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INSTITUTE FOR STUDENT ACHIEVEMENT
Probability Unit
Aligned to the Common Core State Standards
Developed by Dr. Jonathan Katz
ISA Mathematics Coach
Dear Mathematics Teacher,
What is mathematics and why do we teach it? This question drives the work of the math
coaches at ISA. We love mathematics and want students to have the opportunity to begin to
have a similar emotion. We hope this unit will bring some new excitement to students.
This unit was originally written seven years ago and has gone through several iterations. Now it
has been redesigned to align with the Common Core State Standards. Essential to this work is
an inquiry approach to teaching mathematics where students are given multiple opportunities
to reason, discover and create. Problem solving is the catalyst to the inquiry process so as you
look closely at this unit you will see students constantly put in problem solving situations where
they are asked to think for themselves and with their classmates.
The first four Common Core Standards of Practice are central to this unit. Through the constant
use of problematic situations students are being asked to develop perseverance and
independent thought, to reason abstractly and quantitatively, and to critique the reasoning of
others. Throughout the Teacher Guides in this unit we’ve highlighted some places where the
Mathematical Practices are expressed. The Mathematical Practices will be denoted with MP
followed by a number indicating which specific Mathematical Practice is being expressed. As an
example MP2 will refer to Mathematical Practice 2: Reason abstractly and quantitatively.
Mathematical modeling is present throughout the unit as students are asked to analyze
different real world situations and represent them mathematically. Students are also asked to
create models including the final project which is to create a fair game based on the principles
of probability.
The other four Standards of Practice are also present in this unit. Two of them are central to the
inquiry approach. You will see these two statements in the last two standards.


Mathematically proficient students look closely to discern a pattern or structure.
Mathematically proficient students notice if calculations are repeated, and look for
general methods and shortcuts.
We believe, as do many mathematicians, that mathematics is the science of patterns. This
underlying principle is present in all the work we do with teachers and students. In this unit you
will see that students are often asked to discern a pattern within a particular situation. This
leads students to make conjectures and possibly generalizations that are both conceptual and
procedural.
Thank you for looking at this unit ad we welcome feedback and comments.
Sincerely,
Dr. Jonathan Katz
(For the ISA math coaches)
2|P age
Probability Unit
Essential Questions: What does it mean to be fair?
Does probability help you to make predictions about the world?
Final Assessment: Creating a Fair Game based on Principles of Probability
Interim Assessments/ Performance Tasks
The Copy Machine - Lesson 1
New York City Area Codes - Lesson 2
Create a Problem Whose Answer is 8! - Lesson 2
Are these problems the same or different? - Lesson 3
Lotto 20 - Lesson 3
Topping Trauma - Lesson 4
The Spinner Game - Homework - Lesson 6
Which is the Smartest Bet? - Lesson 7
Write Your Own Game Situation - Lesson 8
Is it a Fair Game? - Lesson 10
An Argument about Probability - Lesson 11
An Experiment about Bernoulli’s Law: Is it true? - Lesson 11
What concepts and procedures will be taught?
 Counting principle
 Permutations and Combinations
 Simple Probability
 Compound Probability
 Using and and or.
 Independent and Dependent Events
 Mutually Exclusive and Overlapping Events
 Using a tree diagram and sample space
 Experimenting with Bernoulli’s Law
What enduring understandings will students have?
 Probability can help a person make predictions about the world
 There is an important relationship between theoretical and experimental probability
and the greater the amount of trials the more likely you will get to the expected
(theoretical) probability.
 If you really understand probability you will probably not want to gamble.
 You can discover ideas and procedures of probability through looking for the
underlying patterns within a particular situation.
3|P age
Common Core Content Standards in the Unit
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.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.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.
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.
S-MD.5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff
values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected
winnings from a state lottery ticket or a game at a fast-food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example,
compare a high-deductible versus a low-deductible automobile insurance policy using various,
but reasonable, chances of having a minor or a major accident.
S-MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random
number generator).
Common Core State Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on
important “processes and proficiencies” with longstanding importance in mathematics
education. The first of these are the NCTM process standards of problem solving, reasoning and
proof, communication, representation, and connections. The second are the strands of
mathematical proficiency specified in the National Research Council’s report Adding It Up:
adaptive reasoning, strategic competence, conceptual understanding (comprehension of
mathematical concepts, operations and relations), procedural fluency (skill in carrying out
procedures flexibly, accurately, efficiently and appropriately), and productive disposition
(habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
belief in diligence and one’s own efficacy).
4|P age
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous
problems, and try special cases and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their progress and change course if
necessary. Older students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get the information
they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on
using concrete objects or pictures to help conceptualize and solve a problem. Mathematically
proficient students check their answers to problems using a different method, and they
continually ask themselves, “Does this make sense?” They can understand the approaches of
others to solving complex problems and identify correspondences between different
approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it
symbolically and manipulate the representing symbols as if they have a life of their own,
without necessarily attending to their referents—and the ability to contextualize, to pause as
needed during the manipulation process in order to probe into the referents for the symbols
involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not
just how to compute them; and knowing and flexibly using different properties of operations
and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account
the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is.
Elementary students can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though
5|P age
they are not generalized or made formal until later grades. Later, students learn to determine
domains to which an argument applies. Students at all grades can listen or read the arguments
of others, decide whether they make sense, and ask useful questions to clarify or improve the
arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. In early grades, this might be as simple as
writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high
school, a student might use geometry to solve a design problem or use a function to describe
how one quantity of interest depends on another. Mathematically proficient students who can
apply what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those
relationships mathematically to draw conclusions. They routinely interpret their mathematical
results in the context of the situation and reflect on whether the results make sense, possibly
improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their
grade or course to make sound decisions about when each of these tools might be helpful,
recognizing both the insight to be gained and their limitations. For example, mathematically
proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can
enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels are
able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to
explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use
clear definitions in discussion with others and in their own reasoning. They state the meaning of
the symbols they choose, including using the equal sign consistently and appropriately. They
are careful about specifying units of measure, and labeling axes to clarify the correspondence
with quantities in a problem. They calculate accurately and efficiently, express numerical
6|P age
answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach
high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young
students, for example, might notice that three and seven more is the same amount as seven
and three more, or they may sort a collection of shapes according to how many sides the
shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in
preparation for learning about the distributive property. In the expression x2 + 9x + 14, older
students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
existing line in a geometric figure and can use the strategy of drawing an auxiliary line for
solving problems. They also can step back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a
square and use that to realize that its value cannot be more than 5 for any real numbers x and
y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for
general methods and for shortcuts. Upper elementary students might notice when dividing 25
by 11 that they are repeating the same calculations over and over again, and conclude they
have a repeating decimal. By paying attention to the calculation of slope as they repeatedly
check whether points are on the line through (1, 2) with slope 3, middle school students might
abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when
expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the
general formula for the sum of a geometric series. As they work to solve a problem,
mathematically proficient students maintain oversight of the process, while attending to the
details. They continually evaluate the reasonableness of their intermediate results.
7|P age
Probability Unit
Lesson 1
Teacher Guide
Aim: Can I come up with a strategy to solve a particular type of problem?
Opening Activity:
Evaberg
Lisaville
Waltertown
waltertown
Jarelle
City
In the picture above, each of the thin lines is a road between different towns and cities. Use
the diagram above to answer the questions below:
1. In how many ways can you travel from Lisaville to Waltertown?
2. In how many ways can you travel from Lisaville to Jarelle City?
3. In how many ways can you travel from Lisaville to Evaberg?
4. Based on your answers above, do you see a shortcut that you could use to count the number
of ways from Lisaville to Evaberg? Think carefully about the numbers used and look for a
pattern that exists. Write your thoughts below:
[MP8 – by asking the students “do you see a shortcut” students will need to look at the pattern in questions 1-3 in
order to create a shortcut that will get them thinking along the lines of the Counting Principle]
(To the Teacher: Use the students thinking to have a discussion about the different possibilities. It should
lead to the notion of the counting principle that the student can observe from the map. If they grasp the
idea well they can then build towards the next set of problems.)
8|P age
Second Activity: Different Tasks You Can Try With Students
(To the Teacher: You can try as many problems as you like. Choose the ones that you believe will help
your students develop deeper conceptual and procedural understanding of the counting principle. I
would have them work on question 1a and 1b then stop and have a discussion. How is this problem
related to the map problem? What strategies can I use to answer these questions?)
[MP 1,2,3 – In these two questions for discussion students have to make sense of both problems (MP1), separate
the strategy from the specific numbers needed to answer 1a & 1b in order to connect it back to the map problem
(MP2), and they’ll need to be able to construct an argument for their position based on what their understanding
of the problem (MP3)]
For each of the problems below, show all of your work. You don’t need to show calculations
if you use a calculator, but you must show how you set up the problem to solve it.
1. You have just opened an ice cream parlor. You make your own homemade ice cream. You
have twenty different flavors. Your ice cream parlor offers, in addition to the twenty flavors,
three types of syrup (chocolate, vanilla and strawberry) and five different toppings (walnuts,
sprinkles, pineapples, cookie crumbs and whipped cream). How many different sundaes are
possible if:
a. You use one scoop of ice cream, one syrup and one topping?
b. You use two scoops of the same flavored ice cream, one syrup and one topping?
