence by computing
conditional probabilities.
Analyze two-way tables to determine independence and conditional probability.

Skills Based Task:
1.
Find the probability that
Gender
Summer Summer
Total
a randomly selected
School
Job
student attends summer
school.
Girls
25
20
2. Find the probability that
Boys
35
20
a student is a boy given
that they attend summer
Total
school.
3. Find the probability that a randomly selected student is a boy who
attends summer school.
4. Are the events “Attending Summer School” and “Boys” independent?
Justify you answer.
Problem Task:
Select two categorical variables
and conduct research to answer
various probability questions and
determine independence. Write a
“newsworthy” article for the
school newspaper that interprets
the interesting relationships
between the events.
Some Useful Websites:
Data and Story Library (DASL): http://lib.stat.cmu.edu/cgibin/dasl.cgi?query=Contingency+table&submit=Search!&metaname=nethods&sort=swishrank
NUCC | Secondary II Math 31
Unit 11.3
Titanic Tragedy, Part I
A Develop Understanding Task 3a
Name_____________________________________
Hour___________
Data from http://en.wikipedia.org/wiki/RMS_Titanic
Art by Willy Stower
The RMS Titanic had 2,224 people on board ship (including the children), yet there was lifeboat
space for only 1,178. Various factors on that fateful day also made it so many lifeboats were not
full when they were launched.
Women
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Number
aboard
144
93
165
23
425
Number
saved
140
80
76
20
316
Men
Number
lost
4
13
89
3
109
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Number
aboard
175
168
462
885
1690
Number
saved
57
14
75
192
338
Number
lost
118
154
387
693
1352
Note: in addition to the adults there were 109 children, undifferentiated by gender, of which 56 died and 53 survived.
Create a variety of graphical representations to show relationships between survival and gender.
Historically when populations are in danger, societies protect women. Does the Titanic data
support this priority? In other words, is survival rate independent from gender? Use your
graphical representations and numerical evidence to support your conclusions.
NUCC | Secondary II Math 32
Unit 11.3
Titanic Tragedy, Part II
A Develop Understanding Task 3b
Name_____________________________________
Hour___________
Class of travel may also have influenced the survival rate, where first-class passengers received
special treatment in boarding the lifeboats, while some other passengers were prevented from
boarding because of lack of space.
Was the survival rate of the passengers’ independent from their class of travel? Give evidence to
show that class of travel was or was not independent from survival rate. Do not include the crew
member data in your analysis.
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Women
Number
Number
aboard
saved
144
140
93
80
165
76
23
20
425
316
Number
lost
4
13
89
3
109
Person
category
1st Class
2nd Class
3rd Class
Crew
Total
Men
Number
Number
aboard
saved
175
57
168
14
462
75
885
192
1690
338
Number
lost
118
154
387
693
1352
NUCC | Secondary II Math 33
Unit 11.3
Titanic Tragedy, Part III
A Develop Understanding Task 3c
Name_____________________________________
Hour___________
Suppose we are in an alternate universe where gender was not a factor influencing survival on
the Titanic.
Create values for the table showing there is no influence on survival based on gender.
Survived
Died
Total
Men
1690
Women
425
Total
654
1461
2115
Explain clearly why the values you chose indicate there is independence between gender and
survival rate.
NUCC | Secondary II Math 34
Unit 11.3
Screaming and Throwing Things
A Solidify Understanding Task 3d
Name_____________________________________
Hour___________
Do people find greater stress relief from throwing things
or screaming loudly?
Feel Better
Don’t Feel Better
Total
Throwing things
112
28
140
Screaming loudly
224
56
280
Total
336
84
420
NUCC | Secondary II Math 35
Unit 11.3
Solutions:
NUCC | Secondary II Math 36
Unit 11.4
11.4 CATEGORICAL DATA
Teacher Notes
Time Frame:
Materials Needed: Categorical Data Task Sheet, Calculators
Purpose: To reinforce the connection between different representations of data and the concept
of independence.
Skills: Students will be able to represent data in a two-way table, Venn diagram, tree-diagram,
and using “Probability Notation.” Students will be able to compute conditional probabilities and
determine if events are independent or dependent.