2. How many different types of homes are available if a builder offers one basic floor plan,
three styles of roofs and two styles of bricks?
3. Radio stations each have a four-letter call symbol. If a station is west of the Mississippi
River, it always starts with a K. If the station is east of the Mississippi River, the letters always
start with a W. For example, Mr. Sanchez’s favorite station is WNYC (93.9 FM). How many
different four letter radio stations call letters can be made if:
9|P age
a. The first letter must be W?
b. The first letter must be W and no letter can repeat?
c. The first letter is K or W, no letter can be repeated and the last letter is R?
4. Delaware has license plates consisting of three letters followed by three digits. (Remember,
there are 26 different letters and 10 different digits).
a. How many different license plates are possible?
b. How many different license plates can be formed if no repeats are allowed?
c. How many different license plates are possible if zero can’t be used and you only use
consonants (not the five vowels of A,E,I,O and U)?
(To the Teacher: Here is a task you might want to use at the end of the period to assess student
understanding of the counting principle with an added piece. Or you can do it at another time)
Performance Task: The Copy Machine
The combination to the Hudson High School copy machine is a five-digit code consisting of
numbers. All numbers can be repeated. If it takes 10 seconds per attempt to try a code, and
Akeem claims that he is 100% sure can figure out the code in less than one week, is he lying or
not? Prove your answer mathematically.
[MP 1, 2, 3a, 4 – Students are herein asked to model the total number of possible 5 digit codes (MP4). Through the
process of modeling they’ll need to make sense of the problem (MP1) and contextualize their conclusion within
the time parameters of the question (MP2). Lastly, they’ll have to assemble a viable argument for their conclusion
based on the evidence from their work and decide if Akeem’s claim is viable (MP3)]
10 | P a g e
Probability Unit
Student Activity Sheet
Lesson 1
Name_______________________
Date________________________
Opening Activity:
Evaberg
Lisaville
Waltertown
waltertown
Jarelle
City
In the picture above, each of the thin lines is a road between different towns and cities. Use
the diagram above to answer the questions below:
1. In how many ways can you travel from Lisaville to Waltertown?
2. In how many ways can you travel from Lisaville to Jarelle City?
3. In how many ways can you travel from Lisaville to Evaberg?
4. Based on your answers above, do you see a shortcut that you could use to count the number
of ways from Lisaville to Evaberg? Think carefully about the numbers used and look for a
pattern that exists. Write your thoughts below:
11 | P a g e
Second Activity: Different Tasks Using the Counting Principle
For each of the problems below, show all of your work. You don’t need to show calculations
if you use a calculator, but you must show how you set up the problem to solve it.
1. You have just opened an ice cream parlor. You make your own homemade ice cream. You
have twenty different flavors. Your ice cream parlor offers, in addition to the twenty flavors,
three types of syrup (chocolate, vanilla and strawberry) and five different toppings (walnuts,
sprinkles, pineapples, cookie crumbs and whipped cream). How many different sundaes are
possible if:
a. You use one scoop of ice cream, one syrup and one topping?
b. You use two scoops of the same flavored ice cream, one syrup and one topping?
2. How many different types of homes are available if a builder offers one basic floor plan,
three styles of roofs and two styles of bricks?
3. Radio stations each have a four-letter call symbol. If a station is west of the Mississippi
River, it always starts with a K. If the station is east of the Mississippi River, the letters always
start with a W. For example, Mr. Sanchez’s favorite station is WNYC (93.9 FM). How many
different four letter radio stations call letters can be made if:
a. The first letter must be W?
b. The first letter must be W and no letter can repeat?
12 | P a g e
c. The first letter is K or W, no letter can be repeated and the last letter is R?
4. Delaware has license plates consisting of three letters followed by three digits. (Remember,
there are 26 different letters and 10 different digits).
a. How many different license plates are possible?
b. How many different license plates can be formed if no repeats are allowed?
c. How many different license plates are possible if zero can’t be used and you only use
consonants (not the five vowels of A,E,I,O and U)?
13 | P a g e
Performance Task: The Copy Machine
The combination to the Hudson High School copy machine is a five-digit code
consisting of numbers. All numbers can be repeated. If it takes 10 seconds per
attempt to try a code, and Akeem claims that he is 100% sure can figure out the
code in less than one week, is he lying or not? Prove your answer mathematically
14 | P a g e
Probability Homework Assignment #1
1) How many different types of homes are available if a builder offers a choice of basic floor
plans, 3 roof styles and 2 exteriors?
2) A menu offers a choice of 3 salads, 8 main dishes and 5 deserts. How many different meals
consisting of one salad, one main dish and one desert are available?
3) How many different 4 letter radio stations call letters can be made if
a) the first letter must be K or W and no letter can be repeated?
b) the first letter is K or W and repeats are allowed?
c) the first letter is K or W, there are no repeats and the last letter is R?
4) A certain state has license plates consisting of 4 letters followed by 2 digits or 2 digits
followed by 4 letters.
a) How many different license plates are possible?
b) How many different license plate numbers can be formed if no repeats are allowed?
c) How many different license plates are possible if zero can’t be used and only consonants can
be used?
15 | P a g e
Probability Unit
Lesson 2
Teacher Guide
Aim: How do I use factorial and how is it related to the counting principle?
Opening Activity: New York City Area Codes
(To the Teacher: This can be used as a group performance task. How do students work together on a
problem that asks them to think in a new way based on the counting principle?)
Group Performance Task:
Why does New York City need three different area codes?
Remember that all telephone numbers have seven digits and the first digit cannot be zero?
Show all your work and thinking. Be ready to defend your answer.
(To the Teacher: Have a few students present their thinking and reasoning along with their solution. Be
purposeful in who you choose to speak. There should be different ways of thinking presented so students
can comment on their classmates’ work or ask their classmates’ questions.)
[MP1, 3, 4 – In order for students to formulate a defensible answer to this question they will have made sense of
the question enough to form a mental model from which they will formulate their answers. Listen for what
underlying assumptions students are making, as these will be clues for what could hang them up in the activities to
follow.]
Second Activity: (To the Teacher: This can be used if you want further evidence that students grasp the
counting principle.)
2) How many seven digit telephone numbers are possible if the first digit cannot be zero or one
and
a) only odd digits may be used?
b) the telephone number must be a multiple of ten?
c) no repetitions are allowed?
Third Activity: Investigation: (To the Teacher: This activity was created to see if students can be
pattern hunters in trying to figure out the meaning of factorial. You might have to give them hints in the
forms of questions.)
Can you discover what the symbol “!” means?
Using the calculator find: (Tell students that ! is called factorial).
1!
2!
3!
16 | P a g e
4!
5!
What do you think 6! will equal to? Why?
[MP7 – Students will need to make effective use of what they found in working with factorials 1 – 5.]
Can you make a general statement about what “!” means? (To the Teacher: A class discussion
would be useful here.)
[MP8 – In order to state generally what the factorial is doing they’ll have to make use of repeated reasoning.]
Then what do these equal to:
a) 6!4!
b)
c)
(To the Teacher: You can have the students write these out to show what they mean and with questions
b and c show how to cancel. They should also be able to do this on the calculator.)
Fourth Activity
How many different five letter arrangements can you make using the letters a, f, k, g, v?
(To the Teacher: Now let students play with this problem…The goal of this activity is to see how students
can think about this problem by using the counting principle and see how this principle is related to the
idea of factorial.)
(To the Teacher: Have students return to their activity sheet and work on the following:)
a. In how many ways can I line up all 6 books on a shelf?
b. Given the digits 0-9 how many four digit numbers can write using any of those digits?
(Can you use factorial for this problem?)
Performance Task
Create a problem whose answer is 8!?
What method did you use to create the problem?
[MP 1, 2, 4 – By starting with the answer, 8!, this will be an opportunity for students to go from decontextualized
situation to a contextualized situation as they create the context.]
17 | P a g e
Probability Unit
Student Activity Sheet
Lesson 2
Name_______________________
Date________________________
Group Performance Task:
Why does New York City need three different area codes?
Remember that all telephone numbers have seven digits and the first digit cannot
be zero? Show all your work and thinking. Be ready to defend your answer
18 | P a g e
Now try these:
1) How many seven digit telephone numbers are possible if the first digit cannot be zero or one
and
a) only odd digits may be used?
b) the telephone number must be a multiple of ten?
c) no repetitions are allowed?
Second Activity: An Investigation
Can you figure out what the symbol “!” means. In math we call “!” factorial.
Using the calculator find:
1!
2!
3!
4!
5!
3) What do you think 6! will equal to? Figure it out without using the calculator then check if
you are correct.
4) Can you make a general statement about what “!” means?
19 | P a g e
Third Activity
Question: What is the relationship between factorial and the counting principle? Do the
following problem and use it to explain the relationship/
How many different five letter arrangements can you make using the letters a, f, k, g, v?
Fourth Activity
Now work on these two. Be ready to justify your results.
a. In how many ways can I line up all 6 books on a shelf?
b. Given the digits 0-9 how many four digit numbers can write using any of those digits?
(Can you use factorial for this problem?)
20 | P a g e
Performance Task:
Create a problem whose answer is 8!.
What method did you use to create the problem?
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Probability Homework Assignment # 2
Evaluate the following factorials and permutations.
1) 6!
2) 7! 4!
3) 8! 3!
5!
4) 6!3!
(6-2)!
5) In how many ways can 6 people sit in 6 chairs?
6) In how many ways can 6 people sit in two chairs?
7) In how many ways can you arrange all 7 books on a shelf?
8) In a club there are 15 members. In how many ways can a slate of president, vice president
and secretary be chosen?
9) Write a problem whose answer is 4!.
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Probability Unit
Lesson 3
Teacher Guide
Aim: Can you tell the difference between these two problems?
Opening Activity: Are these problems the same or different?
With your partner look at these two problems, work on them together and try to solve them.
Are these problems are the same or different? Give mathematical evidence.
(To the Teacher: In the discussion encourage and challenge students to use their mathematical evidence)
1) Five students met and shook each others hand once. How many handshakes were there?
2) In how many ways can five children sit in two different chairs?
[MP 1, 2, 3, 4, 7 – Students will need to understand and model both situations in order to gather data for the
comparison, which is set up in overarching question for the opening task. When creating their argument for their
position on the overarching question students will have gone through some level of decontextualizing in order to
make the necessary comparison.]
[To the Teacher: You can have students act out the handshake problem and the chair problem to help
clarify the differences. Lead a class discussion answering the question: What is the difference between
the two problems? By introducing both concepts together the contrast becomes a teaching aid to help
students understand the unique characteristics of a permutation and a combination. This should lead to
the following definitions.
Permutations are arrangements of a given number of elements in which the order makes a different
arrangement.
Combinations are arrangements of a given number of elements in which the order does not make a
different arrangement. (e.g. handshake problem)]
[To the Teacher: Here is guide to how you might develop the lesson:
Today we will focus on permutation problems. Look at the two methods.
Your job is to find the connection between the counting principle and the permutation formula.
How can you use the counting principle to answer the seating arrangement problem?
(5 * 4 = 20)
Now you are going to look at a formula that can be used for permutation problems?
Your questions are, “Why does this formula make sense? How is it related to the counting principle?”
To understand formula---- n stands for number of elements and r stands for how many at a time.
=
n Pr
n!
.
23 | P a g e
(n – r)!
Show with team member problem….. n = 5 and r =2 so
=
5! .
5 P2
(5 – 2)!
5*4*3*2*1
3*2*1
=
5 * 4 = 20
Why did the two methods give us the same result? How are they related to each other?
What is going on in the formula?)
(To The Teacher: You want the students to see that the numerator represents the arrangements if all the
elements will be used in every outcome and the denominator represents what we want to eliminate
when all elements are not used in every outcome. If the formula makes sense to a student he should be
able to use it or the counting principle. Of course when more complex problems occur the formula might
be the most efficient way to think.)
Second Activity
Write about the two methods in solving a permutation problem. How are the two methods
connected to each other?
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______________________________________________________________________________
________________________________________________
(To the Teacher: You want to share with the students how we can use the calculator to find the result of
a permutation problem.
5 P2-----Press 5
Press MATH
Go to PRB by moving > three places
Go to n Pr
Press Enter
Press 2
Press Enter)
24 | P a g e
A little practice:
Find a) 10P4
b) 20P12
Third Activity: A Group Problem
Seven students from the class were picked to go on a television show. In how many ways can
they sit on the stage which has eight seats if
a) two people can appear at one time?
b) three people can appear at a time?
c) all seven can appear at a time?
Performance Task: Lotto 20
The class has decided to play LOTTO 20. We have to pick four numbers that will come up in the
right order but numbers can not appear twice. It costs $2 per ticket. You can win $10,000 if you
pick the right number. Does it make sense to play LOTTO 20? Defend your answer
mathematically
[MP 1, 3, 4 – Students will be modeling this situation with the probabilistic tools they have developed so far. Their
final answer is an argument with mathematical evidence that supports their position.]
25 | P a g e
Probability Unit
Student Activity Sheet
Lesson 3
Name_______________________
Date________________________
Opening Task: Are these problems the same or different?
With your partner look at these two problems, work on them together and try to solve them.
Are these problems are the same or different? Give mathematical evidence.
1) Five students met and shook each others hand once. How many handshakes were there?
2) In how many ways can five children sit in two different chairs?
Are these problems the same or different? Why?
Second Activity
Write about the two methods in solving a permutation problem. How are the two methods
connected to each other?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
________________________________________________
A little practice:
Find a) 10P4
b) 20P12
26 | P a g e
Third Activity : A Group Problem
Seven students from the class were picked to go on a television show. In how many ways can
they sit on the stage which has eight seats if
a) two people can appear at one time?
b) three people can appear at a time?
c) all seven can appear at a time?
27 | P a g e
Performance Task: LOTTO 20
The class has decided to play LOTTO 20. We have to pick four numbers that will come up in the
right order but numbers can not appear twice. It costs $2 per ticket. You can win $10,000 if you
pick the right number. Does it make sense to play LOTTO 20? Defend your answer
mathematically
28 | P a g e
Probability Homework Assignment # 3
Evaluate the following factorials and permutations.
1) 8P4
2) 15P5
3) What will be the answer for the following
n!
(n – 1)!
Justify your answer.
.
4) In how many ways can 8 people sit in three chairs?
5) In how many ways can you arrange all 10 books on a shelf?
6) If Lehman College offers 200 courses 20 of which are in mathematics and your counselor
arranges your schedule of 4 courses by random selection how many schedules are possible that
do not include a math course?
7) Find the number of different arrangements of the letters in the word hospital if you could
use three or four letters at a time.
8) You forgot the combination for your Yale lock. The numbers on your lock go from 0 to 40.The
combination is made up of three numbers All numbers can be repeated and the order makes a
difference. If it took you 10 seconds to try each combination Chris said it would take us at least
10 days to try all the combinations. Is he right?
29 | P a g e
Probability Unit
Lesson 4
Teacher Guide
Aim: How do we work with combinations?
Opening Activity: Revisit yesterday’s two problems.
We did two problems yesterday. What made them the same and what made them different?
How can we use them to come up with a method to solve combination problems? Solve the
first problem by both listing and using a mathematical formula. Solve the second one by listing.
1) In how many ways can five children sit in two different chairs?
2) Five students met and shook each others’ hand once. How many handshakes were there?
Could you come up with a method to solve the second problem without listing?
(To the Teacher: Students might see that if you divide by 2 you get the combination result. They might
say then for combinations you would always need to divide the permutation answer by two.
(To the Teacher: Now you want to help students see the combination formula and these questions should
guide you.)
There is a combination formula. How is it related to the permutation formula?
Why does this formula work? How is it related to the listing approach?
Since combinations are similar to permutations the formula will look similar except that we don’t want
groups counted more than once in combinations.
nCr=
n!
.
(n – r)! r!
Notice the difference…. We have an extra r! in the denominator to avoid the repetitions.
So what is 5C2 equal to?
=
5!
.
5C2
(5-2)! 2!
= 5 *4*3*2*1
3*2*1*2*1
After canceling we are left with 10. The same result we just got.
Why does this formula work?
30 | P a g e
(To the Teacher: You want students to see that it is like a permutation but you want to avoid the
repetition. So you must divide the permutation result by the factorial of the size of the grouping. Why
does that make sense?)
Second Activity: Try this combination problem and solve it using both the listing method and the
formula.
How many groups of three can I make from 4 people?
Did you get the same result using both methods? Can you talk again about why the formula
for combinations worked?
(To the Teacher: Again you want to see how students are making sense of combinations as an idea and
why the formula makes sense. Why did we need to divide by 6 in this case? You could handout the paper
below. A good discussion can arise.
How many groups of three can I make from 4 people?
Read and Observe:
If this was as permutation problem there would be 24 different possible arrangements
ABC
BAC
CAB
DAB
ABD
BAD
CAD
DAC
ACB BCA
CBA DBA
ACD
BCD
CBD DBC
ADB BDA
CDA
DCA
ADC BDC CDB
DCB
But since it is a combination problem we don’t want the repetition so we crossed them out
ABC
BAC
CAB
DAB
ABD BAD
CAD
DAC
ACB BCA
CBA
DBA
ACD
BCD
CBD
DBC
ADB BDA
CDA
DCA
ADC BDC CDB
DCB
What does this tell us about the relation of permutations to combinations?
How can you use the results from a permutation to find the number of combinations?)
(To the Teacher: Then you want the students to see what you would do on the calculator)
8C5
Press 8
Press MATH
Go to PRB moving > three places
Go to nCr
Press ENTER
Press 5
31 | P a g e
Press ENTER
(To the Teacher: Now have a discussion again: What is the difference between a combination and a
permutation? Students might need to try two more examples to compare combinations and
permutations and see the different results. )
Third Activity: PROBLEMS to be done with a partner
Here are two problems one is a combination and one is a permutation. Can you find the
answers for each of them? Be ready to explain and justify your results.
[MP 1, 2, 3, 4, 7 – Students will need to understand and model all the situations in order to gather data for the
comparison, which is the focus of the overarching question for these problems. When creating their argument
students will have gone through some level of decontextualizing in order to make the necessary comparison.]
a. You are playing cards with your friends. A deck of cards has 52 cards. Each player is
given 5 cards. How many different hands are possible?
b. Using the letters A, S, D, F, G, H, I, R how many different arrangements are possible if
each letter can be used only once?