Core Standards Focus S.CP.2 Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their probabilities, and use this
characterization to determine if they are independent.
S.CP.4 Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space and decide
if events are independent and to approximate conditional probabilities.
Related Standards Focus S.CP.3 Understand the conditional probability of A given B as P(A
and B)/P(B), and interpret the independence of A and B as saying that the conditional probability
of A given B is the same as the probability of A, and the conditional probability of B given A is
the same as the probability of B.
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also
belong to A, and interpret the answer in terms of the model.
Vocabulary:
Tree Diagram – A graphical representation of probabilities or numbers where each branch of the
tree adds a level of conditionality or possibility.
Launch (Whole Class): Treat the four parts of the task as a puzzle with each section having
missing pieces. Distribute the tasks to the students.
NUCC | Secondary II Math 37
Unit 11.4
NUCC | Secondary II Math 38
Unit 11.4
Explore (Individual, small group or pairs): Students complete the missing parts. The goal is to
solidify the connections between the multiple representation s that have been developed.
Discuss (Whole Class or Group): Allow students to verify solutions and self-correct. Discuss
that the purpose of many surveys and research is to establish either dependence or independence
between variables.
NUCC | Secondary II Math 39
Unit 11.4
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.2: Understand that two events A and B are independent if the probability of A
and B occurring together is the product of their probabilities, and use this characterization to
determine if they are independent.
Concepts and Skills to Master
 Use appropriate probability notation for individual events as well as their intersection (joint
probability).
 Calculate probabilities for events, including joint probabilities, using various methods (e.g.
Venn diagram, frequency table).
 Compare the product of probabilities for individual events ( P ( A) AP ( B )) with their joint
probability ( P( A_ B)).
 Understand that independent events satisfy the relationship P( A) AP( B)  P( A_ B).
Critical Background Knowledge
 Understand basic properties of probability. (7.SP.5)
 Approximate probabilities of chance events through experiment. (7.SP.6)
 Use Venn diagrams (II.4.S.CP.1) and two-way frequency tables. (I.S.ID.5)
 P( A_ B) is the equivalent of the probability of event A and event B occurring together.
(II.4.S.CP.1)
Academic Vocabulary
joint probability, intersection, event independent events, P( A), P( A_ B), P ( A and B )
Suggested Instructional Strategies
 Convert frequencies from a Venn diagram or a two-way frequency table into probabilities
with correct notation.
 Generate a two-way frequency table to describe characteristics of your class (e.g., gender and
eye color) and use the table to determine if eye color and gender are independent.
 Compare experimental results to theoretical (long run) probabilities.
Skills Based Task:
Problem Task:
When rolling two dice:
Roll a pair of dice 100 times and keep track of
5) What is the probability of rolling a sum
the outcomes. Find pairs of events that are
that is greater than 7?
independent and pairs that are not. Justify your
6) What is the probability of rolling a sum
conclusions. (For example, the probability of
that is odd?
rolling doubles and the probability of rolling 7
7) What is the probability of rolling a sum
vs. the probability of rolling doubles and the
that is greater than 7 and is odd?
probability of rolling a sum that is less than 4.)
8) Are the events rolling a sum greater than 7
and rolling a sum that is odd independent?
Justify.
Some Useful Websites:
www.youtube.com/watch?v=mkDzmI7YOx0
www.youtube.com/watch?v+4lvkf53g16A
www.onlinemathlearning.com/tree-diagram.html
NUCC | Secondary II Math 40
Unit 11.4
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.4: Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to decide if events
are independent and to approximate conditional probabilities. (For example, collect data from a random
sample of students in your school on their favorite subject among math, science and English. Estimate the
probability that a randomly selected student from your school will favor science given that the student is
in tenth grade. Do the same for other subjects and compare the results.)
Concepts and Skills to Master


Model real-life data using two-way frequency tables.
Recognize that the conditional probability, represents the joint probability for A and B divided by the
marginal probability of B.

Use P( A-B ) 

Apply the definition of independence to a variety of chance events as represented by a two-way
frequency table.
P( A_ B )
to calculate conditional probabilities from a two-way frequency table.
P( B )
Critical Background Knowledge

Summarize categorical data in a variety of ways. (I.4.S.ID.5)
 Understand what it means for two events to be independent. (II.S.CP.2)
Academic Vocabulary
conditional, independence, joint probability ( P( A_ B)), conditional probability ( P( A_ B)), marginal
probability
Suggested Instructional Strategies

Construct two-way tables based on data from news media and investigate independence by computing
conditional probabilities.
Analyze two-way tables to determine independence and conditional probability.

Skills Based Task:
5.
Find the probability that
Gender
Summer Summer
Total
a randomly selected
School
Job
student attends summer
school.
Girls
25
20
6. Find the probability that
Boys
35
20
a student is a boy given
that they attend summer
Total
school.
7. Find the probability that a randomly selected student is a boy who
attends summer school.
8. Are the events “Attending Summer School” and “Boys” independent?
Justify you answer.
Problem Task:
Select two categorical variables
and conduct research to answer
various probability questions and
determine independence. Write a
“newsworthy” article for the
school newspaper that interprets
the interesting relationships
between the events.
Some Useful Websites:
Data and Story Library (DASL): http://lib.stat.cmu.edu/cgibin/dasl.cgi?query=Contingency+table&submit=Search!&metaname=nethods&sort=swishrank
NUCC | Secondary II Math 41
Unit 11.4
Representation of Categorical Data, Part I
A Practice Understanding Task 4a
Name_____________________________________
What’s your favorite color?
Hour___________
When asked this question, the most popular color named was blue.
Symbols
Key:
Male = M
Blue = B
2-way Table
Female = F
Not Blue = N
Blue
Not Blue
Total
Sample size = 200
Male
P(B) = 84/200
P(M) = 64/200
Female
P(F|B) = 48/84
Total
P(B|F) =
P(MB) =
Is color preference independent of gender?
How do you know?
P(MB) =
Venn Diagram
Tree Diagram
NUCC | Secondary II Math 42
Unit 11.4
Representation of Categorical Data, Part II
A Practice Understanding Task 4b
Name_____________________________________
Hour___________
Are you a lefty or a righty?
Symbols
Key:
Male = M
Lefty = L
2-way Table
Female = F
Righty = R
Lefty
Righty
Total
Sample size =
Male
P(L) =
P(M) =
Female
P(F) =
Total
P(L|F) =
Is “handedness” independent from gender?
How do you know?
P(L|M) =
In this sample are there equal proportions of
males and females who are left handed?
Explain.
Venn Diagram
Tree Diagram
NUCC | Secondary II Math 43
Unit 11.4
Representation of Categorical Data, Part III
A Practice Understanding Task 4c
Name_____________________________________
Hour___________
Do you eat breakfast or not?
Symbols
Key:
Male = M
Eats Breakfast = E
D
2-way Table
Female = F
Doesn’t Eat Breakfast =
Sample size =
Eats
Doesn’t
Total
Male
P(E) =
Female
P(E|M) =
685
P(EM)
Total
P(E|F) =
Is eating breakfast (or not) independent from
gender? How do you know?
P(EF)
Venn Diagram
Tree Diagram
NUCC | Secondary II Math 44
Unit 11.4
Representation of Categorical Data, Part IV
A Practice Understanding Task 4d
Name_____________________________________
Hour___________
The Humane Society likes to keep track of the percent of people
who are dog and cat owners. Some people own only dogs, some
only own cats, and some own both.
The Human Society reports that 13% of households own dogs
and cats, 33% own cats, 39% own dogs, and 41% don’t own
either.
Is ownership of dogs vs. cats independent? Justify your response with appropriate
representations of the data and numerical calculations.
NUCC | Secondary II Math 45
Unit 11.4
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
1. Draw a tree diagram representing the outcomes for flipping a coin then tossing a dice.
a. How many outcomes are there? _______
b. What is the probability of getting a heads, then an even number? _____
2. The probability that a student passes the driver’s ed. written test is 62%. The probability that a student
passes the driving part of the test is 86%. Draw a tree diagram showing the different outcomes (two
branches then two more branches).
a. Find the end probability for each of the branches of your diagram.
b. Add up all the end probabilities. They add up to_____
c. What is the probability that a student passes both tests?
d. What is the probability that a student passes only one of the tests?
3. Every day that you drive to school you break the speed limit. The probability that you won’t get a
ticket is 98%. Wahoo! You think you will never get a ticket! You will make 180 trips to school this
year.
a. What is the probability that you won’t get any tickets for the whole year?