Performance Task: Topping Trauma
The local ice cream parlor is running a special for the week. They are offering a 1or 2-topping sundae for the price of $0.99. There are 10 toppings from which to
choose.
You and your friends, Ben and Jerry, decide to walk to the parlor after school to
get a sundae. Ben and Jerry get in a huge argument along the way. Ben wants to
stop for a $0.99 sundae every day to try all of the different combinations. Jerry
claims that Ben was not listening in math class, and that there is no way that each
of them could eat all the different combinations of sundaes in one week. Your
friends turn to you and ask what you think! Thankfully, your mother drives up and
offers you a ride home. You jump in the car and tell Ben and Jerry that you will let
them know tomorrow who is right. What will you say? Back up your statement
mathematically.
[MP 1, 3, 4 – Students will be modeling this situation with the probabilistic tools they have developed so far. Their
final answer is an argument with mathematical evidence that supports their position.]
32 | P a g e
Probability Unit
Student Activity Sheet
Lesson 4
Name_______________________
Date________________________
Revisit yesterday’s two problems.
We did two problems yesterday. What made them the same and what made them different?
How can we use them to come up with a method to solve combination problems? Solve the
first by both listing and using a mathematical formula. Solve the second one by listing.
1) In how many ways can five children sit in two different chairs?
2) Five students met and shook each others’ hand once. How many handshakes were there?
3) Could you come up with a method to solve the second problem without listing?
4) We have just discussed a method to solve combination problems using a formula. Can you
explain the method and why it works? Include in your discussion how this method is similar to
but different from the permutation formula.
33 | P a g e
Second Activity: Try this combination problem and solve it using both the listing method and the
formula.
How many groups of three can I make from 4 people?
Did you get the same result using both methods? Can you talk again about why the formula
for combinations worked?
34 | P a g e
PROBLEMS to be done with a partner
Here are two problems one is a combination and one is a permutation. Can you find the
answers for each of them? Be ready to explain and justify your results.
c. You are playing cards with your friends. A deck of cards has 52 cards. Each player is
given 5 cards. How many different hands are possible?
d. Using the letters A, S, D, F, G, H, I, R how many different arrangements are possible if
each letter can be used only once?
35 | P a g e
Performance Task: Topping Trauma
The local ice cream parlor is running a special for the week. They are offering a 1or 2-topping sundae for the price of $0.99. There are 10 toppings from which to
choose.
You and your friends, Ben and Jerry, decide to walk to the parlor after school to
get a sundae. Ben and Jerry get in a huge argument along the way. Ben wants to
stop for a $0.99 sundae every day to try all of the different combinations. Jerry
claims that Ben was not listening in math class, and that there is no way that each
of them could eat all the different combinations of sundaes in one week. Your
friends turn to you and ask what you think! Thankfully, your mother drives up and
offers you a ride home. You jump in the car and tell Ben and Jerry that you will let
them know tomorrow who is right. What will you say? Back up your statement
mathematically.
36 | P a g e
Probability Homework Assignment #4
Combinations with one permutation.
1) Evaluate each of the following:
A) 8C3
B) 6C1
C) 12C5
D) 6C0
E)12C12
2) How many different three-card hands can be dealt from an ordinary 52-card deck?
3) In a game of musical chairs 7 children will sit in 6 chairs (with one child left out). In how many
ways can the 7 children find seats?
4) In the Student Government there are 12 males and 14 females. A three-person delegation is
to be selected to attend a convention.
a) How many delegations are possible?
b) How many delegations could have all males?
c) How many delegations could be all female?
e) How many delegations are possible if Liliana is required to go?
5) There are 4 girls and 5 boys who are members of the chess club. How many games are to be
played if each member is to play each other once?
37 | P a g e
Probability Unit
Lesson 5
Teacher Guide
Aim: What can we learn from experimental probability and how does it relate to theoretical
probability?
[MP2 – Throughout this lesson the students will be transitioning between the abstract (theoretical) and the
concrete (empirical) and need to make sense of both within the context of this investigation.]
Opening Activity: Questions to Start Off the Lesson
Why do we do experiments in science? Why would we do them in mathematics?
Put the students into groups of 3 or four and have them do the following activity.
Second Activity:
You are going to do an investigation with candies. In this investigation you are going to look at
theoretical and experimental probability and compare them.
Step 1
Use the table below that lists the different candy colors to record the results of each trial.
Experimental Outcomes
Color
Red
Blue
Total
Tally
40
Frequency
Experimental
Outcomes
(Relative
Frequency)
xxxxxxxxx
xxxxxxxxxx
xxxxxxxxxx
xxxxxxx
Percent
Put the colored candies in a paper bag then randomly select a candy by reaching into the bag
without looking and removing one candy. Record the color as a tally mark then replace the
candy into the bag before the next person reaches in. Take turns removing, tallying the color,
replacing the candy for a total of 40 trials.
Now record the frequency of each outcome on your table.
Step 2
From the experimental frequencies and the total number of trials (40), you can calculate the
relative frequency or observed probability of each color. For instance the observed probability
of removing a red candy will be:
38 | P a g e
Number of red candies
Total number of candies
Find the relative frequencies for each color and record it on the chart
(To the Teacher: Have students read the following statement and write down their interpretation. You
should have a discussion to ensure that students are getting comfortable with the language.)
Read the following statement:
When all the candies are put into a bag, drawing one candy from the bag has several possible
outcomes. The possible outcomes are the different colors listed on your table. Each color is
equally likely to be drawn but some have a higher probability of being drawn.
Put in your own words what the prior statement means.
______________________________________________________________________________
______________________________________________________________________________
____________________________________________________________
Step 3
Make a second table.
Theoretical Outcomes
Color
Number of
candies
counted
Theoretical
probability
Percent
Red
Orange
TOTAL
Now you are going to find the theoretical probability. This means the probability calculated by
analyzing a situation rather than performing an experiment. If the outcomes are equally likely
then the theoretical probability of a particular outcome is:
P(particular outcome) = total number of favorable outcomes
Total number of outcomes
You will work with this idea now.
Step 4
Dump out all the candies and count the number of candies of each color. Record the
information in the top row.
Step 5
39 | P a g e
Use the known quantities in the first row to calculate the theoretical probability of drawing
each color. For example:
P(Red)=
number of red candies in the bag .
Total number of candies in the bag
Now complete the table finding the theoretical probability of each color then finding the
percent.
(To the Teacher: This is now the most important part of the lesson. How will students analyze the results?
What do the results mean and what do they tell us about experimental versus theoretical probability?
You will want to lead a discussion based on the student analysis.)
Third Activity
Now you are going to analyze the results on your two tables.
Look closely at both tables and answer the following questions for each one.
1) Is one color most likely to be drawn? Did this happen in your experiment? Explain your
answer.
2) Is one color least likely to be drawn? Did this happen in your experiment? Explain your
answer,
3) What was the sum of all the theoretical probabilities? What was the sum of all the empirical
probabilities? Explain why.
4) Were the results for both tables the same? Why?
Closing Writing Activity
Now you are going to again observe the tables closely then write a paragraph comparing your
results in the experimental probability with your results in the theoretical probability. Make
sure you answer the question, “How are experimental and theoretical probability related to
each other?”
40 | P a g e
Probability Unit
Student Activity Sheet
Lesson 5
Name_______________________
Date________________________
Experimental and Theoretical Probability
Opening Activity: Questions to start off the Lesson
Why do we do experiments in science? Why would we do them in mathematics?
Second Activity: An Investigation
You are going to do an investigation with candies. In this investigation you are going to look at
theoretical and experimental probability and compare them.
Step 1
Use the table below that lists the different candy colors to record the results of each trial.
Experimental Outcomes
Color
Tally
Red
Blue
Total
40
Frequency
Experimental
Outcomes
(Relative
Frequency)
xxxxxxxxx
xxxxxxxxxx
xxxxxxxxxx
xxxxxxx
Percent
Put the candies in a paper bag then randomly select a candy by reaching into the bag without
looking and removing one candy. Record the color as a tally mark then replace the candy into
the bag before the next person reaches in. Take turns removing, tallying the color, replacing the
candy for a total of 40 trials.
Now record the frequency of each outcome on your table.
41 | P a g e
Step 2
From the experimental frequencies and the total number of trials (40), you can calculate the
relative frequency or observed probability of each color. For instance the observed probability
of removing a red candy will be:
Number of red candies
Total number of candies
Find the relative frequencies for each color and record you results on the table.
IMPORTANT: Read the following statement
When all the candies are put into a bag, drawing one candy from the bag has several possible
outcomes. The possible outcomes are all the different results you can get when you pick a
candy. Each color is equally likely to be drawn but some have a higher probability of being
drawn.
Put in your own words what the prior statement means.
______________________________________________________________________________
______________________________________________________________________________
____________________________________________________________
Step 3
Make a second table.
Theoretical Outcomes
Red
Orange
TOTAL
Number of
candies
counted
Theoretical
probability
Percent
Now you are going to find the theoretical probability. This means the probability calculated by
analyzing a situation rather than performing an experiment. If the outcomes are equally likely
then the theoretical probability of a particular outcome is:
P(particular outcome) = total number of favorable outcomes
Total number of outcomes
42 | P a g e
You will work with this idea now.