b. What is the probability that you will get at least one ticket?
Set
4. Your drawer is full of socks that aren’t paired up. You have 6 red socks, 4 black socks, and 5 green
socks (you lost one). You get dressed in the dark because you don’t want to wake up your puppy.
What is the probability that you choose two matching socks?
NUCC | Secondary II Math 46
Unit 11.4
5. The probability that a student plays on the school basketball team is 10%. If a student plays high
school ball, the probability they play college ball is 8%. If they play college ball, the probability that
they play professional basketball is 3%. What is the probability that a student will play at all levels of
basketball (high school, college, and professional?)
Go!
6. Chris’s kidneys have failed and he is awaiting a kidney transplant. His doctor gives him this
information for patients in his condition: 90% survive the transplant, and 10% die. The transplant
succeeds in 60% of those who survive, and the other 40% must return to kidney dialysis. The
proportions who survive five years are 70% for those with a new kidney and 50% for those who
return to dialysis. Chris draws a tree diagram representing the different outcomes.
a. What is the probability that Chris will survive more than 5 years and that his new kidney will
work?
b. What is the total probability that Chris will survive more than 5 years?
c. What is the total probability that Chris will not survive more than 5 years?
d. There are a lot of risks to having a kidney transplant. If you were Chris, what would you decide?
7. In the Parade Magazine on Dec. 25, 2011, The “Ask Marilyn” Sunday question was: I manage a
drug-testing program for an organization with 400 employees. Every 3-months a random number
generator selects 100 names for testing. Then these names go back into the pool. Obviously, the
probability of an employee being chosen in one quarter is 25%. But what’s the likelihood of being
chosen over the course of a year?
Marilyn’s answer was: “The probability remains 25%, despite the repeated testing. One
might think that as the number of tests grows, the likelihood of being chosen increases, but
as long as the size of the pool remains the same, so does the probability. Goes against your
intuition, doesn’t it?
a. What was wrong with Marilyn’s reasoning?
b. What needs to be changed in Marilyn’s answer to make it correct?
NUCC | Secondary II Math 47
Unit 11.4
Solutions:
NUCC | Secondary II Math 48
Unit 11.5
11.5 DEPENDENT & INDEPENDENT EVENTS
Teacher Notes
Time Frame:
Materials Needed: Dependent & Independent Task Sheet, Calculators
Purpose: The purpose of this task is to solidify student understanding of dependent and
independent events, specifically being able to compute the probability of various dependent and
independent events, applying the general multiplication rule.
Skills: Students will be able to determine the probability of various events, both dependent and
independent.
Core Standards Focus S.CP.1 Describe events as subsets of a sample space (the set of
outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or
complements of other events (“or”, “and”, “not”).
S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the
answer in terms of the model.
Related Standards Focus S.CP.4 Construct and interpret two-way frequency tables of data
when two categories are associated with each object being classified. Use the two-way table as a
sample space and decide if events are independent and to approximate conditional probabilities.
S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also
belong to A, and interpret the answer in terms of the model.
Vocabulary:
General Multiplication Rule: P(A and B) = P(A)P(B|A) = P(B)P(A|B)
Launch (Whole Class): This is a teacher lead task based activity, but will solidify understanding
with the students. Provide students with examples of problems where they have to find the
probability of events occurring that are both dependent and independent. The “Ready, Set, Go”
assignment can be referenced.
Note: No worksheet to hand to students.
NUCC | Secondary II Math 49
Unit 11.5
Explore (Individual, small group or pairs): For example: Find the probability of drawing from
a standard 52-card deck a King, then a 10.
 4  4 
Without Replacement: P  K then 10      
 52   51 
With Replacement:
 4  4 
P  K then 10     
 52  52 
Discuss (Whole Class or Group): Be sure to note how the sample space changes for dependent
events and doesn’t for independent events. Also note that the probability of events where
elements exist in both A and B are not addressed in this section.
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 50
Unit 11.5
Mathematics Content
Cluster Title: Understand independence and conditional probability and use them to
interpret data.
Standard S.CP.1: Describe events as subsets of a sample space (the set of outcomes) using
characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
other events (“or,” “and,” “not”).
Concepts and Skills to Master