Step 4
Dump out all the candies and count the number of candies of each color. Record the
information in the top row.
Step 5
Use the known quantities in the first row to calculate the theoretical probability of drawing
each color. For example:
P(Red)=
number of red candies in the bag .
Total number of candies in the bag
Now complete the table finding the theoretical probability of each color then finding the
percent.
43 | P a g e
Third Activity
You are now going to analyze the results on you’re your two tables
Look closely at both tables and answer the following questions for each one. Explain if you used
one of the table or both tables to answer the question.
1) Is one color most likely to be drawn? Did this happen in your experiment? Explain your
answer.
2) Is one color least likely to be drawn? Did this happen in your experiment? Explain your
answer,
3) What was the sum of all the theoretical probabilities? What was the sum of all the empirical
probabilities? Explain why.
4) Were the results for both tables the same? Why?
44 | P a g e
Closing Writing Activity
Now you are going to again observe the tables closely then write a paragraph comparing your
results in the experimental probability with your results in the theoretical probability. Make
sure you answer the question, “How are experimental and theoretical probability related to
each other?”
45 | P a g e
Probability Unit
Lesson 6
Teacher Guide
Aim: How do I understand the different ideas within simple probability?
(To the Teacher: This lesson is broken up into short activities that develop a concept within probability.
You can have students work on each activity followed by a short discussion. The goal is get students to
develop conceptual and procedural understanding independently or with their partners. Hand out to
groups of two a spinner numbered 1 – 8.)
Opening Activity:
You are going to be given a spinner and different events to think about in order for you to
develop a deeper understanding of probability.
You are going to work with theoretical probability. Do you remember what that means?
[MP 6 – This opportunity to explain what theoretical probability means will require the student to attend to some
level of precision in order to effectively communicate their thinking.]
Theoretical probability is the probability calculated by analyzing a situation rather than
performing an experiment. If the outcomes are equally likely then the probability of a particular
event is:
Number of favorable outcomes .
Total number of possible outcomes
Look at your spinner.
What is the probability of spinning an 8?
Another way of asking the question is: Find P(8).
Since there is one favorable outcome and there are eight total outcomes the
P(8) = 1/8.
Now in your group do the following:
Find the answer and be ready to justify your solution
1) P(6)
2) P(even number)
3) P(prime number)
(To the Teacher: Stop if necessary to see if you need to go over. Have students explain.)
[MP 3, 6 – Giving students the opportunity to justify their solution for their classmates.]
Second Activity
Now look at these two examples:
46 | P a g e
Find:
4) P(not even)
5) P(not prime)
How did you find the answer?
Can you make a general statement about how to find P(not an event)?
If P(event) = x then what is P(the event not occurring)? Explain your reasoning.
If the probability of rain is 70% what is the probability it won’t rain?
(To the Teacher: This would be a good time to bring people together to see their understanding of an
event not occurring..)
Third Activity
Now look at these four events:
6) P(all numbers less than nine)
7)P(an integer)
8)P(all numbers greater than nine)
9) P(not an integer)
Look closely at the results of the four questions then try to answer the following questions.
What do you notice about the examples and the answers?
What would be the probability of an event definitely happening?
What would be the probability that an event will never happen?
Can a probability be greater than one? Why
Can a probability be less than zero? Why?
[MP 8 – The above questions give students the opportunity to enter into the process of abstracting what the
probabilistic bounds are and why those are reasonable.]
0____________________________1
impossible
definite
47 | P a g e
In your own words explain the diagram above
______________________________________________________________________________
______________________________________________________________________________
____________________________________________________________
(To the Teacher: You might stop here to have students talk about what they wrote and understand about this
important idea.)
[MP 3, 6 – Giving students the opportunity to discuss how they conceptualize the meaning of the diagram as it
relates to probability.]
Fourth Activity
Now look at these two examples:
P(even and red)
P(even or red)
How are these two examples different? How does it affect the probability?
What is the difference between and and or?
(To the Teacher: Possible Explanation:
Even and red is a favorable outcome if the spinner lands on a segment that is both red and
even. If it lands on a red 4 it is favorable but if it lands on a red 7 it is not favorable.
Even or red is a favorable if it lands on a red segment that is even or odd, or an even number
that is any color.
In other words for an event that uses and both things have to be true for it to be favorable. For
an event that uses or only one of the things has to be true for it to be favorable.)
Fifth Activity
You are going to answer the following questions based on the spinner we used in class today
then you are going to write a paragraph explaining the different ideas you learned and anything
you feel confused about. This will be collected
1) Using your spinner find each of the following.
a) P(7)
b)P(less than three)
c)P(not red)
d)P(even or odd)
48 | P a g e
e)P(even and odd)
f)P( 6 or 3 or blue)
g)P(not 3 and not blue)
Homework #5: Performance Task
The Spinner Game
Kwame and Augustina are playing a spinner game. The spinner is divided up so that the arrow
will land on Kwame’s area 1/5 of the time and Augustina’s area 4/5 of the time.
Kwame pays Augustina $0.30 when the spinner lands in her area and Augustina pays Kwame
$1.50 when it lands in his area. Is that fair? Why? Who would probably have the most money if
they played this game 50 times? How could you make the game fair?
[MP 1, 2, 3, 4, 7 – Students will need to understand and model each players situation in order to gather data for
the comparison found in the overarching question, “Is that fair?” When creating their argument for students will
have gone through some level of decontextualizing in order to make the necessary comparison.]
49 | P a g e
Probability Unit
Student Activity Sheet
Lesson 6
Name_______________________
Date________________________
You are going to be given a spinner and different events to think about in order for you to
develop a deeper understanding of probability.
Today you are going to work with theoretical probability. Do you remember what that means?
Theoretical probability is the probability calculated by analyzing a situation rather than
performing an experiment. If the outcomes are equally likely then the probability of a
particular event is:
Number of favorable outcomes.
Total number of possible outcomes
Look at your spinner.
What is the probability of spinning an 8?
Another way of asking the question is: Find P(8).
Since there is one favorable outcome and there are eight total outcomes the
P(8) = 1/8.
Opening Activity
Now in your group do the following:
Find
1) P(6)
2)P(even number)
3)P(prime number)
50 | P a g e
Second Activity
Now look at these two examples. We want you to discover what happens in probability when
we use the word “not.”
Find:
4) P(not even)
5) P(not prime)
How did you find the answer?
Can you make a general statement about how to find P(not an event)?
If P(event) = x then what is P(the event not occurring)? Explain your reasoning/
If the probability of rain is 70% what is the probability it won’t rain?
51 | P a g e
Third Activity
Now we want you to continue to be like a mathematician and discover a new idea:
TRY THESE:
6) P(all numbers less than nine)
7)P(an integer)
8)P(all numbers greater than nine)
9) P(not an integer)
Look closely at these four examples.
What do you notice about the examples and
the answers?
What would be the probability of an event definitely happening?
What would be the probability that an event will never happen?
Can a probability be greater than one? Why
Can a probability be less than zero? Why?
Statement: All probability fits on the following number line:
0____________________________1
impossible
definite
In your own words explain the diagram above
______________________________________________________________________________
______________________________________________________________________________
____________________________________________________________
52 | P a g e
Fourth Activity
We now have a new idea for you to find out. Try these examples
P(even and red)
P(even or red)
How are these two examples different? What is the difference between and and or?
Fifth Activity
You are going to answer the following questions based on the spinner we used in class today
then you are going to write a paragraph explaining the different ideas you learned and anything
you feel confused about. This will be collected
a) P(7)
b)P(less than three)
c)P(not red)
d)P(even or odd)
e)P(even and odd)
f)P( 6 or 3 or blue)
g)P(not 3 and not blue)
53 | P a g e
Probability Homework Assignment #5
Performance Task
The Spinner Game
Kwame and Augustina are playing a spinner game. The spinner is divided up so that the arrow
will land on Kwame’s area 1/5 of the time and Augustina’s area 4/5 of the time.
Kwame pays Augustina $0.30 when the spinner lands in her area and Augustina pays Kwame
$1.50 when it lands in his area. Is that fair? Why? Who would probably have the most money if
they played this game 50 times? How could you make the game fair?
54 | P a g e
Probability Unit
Lesson 7
Teacher Guide
Aim: How do you work with two different events at the same time?
(To the Teacher: The objective in playing the GAME of SEVEN is to use it as a means of beginning to
understand compound probability. What is different if we work with two different events at a time? You
are going to put the students into groups of 2.)
Opening Activity: GAME of SEVEN
To the students:
You are going to play a game rolling two dice. You want to keep rolling until you get a sum of
seven. You need to record the number of rolls it took to get a sum of seven.
Example: Dice came up 3 and 2
Dice came up 5 and 4
Dice came up 6 and 2
Dice came up 4 and 3
IT TOOK 4 ROLLS TO GET A SUM OF SEVEN
Before you start the game predict the average number of rolls it will take to get a sum of
seven. Write a sentence explaining your prediction.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
________________________________________________
[MP 1, 2, 3, 4, 6, 7 – Students will have some operational model they’ll base their answer on, which is in
accordance with their understanding of the problem. Regardless of the accuracy of their operational model in
communicating their prediction they’ll have opportunity to leverage what they’ve learned thus far in the
explanation they provide for their prediction with varying degrees of precision.]
55 | P a g e
You are going to play the game ten times. Record the number of rolls needed each time to get a
sum of seven.
Game #
1
2
3
4
5
6
7
8
9
10
Number of rolls needed to get a sum of 7
Now find the average number of rolls by adding the number of rolls and dividing by 10.
Did this get close to your prediction?
(To the Teacher: Bring class together and list all the results from the different groups. Discuss the
meaning of the results.)
What was the average of the predictions from the different groups?
Which do you think was a better prediction the class average or your prediction? Why?
Now let’s see what would have been a good prediction.
You are going to make a sample space showing all the possible outcomes when you roll two six
sided dice numbered one to six
I’ll get you started and you complete it.
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SAMPLE SPACE
1-1 2-1 3-1
1-2
1-3
1-4
1-5
1-6
4-1
5-1
6-1
How many possible outcomes are there?
How could I have gotten that answer using the counting principle? EXPLAIN
[MP 2, 3, 7 – Here students can bridge from the quantitative to the abstract and have the opportunity to utilizing
their understanding of previous experiences to connect the counting principle to the generation of the sample
space.]
Now look at your sample space and complete this table.
SUM
2
3
4
5
6
7
8
9
10
11
12
# of favorable outcomes
PROBABILTY
Look at your table .
Do you see any patterns? Write about them.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
[MP 7 – Patterns are manifestations of particular structures.]
Look at the sum of seven on your table. How does the probability compare to what happened
when you played the Game of Seven? How does it compare with the class average? Explain
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what your results tell us about probability.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
_____________________________________________
(To the Teacher: Bring the class together. Discuss students’ observations. Use the list from the games
played to analyze the compound probability. Compare theoretical with empirical probability. What
probably occurred is that the class average was closer to the theoretical probability although some
students might have guessed 6. We would want students to think about why would the class average be
closer?)
Second Activity: A Little Practice
Use the sample space or your table to answer these questions.
P(double)
P(sum is even)
P(sum is less than 6)
P(sum is not prime)
P(each die has an even number)
Closing Performance Task: Which is the smartest bet?
You are making a bet with your friend. Now that you know about dice you can be smart about
it. The rule says that if you win you get ten times the amount you bet.
Which bet makes more sense?
a) To bet $0.50 on a sum of seven 100 different times
b) To bet $5.00 on a sum of three 100 different times.
c) Not to bet at all
Explain your answer mathematically.
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Probability Unit
Student Activity Sheet
Lesson 7
Name_______________________
Date________________________
Opening Activity: GAME of SEVEN
You are going to play a game rolling two dice. You want to keep rolling until you get a sum of
seven. You need to record the number of rolls it took to get a sum of seven.
Example: Dice came up 3 and 2
Dice came up 5 and 4
Dice came up 6 and 2
Dice came up 4 and 3
IT TOOK 4 ROLLS TO GET A SUM OF SEVEN
Before you start the game predict the average number of rolls it will take to get a sum of
seven. Write a sentence explaining your prediction.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
________________________________________________
You are going to play the game ten times. Record the number of rolls needed each time to get a
sum of seven.
Game #
1
2
3
4
5
6
7
8
9
10
Number of rolls needed to get a sum of 7
Now find the average number of rolls by adding the number of rolls and dividing by 10.
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Did this get close to your prediction? Why?
What was the average of the predictions from the different groups in the class?
Which do you think was a better prediction the class average or your prediction? Why?
Now let’s what would have been a good prediction.
You are going to make a sample space showing all the possible outcomes when you roll two six
sided dice numbered one to six.
I’ll get you started and you complete it.
SAMPLE SPACE
1-1 2-1 3-1
1-2
1-3
1-4
1-5
1-6
4-1
5-1
6-1
How many possible outcomes are there?
How could I have gotten that answer using the counting principle? EXPLAIN
_________________________________________________________________
__________________________________________________________________
Now look at your sample space and complete this table.
SUM
2
3
4
5
6
7
8
9
10
# of favorable outcomes
PROBABILTY
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11
12
Look at your table .
Do you see any patterns? Write about them.
Which sums have the greatest probability? Least probability?
Look at the sum of seven on your table. How does the probability compare to what happened
when you played the Game of Seven? How does it compare with the class average? Explain
what your results tell us about probability.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Second Activity: A Little Practice
Use the sample space or your table to answer these questions.
P(double)
P(sum is even)
P(sum is less than 6)
P(sum is not prime)
P(each die has an even number)
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Closing Performance Task: Which is the Smartest Bet?
You are making a bet with your friend. Now that you know about dice you can be smart about
it. The rule says that if you win you get ten times the amount you bet.
Which bet makes more sense?
a) To bet $0.50 on a sum of seven 100 different times
b) To bet $5.00 on a sum of three 100 different times.
c) Not to bet at all
Explain your answer mathematically.
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Probability Homework Assignment #6
You have one fair die and one spinner. The die has eight sides numbered 1 to 8.
The spinner is divided into four equal segments numbered 1 to 4.
1) Create a sample space that shows all the possible outcomes if you roll the die
and spin the spinner at the same time.
2) Use the sample space to find:
a) P(sum of 6)
b) P(sum is less than eight)
c) P(sum is odd)
d) P(the die and the spinner each have even numbers)
e) P(sum is 7 or 11 or 12)
3) Which sum has the greatest probability? least probability?
4) Which has a greater chance of happening? Explain your answer
a) The spinner lands on one and the number on the die is odd
or
b) The number on the die is one and the spinner lands on an odd number.
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Probability Unit
Lesson 8
Teacher Guide
Objective: To create a game situation using the spinner and the die
Do Now
Have students use the sample space from their homework assignment to complete the
following table:
SUM
PROBABILITY
2
3
4
5
6
7
8
9
10
11
12
You are going to work with a partner to create a situation in a game using the die and the
spinner based on the probabilities you have discovered.
HERE IS AN EXAMPLE OF HOW YOU CAN DO IT
1) What events am I going to use in the game situation?
I am creating a game situation using a sum of 2 and a sum of 6.
2) What information do I need to know?
Necessary Information: Since there is one way of getting a sum of 2 and there are 32 possible
outcomes P(sum of 2) =1/32.
Since there is four ways of getting a sum of 6 and there are 32 possible outcomes then P(sum of
6) = 4/32 or 1/8.
3) What probability facts about the relationship of the two events do I need to know?
Fact: The probability of getting a sum of six is four times greater than getting a sum of two
because 4/32 is four times greater than 1/32.
4) What is the game situation?
If you predict getting a 2 and are correct you win $100 and if you are incorrect you lose $2
If you predict getting a six and are correct you win $25 and if you are incorrect you lose $8.
Do you understand this situation? Explain where the different money amounts came from.
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(To the Teacher: Lead a discussion about the different money amounts and how they make the game
fair. When you feel people are ready have them work on creating their own situation based on the
spinner and the die the students worked on today.)
Performance Task: Write your own Game Situation
If you feel that you understand the game situation, then you can begin to write your own
situation based on the spinner and the die.
(To the Teacher: When students have completed writing their situation have them present it to the class.
There should be a discussion to see if the situations mathematically make sense.)
[MP 1, 2, 3, 4, 7 – Through the creation process students will have the opportunity to go between the concrete
(the game itself) and the abstract (the probabilistic underpinnings for their game).]
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Probability Unit
Student Activity Sheet
Lesson 8
Name_______________________
Date________________________
Can you create a game situation?
Preparation for creating your own game.
Last night you worked on problems that utilized one die and one spinner.
Use the sample space from your homework assignment to complete the following table:
SUM
PROBABILITY
2
3
4
5
6
7
8
9
10
11
12
You are going to now create a situation in a game using the die and the spinner based on the
probabilities you have discovered.
HERE IS AN EXAMPLE OF HOW YOU CAN DO IT
1) What events am I going to use in the game situation?
I am creating a game situation using a sum of 2 and a sum of 6.
2) What information do I need to know?
Necessary Information: Since there is one way of getting a sum of 2 and there are 32 possible
outcomes P(sum of 2) =1/32.
Since there is four ways of getting a sum of 6 and there are 32 possible outcomes then P(sum of
6) = 4/32 or 1/8.
3) What probability facts about the relationship of the two events do I need to know?
Fact: The probability of getting a sum of six is four times greater than getting a sum of two
because 4/32 is four times greater than 1/32.
Does this make sense? Why? ________________________________________________
________________________________________________________________________
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________________________________________________________________________
4) What is the game situation?
If you predict getting a 2 and are correct you win $100 and if you are incorrect you lose $2
If you predict getting a six and are correct you win $25 and if you are incorrect you lose $8.
Do you understand this situation? Does it make sense? Is it fair? Why?_______________
________________________________________________________________________
______________________________________________________________________________
_________________________________________________________________
________________________________________________________________________
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Performance Task: Write your own Game Situation
If you feel that you understand the game situation, then you can begin to write your own
situation based on the spinner and the die.