Use correct set notation, with appropriate symbols and words, to identify sets and subsets
within a sample space.
Identify an event as a subset of a set of outcomes (a sample space).
Draw Venn diagrams that show relationships (unions, intersections, or complements)
between sets within a sample space.
Critical Background Knowledge

Represent sample spaces. (7.SP.8).
Academic Vocabulary
Sample space, subset, outcome, union, intersection, complement, 1, c , Ac , A' ,  A, A (Note:
Various notations are commonly used for complement.)
Suggested Instructional Strategies


Create and use Venn diagrams to illustrate relationships between sample spaces and events.
Perform chance experiments, such as rolling dice or tossing coins, to generate sample spaces
and identify events within the same spaces.
Skills Based Task:
Describe the event that the summing two
rolled dice is larger than 7 and even, and
contrast it with the event that the sum is larger
than 7 or even.
Problem Task:
Create a Venn diagram to display the
information in the table. Describe the set of
students who have a curfew but don’t do chores
as a subset of the group.
Chores: Yes
Chores: No
Total
Curfew:
Yes
13
12
25
Curfew:
No
5
3
8
Total
18
15
Some Useful Websites:
www.mathgoodies.com/lessons/vol6/independent_events.html
www.khanacademy.org/math/.../v/independent-events-2
www.mathsisfun.com/data/probability-events-independent.html
www.regentsprep.org/Regents/math/ALGEBRA/APR6/Lindep.htm
NUCC | Secondary II Math 51
Unit 11.5
Mathematics Content
Cluster Title: Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
Standard S.CP.7: Apply the Addition rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret
the answer in terms of the model.
Concepts and Skills to Master


Define the probability of event (A or B) as the probability of their union.
Understand and use the formula (P(A or B) = P(A) + P(B) – P(A and B).
Critical Background Knowledge

Find probabilities of compound events. (7.SP.8)
Academic Vocabulary
Or, and, P(A)l, ^ , _
Suggested Instructional Strategies

Make a connection between the formula for the addition rule and a probability model.
Skills Based Task:
Problem Task:
Given the following table, which includes
data regarding boating preferences of boys
and girls, use the Addition Rule to find P(L ^
G).
Sally shaded the following Venn diagram to
illustrate the Addition Rule. What was wrong
with her reasoning? How could you represent
the addition rule
pictorially?
Lake (L)
River (R)
21
29
Boys (B)
32
Some Useful Websites:
18
Girls (G)
NUCC | Secondary II Math 52
Unit 11.5
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
A bag contains 8 red chips, 4 blue chips, 10 green chips, and 6 yellow chips. Consecutive
draws are made with replacement. Find the probability of each event.
1. red first, red second
2. yellow first, green second
3. blue first, blue second, blue third
4.
green first, red second, blue third,
yellow fourth
Set
Two cards are drawn from a standard 52-card playing deck with replacement. Find the
probability of each event.
5. Drawing a queen, then a king
6. Drawing a red card, then a black card
7. Drawing an Ace, then another Ace
8.
Drawing a 2 or 3, then the 5, 6, or 7 of
diamonds
Go!
A bag contains 8 red chips, 4 blue chips, 10 green chips, and 6 yellow chips. Consecutive
draws are made without replacement. Find the probability of each event.
9. red first, red second
10. yellow first, green second
11. blue first, blue second, blue third
12. green first, red second, blue third,
yellow fourth
NUCC | Secondary II Math 53
Unit 11.5
Two cards are drawn from a standard 52-card playing deck without replacement. Find the
probability of each event.
13. Drawing a queen, then a king
14. Drawing a red card, than a black card
15. Drawing an Ace, then another Ace
16. Drawing a 2 or 3, then the 5, 6, or 7 of
diamonds
NUCC | Secondary II Math 54
Unit 11.5
Solutions:
NUCC | Secondary II Math 55
Unit 11.6H
11.6H PERMUTATIONS & COMBINATIONS
Teacher Notes
Time Frame:
Materials Needed: Permutations & Combinations Task Sheet, Calculators
Purpose: The purpose of this task is to practice understanding of permutations and
combinations. To help students recognize the difference between permutations and combinations
and to be able to use both in the calculation of probabilities.
Skills: Students will be able to:
 Determine the number of permutations and combinations that are possible for various
scenarios, understanding the difference between a permutation and a combination.
 Compute the probability of various events that involve permutations and combinations.
 Determine whether or not to use a permutation or a combination to determine the
probability of an event occurring.
Core Standards Focus S.CP.8 Apply the general Multiplication Rule in a uniform probability
model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
S.CP.9 Use permutations and combinations to compute probabilities of compound events and
solve problems.
Related Standards Focus S.CP.6 Find the conditional probability of A given B as the fraction
of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
Vocabulary:
General Multiplication Rule: P(A and B) = P(A)P(B|A) = P(B)P(A|B)
Factorial: The symbol ! typically used to represent an integer multiplied by each consecutively
decreasing integer down to 1.
n!
Combination: The combinations of n objects taken r at a time is given by n Cr 
r ! n  r !
Permutation: The permutations of n objects taken r at a time is given by n Pr 
n!
 n  r !
Launch (Whole Class): This is a teacher lead task activity, but will practice understanding with
the students. Provide students with examples of problems where they have to find the probability
of events occurring that are both permutations and combinations. The “Ready, Set, Go!”
assignment can be referenced.
Note: No worksheet to hand students.
NUCC | Secondary II Math 56
Unit 11.6H
Explore (Individual, small group or pairs): Pass out the “Ready, Set, Go!” assignment.
NUCC | Secondary II Math 57
Unit 11.6H
Discuss (Whole Class or Group): Be sure to discuss how to determine the difference between
permutations and combinations with students. A good example is using a Presidency to describe
a permutation and a Committee to describe a Combination. While rearranging positions within a
Presidency which yield a completely different Presidency, rearranging positions within a
Committee will yield the same Committee.
NUCC | Secondary II Math 58
Unit 11.6H
Mathematics Content
Cluster Title: Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
Standard S.CP.8: Apply the general Multiplication Rule in a uniform probability model, P ( A
and B )  P( A) P( B * A) = P(B)P( A* B ), and interpret the answer in terms of the model.
Concepts and Skills to Master




Define the probability of events (A and B) as the probability of the intersection of events A
and B.
Understand P(B* A) to mean the probability of event B occurring when A has already
occurred
Use the Multiplication Rule, P(A and B) = P(A)P(B* A)=P(B)P(A* B), to determine P(A and
B).
Determine the probability of dependent and independent events in real contexts.
Critical Background Knowledge




Probabilities of compound events and tree diagrams. (7.SP.8).
Sample space, sets, subsets, outcomes, events, union, intersection, “and,” “or.” (II.4.S.CP.1)
Conditional probability (II.4.S.CP.3)
Two-way tables. (II.4.S.CP.4)
Academic Vocabulary
uniform probability model, multiplication rule, P(A and B) = P(A)P(B given A) = P(B)P(A* B),
P(A _ B) = P(A)P(B* A) = P(B)P(A* B)
Suggested Instructional Strategies


Apply and interpret the Multiplication Rule to a variety of contextual events.
Illustrate the Multiplication Rule with tree diagrams and two-way tables.
Skills Based Task:
Problem Task:
Given the following table, which includes
The probability that a student passes the written
data regarding boating preferences of boys
portion of a driving test is 62%. The probability
and girls, find the probability that a randomly that a student passes the driving part of the test
chosen student is a girl who prefers lake
is 86%. Draw a diagram to clearly demonstrate
boating.
the probability that a student passes both tests.
Lake (L)
River (R)
Girls (G)
21
29
Boys (B)
32
18
Some Useful Websites:
www.omegamath.com/Data/d2.2.html
www.mathsisfun.com/combinatorics/combinations-permutations.html
http://www.khanacademy.org/math/probability/v/permutations-and-compinations-1
NUCC | Secondary II Math 59
Unit 11.6H
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Evaluate each expression.
1. 7! 5!
2.
5. 10P7
6.
9.
P5
6 P5
12
10.
(7  2)!A5!
0!
8!
3!A5!
3.
P3
7.
50
C7
4 C3
11.
6
7
10
4.
C1
8.
C4 A
11 C7
12.
2!A5!
10!
9
C4
P A P7
8 P4
6 1 10
Set
For a classroom of 15 students, find how many ways each of the following can be selected:
13. 5 students on the first row
14. A class committee of 4 students
15. A 4-student presidency
16. A dodge ball team of 6 students
NUCC | Secondary II Math 60
Unit 11.6H
How many different ways can the letters of the following words be rearranged?
17. football
18. trout
19. vignette
20. Mississippi
21. circus
22. initiate
23. bookkeeper
24. correspondence
Go!
25. Franklin High School has 4 Valedictorians. In how many different orders can they give their
speeches?
26. There are 10 entries for parade floats in the 4th of July parade. How many different parades
can the parade manager organize?
27. Eight colleges are participating in a college fair. Booths are positioned along one wall of the
high school gymnasium. In how many different orders can the booths be arranged?
28. The winners of a contest were awarded $100, $75, $50, and $25. In how many ways could
the prize winners be selected from a group of 25 contestants?
29. In how many ways could 9 players from a class of 24 to be on the class softball team?
NUCC | Secondary II Math 61
Unit 11.6H
A pizza parlor offers a selection of 3 different cheeses and 9 different toppings. In how
many ways can a pizza with the following toppings be made?
30. 1 cheese and 3 toppings
31. 1 cheese and 4 toppings
31. 2 cheeses and 4 toppings
33. 2 cheeses and 3 toppings
NUCC | Secondary II Math 62
Unit 11.6H
Solutions:
NUCC | Secondary II Math 63