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Probability Unit
Lesson 9
Teacher Guide
Aim: How is the probability affected when you combine two events?:
GAME of SEVEN…Revisited
To Students: Do you remember the Game of Seven? Now you are going to play a similar game
called SEVEN OR DOUBLE
You are going to play a game rolling two dice. You want to keep rolling until you get EITHER a
sum of seven or a double. You need to record the number of rolls it took to get a sum of seven
or double.
Example: Dice came up 3 and 2
Dice came up 5 and 4
Dice came up 2 and 2
IT TOOK 3 ROLLS TO GET A SUM OF SEVEN OR A DOUBLE
1) Before you start the game predict the average number of rolls it will take to get EITHER a
sum of seven or a double. Write a sentence explaining your prediction.
You might want to look back at what happened when we played Game of 7 a few days ago.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
You are going to play the game ten times. Record the number of rolls needed each time to get
a sum of seven or a double.
Game #
Number of rolls needed to get a sum of 7
or a double
1
2
3
4
5
6
7
8
9
10
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2) Now find the average number of rolls by adding the number of rolls and dividing by 10.
3) Did this get close to your prediction? Why do you think that happened?
[MP 2, 4, 6 – Reflecting on their empirical results and their prediction is the beginning of determining the accuracy
of the model they were basing their predication on. This explanation is also an opportunity to show what they’ve
learned thus far in the unit.]
4) If we took the class average what do you think would happen? Why?
(To the Teacher: You might want to stop the class and have some groups explain why they made a
certain prediction. Since students have a greater understanding of probability their predictions should
have more basis to it.)
Make a sample space showing all the possible outcomes when you roll two six-sided dice each
numbered 1-6.
5) Looking at your sample space tell what is P(sum of7 or double).
(REMINDER: The word OR means the event is favorable if either outcome happens)
Now let’s look at the two outcomes separately
6) What is P(sum of 7)?
7) What is P(double)
8) Look at your two results. How do they relate to P(sum of 7 or double)?
9) Can you make a general statement about P(Event A or Event B)?
10) Did these two events have any outcomes that are the same?
Definition: We call two events that have no outcomes that are the same “mutually exclusive.”
11) Complete: If two events are mutually exclusive and the two events are called A and B
then
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P(A or B) =__________________________
(To the Teacher: Before you move on make sure students understand. You can have students give other
examples of mutually exclusive events.)
12) Can you come up with an example from the game where two events have at least one
outcome in common?
We call these events “overlapping events.”
Let’s see if you can come with a general rule about overlapping events.
We will use P(sum of 8 or double) Why would we describe these two events as
overlapping?___________________________________________________________________
__________________________________________________________________
[MP 7, 8 – The process of abstracting requires the observations of both patterns as well as repeated reasoning in
order to arrive at a general rule.]
13) Find P(sum of 8 or double)
14) Find P(8)
15) Find P(double)
16) Does the rule you came up with about mutually exclusive events work? What do you need
to do?
17) How can you find P(8 or double)? Explain
Here is a general rule. Do you follow it?
If two events are overlapping then
P(A or B) = P(A) + P(B) – P(A and B)
18) Explain what the rule is saying then show how to use the formula for:
P(sum of 8 or double).
Final Activity:
Write a paragraph describing the meaning of mutually exclusive and overlapping events. Use
examples to support your explanation.
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[MP 2, 3, 6 – Differentiating between mutually exclusive and overlapping events students will give students the
opportunity to reason abstractly and quantitatively as they relate different events to their probabilistic likelihood.]
Probability Unit
Student Activity Sheet
Lesson 9
Name_______________________
Date________________________
Do you remember the Game of Seven? Now you are going to play a similar
game called SEVEN OR DOUBLE.
You are going to play a game rolling two dice. You want to keep rolling until you get EITHER a
sum of seven or a double. You need to record the number of rolls it took to get a sum of seven
or double.
Example: Dice came up 3 and 2
Dice came up 5 and 4
Dice came up 2 and 2
IT TOOK 3 ROLLS TO GET A SUM OF SEVEN OR A DOUBLE
1) Before you start the game predict the average number of rolls it will take to get EITHER a
sum of seven or a double. Write a sentence explaining your prediction.
You might want to look back at what happened when we played Game of 7 a few days ago.
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
You are going to play the game ten times. Record the number of rolls needed each time to get
a sum of seven or a double.
Game #
Number of rolls needed to get a sum of 7
or a double
1
2
3
4
5
6
7
8
9
10
72 | P a g e
2) Now find the average number of rolls by adding the number of rolls and dividing by 10.
3) Did this get close to your prediction? Why do you think that happened?
4) If we took the class average what do you think would happen? Why?
Now let’s see mathematically what happens …...
Make a sample space showing all the possible outcomes when you roll two six-sided dice each
numbered 1-6.
5) Looking at your sample space tell what is P(sum of 7 or double).
(REMINDER: The word OR means the event is favorable if either outcome happens)
Now let’s look at the two outcomes separately
6) What is P(sum of 7)?
7) What is P(double)
8) Look at your two results. How do they relate to P(sum of 7 or double)?
9) Can you make a general statement about P(Event A or Event B)?
10) Did these two events have any outcomes that are the same?
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Definition: We call two events that have no outcomes that are the same “mutually exclusive.”
11) Complete: If two events are mutually exclusive and the two events are called A and B
then
P(A or B) =__________________________
12) Does P(sum of 7 or double) which you just looked at go along with the your prediction
results of the experiment you did at the beginning of the class?
Explain________________________________________________________________________
______________________________________________________________________________
___________________________________________________________
12) Can you come up with an example where two events have at least one outcome in
common? (We call these events as overlapping.)
Let’s see if you can come with a general rule about overlapping events.
We will use P(sum of 8 or double). Why would we describe these two events as overlapping?
(Use the sample space to help you.)
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
13) Find P(sum of 8 or double)
14) Find P(8)
15) Find P(double)
16) Does the rule you came up with about mutually exclusive events work? What do you need
to do?
17) How can you find P(8 or double) without using the sample space? Explain
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Here is a general rule. Do you follow it?
If two events are overlapping then
P(A or B) = P(A) + P(B) – P(A and B)
18) Explain what the rule is saying then show how to use the formula for:
P(sum of 8 or double).
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Final Activity:
Write a paragraph describing the meaning of mutually exclusive and overlapping
events. Use examples to support your explanation.
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Probability Homework Assignment #7
Imagine a set of six sided dice in which every four was replaced by a 7. So each die could roll a
1,2,3,5,6, or 7 with each result equally likely.
Create the sample space.
1) Find P(sum is 10)
2) Find P(sum is less than 7)
3) Find P(sum is 10 or sum is less than 7)
4) Is question 3 an example of mutually exclusive or overlapping? Explain.
5) Find P(sum of dice is a multiple of three)
6) Find P(both dice are the same)
7) Find P(sum is a multiple of three or both dice are the same)
8) Is question 7 an example of mutually exclusive or overlapping? Explain.
9) On the back of the paper create two situations using the two dice. One situation is mutually
exclusive and the other is overlapping. Explain your answer.
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Probability Unit
Lesson 10
Teacher Guide
Aim: What can I learn about dependent and independent events and fairness through playing
the game of Double Yellows?
(To the Teacher: A pair of students will play the Game of Double Yellows. The purpose of the game is to
understand the difference between independent and dependent events and how they affect probability.
This lesson is going to take 90 minutes. The driving question for his lesson: Is this a Fair Game?)
To Students:
You are going to work with a partner. You are going to play the game of Double Yellows. Put
three yellow cubes and one red one in a bag.
Partner 1: Pick a cube. Put it back in the bag. Then pick again.
Partner 2: Pick a cube. Do not put it back into the bag. Then pick again.
Each partner takes turns choosing two cubes as described above. If a partner picks two yellow
cubes, he or she scores a point.
Each partner will take ten turns. The person with the most points wins the game
Before you begin to play answer these questions.
1)Who do you think will win; Partner 1 or Partner 2? Why?
[MP 1, 2, 3, 4, 6, 7 – Students will have some operational model they’ll base their answer on, which is in
accordance with their understanding of the problem. Regardless of the accuracy of their operational model in
communicating their prediction they’ll have opportunity leverage what they’ve learned thus far in the explanation
they provide for their prediction.]
2)Do you think the game is fair? Why?
Now start to play the game…
Round #
1
2
3
4
5
6
7
8
9
10
Partner 1
Partner 2
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TOTAL
Who won the game?
Who won more games in the class Partner 1 or Partner 2?
(To the Teacher: Have each group tell who won the game Partner 1 or 2.)
Was it a fair game? Do you think you know why? Explain using a sample space as part of your
evidence. (To the Teacher: Let students work on this question a while. It would be important to see how
they think about this. Let them share their ideas. You will now follow up to show the mathematical
reason why it was not a fair game.)
[MP 2, 3, 6 – Defining what a fair game is will give students the opportunity to reason with probabilities in making
their case for their definition and allow them to utilize what they’ve learned so far in the unit]
Let’s look at the mathematical reason.
Was the probability of choosing a yellow cube on the first pick the same or different for the two
partners? Explain
Was the probability of choosing a yellow cube on the second pick the same or different for the
two partners? Explain your thinking.
Let’s look at Partner 1
He/She picked a cube, put it back in the bag then picked a cube again?
Did his/her first pick affect the second pick? Why?
Let’s look at all the possible outcomes using a TREE DIAGRAM.
(To the Teacher: This is probably the first time you are using a tree diagram so you will need to support
them )
We will call the cubes Y1, Y2, Y3, and R1
The first pick could be any of the four cubes.
The second pick could be any of the four cubes.
79 | P a g e
How many outcomes do you see?
List them in a sample space
(To the Teacher: Make sure students understand how to read the tree. Have a student come up to
explain.)
What is the probability of two yellows?
Now let’s look at Partner 2
He/She picked a cube, did not put it back in the bag then picked a cube again.
Did his/her first pick affect the second pick? Why?
Let’s look at all the possible outcomes using a TREE DIAGRAM.
We will call the cubes Y1, Y2, Y3, and R1
The first pick could be any of the four cubes.
The second pick could be only three of the four picks. Show in a tree diagram all the possible
outcomes Partner 2 could have.
How many outcomes do you see? List them in a sample space
What is the probability of two yellows?
Compare the results from the game with the probability you just discovered. Did the partner
who won more games in the class go along with the theoretical probability? Explain.
Now let’s see if you can discover a pattern that can help you do these problems without
drawing the tree.
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Look at Partner1
What was P(yellow on the first pick)?
What was P(yellow on the second pick)?
What was P(2 yellows)?
Look at Partner 2
What was P(yellow on the first pick)?
What was P(yellow on the second pick)?
What was P(2 yellows)?
Do you see a connection between the probability on the first and second pick separately to the
probability of both picks combined? Explain?
Do you see a connection to the Counting Principle? How?
[MP 2, 7 – Making connections is an opportunity to reason abstractly and quantitatively while looking for patterns
in the ideas being connected]
Let’s see if you get it?
Using the same game
Find P(picking a yellow then a red with replacement of the first pick)
Find P(picking a yellow then a red without replacement of the first pick)
Go to your tree diagrams. Do your answers match? If they do then you get the idea.
Congratulations…….
Final Activity: Was the Game Fair?
Write a statement that answers the question using mathematical evidence to
justify your answer. Then describe what new idea and procedure you learned
today about probability and when you will need to use it.
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(To the Teacher: Have students share their ideas and explain the general approach to working with
dependent and independent events. You should incorporate the language of independent and dependent
events now that students understand the concept.)
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Probability Unit
Student Activity Sheet
Lesson 10
Name_______________________
Date________________________
Is this a Fair Game?
You are going to work with a partner playing the game of Double Yellows. Your goal today is to
decide if this is a fair game.
To begin put three yellow cubes and one red one in a bag.
Partner 1: Pick a cube. Put it back in the bag. Then pick again.
Partner 2: Pick a cube. Do not put it back into the bag. Then pick again.
Each partner takes turns choosing two cubes as described above. If a partner picks two yellow
cubes, he or she scores a point.
Each partner will take ten turns. The person with the most points wins the game
Before you begin to play answer these questions.
1) Who do you think will probably win; Partner 1 or Partner 2? Why? __________
__________________________________________________________________
2) Do you think the game is fair? Why? ___________________________________
Now start to play the game…
Round #
1
2
3
4
5
6
7
8
9
10
TOTAL
Partner 1
Partner 2
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1) Who won the game?
2) Who won more games in the class Partner 1 or Partner 2?
3) Was it a fair game? Do you think you know why? Explain using a tree diagram and/or
sample space as part of your evidence.
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Let’s look at the mathematical reason in more depth.
1) Was the probability of choosing a yellow cube on the first pick the same or different for the
two partners? Explain _______________________________________________
_______________________________________________________________________
2) Was the probability of choosing a yellow cube on the second pick the same or different for
the two partners? Explain ________________________________________________
________________________________________________________________________
Let’s look at Partner 1
3) He/She picked a cube, put it back in the bag then picked a cube again?
Did his/her first pick affect the second pick? Why? ______________________________
_______________________________________________________________________
_______________________________________________________________________
Design a TREE DIAGRAM for Partner 1.
We will call the cubes Y1, Y2, Y3, and R1
4) How many outcomes do you see?
5) List them in a sample space
6) What is the probability of two yellows?
Now let’s look at Partner 2
7) He/She picked a cube, did not put it back in the bag then picked a cube again.
Did his/her first pick affect the second pick? Why? ________________________
Design a TREE DIAGRAM for Partner 1.
We will call the cubes Y1, Y2, Y3, and R1
9) How many outcomes do you see?
10) List them in a sample space.
11) What is the probability of two yellows?
12) Compare the results from the game with the probability you just discovered. Did the
partner who won more games in the class go along with the theoretical probability you just
found? Explain. _____________________________________________________
______________________________________________________________________
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______________________________________________________________________
Now let’s see if you can discover a pattern that can help you do these problems without
drawing the tree.
Look at Partner1
18) What was P(yellow on the first pick)?
19) What was P(yellow on the second pick)?
20) What was P(2 yellows)?
Look at Partner 2
21) What was P(yellow on the first pick)?
22) What was P(yellow on the second pick)?
23) What was P(2 yellows)?
24)What is the connection between the probability on the first and second pick separately to
the probability of both picks combined? Explain?
25) Do you see a connection to the Counting Principle? How?
Let’s see if you get it?
Using the same game
Find P(picking a yellow then a red with replacement of the first pick)
Find P(picking a yellow then a red without replacement of the first pick)
Go to your tree diagrams. Do your answers match? If they do then you get the idea.
Congratulations……….
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Final Activity: Was the Game Fair?
Write a statement that answers the question using mathematical evidence to
justify your answer. Then describe what new idea and procedure you learned
today about probability and when you will need to use it.
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Probability Homework #8
1) You have three blue marbles and two yellow marbles. You are going to pick a marble, replace
it, and then pick a second marble. Draw a tree diagram and list a sample space showing all the
possible outcomes. (Do on back of paper)
2) Using the sample space find:
P(two blues)
P(two yellows)
P(yellow, then blue)
*P(same color)
3) Now use the formula we learned to answer the following questions.
You have three $1 bill and two $5 bills and one $20 bill in your pocket. You pull out two bills.
Find each probability.
a) P($5 and $1) with replacement
b) P($20 and $1) without replacement
c) P(two one dollar bills) without replacement.
4) A hat contains the names of the students in the class. There are 15 boys and 7 girls. Ms.
O’Keeffe draws two names from the hat without replacing the first name. What is the
probability that she will choose two girls?
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Probability Unit
Lesson 11
Teacher Guide
Aim: The Bernoulli Law: Is it true?
Opening Task: An Argument about Probability
I flipped a coin ten times and got a head 4 times. My friend Eduardo flipped the coin 100 times
got a head 56 times. We had an argument about who got closer to the expected probability.
Who is right? Explain your reasoning
(To the Teacher: An interesting follow-up question might be: If you used your computer to conduct 5000
trials what is the most you can be off and still be closer to the expected probability than the boy who
flipped ten coins?)
[MP 1, 2, 3, 6, 7 – These questions present students with an opportunity to go back and forth between quantitative
and abstract reasoning as they compare these two outcomes making use of their past experiences and
observations of patterns in probability.]
(To the Teacher: Today you are going to have students see if an idea of probability holds true. Make sure
that they understand what they are looking for as they go through the experiment.)
A Story: Jacob Bernoulli, a mathematician who lived in the 17th century, stated that when you
look at probability it is important to understand that the greater amount of trials you do the
closer you will get to the expected probability.
What does that mean? Do you think it is true?
Today you will be doing an experiment that will test the Bernoulli idea.
Performance Task: An Experiment about Bernoulli’s Law: Is it true?
Step 1: You are going to use the calculator to randomly pick a number from 1 to 10.
How do you do it?
1) Go to MATH then PRESS ENTER
2) Move > THREE PLACES to PRB
3) PRESS 5
4) You will see randInt(
5) Type 1,10)
If you press enter the calculator will randomly pick a number from 1 to 10. TRY IT.
Step 2: Now you are going to decide what outcome you are looking for. You might want the
number six, or a number less than 4.
What outcome do you want?
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What is the probability of your chosen outcome?
Step 3: You are going to begin your experiment. You will use the following table to list your
results.
# of
trials
What is the
probability?
(as a fraction)
What is the
probability?
(as a
percent)
How many
favorable
outcomes
are
expected?
How many
favorable
outcomes
really
happened?
Percent of
real
favorable
outcomes
Difference between
expected
and real
outcomes
10
20
50
100
200
How are you going to do this? Let me show you an example.
1) I decide I’m looking for a two to come up.
2) The probability of getting a 2 is 1/10.
3)1/10 = 10%
4) I’m going to have 10 trials so since the probability is 1/10 I expect 1 out of 10 to be a two.
5) I randomly pick ten numbers. How many twos came up? (Let’s say I got two 2’s.)
Now I fill out the chart
10
1/10
10%
1
2
20%
10%
Step 4: You are now going to run this experiment for the different amount of trials given to you.
AFTER YOU HAVE COMPLETED THIS EXPERIMENT ANSWER THE FOLLOWING QUESTIONS.
1) Did Bernoulli’s idea hold true for your experiment? Explain in detail using the results of your
experiment
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2) What do you think would happen if you ran the experiment 1000 times?
Probability Unit
Student Activity Sheet
Lesson 11
Name_______________________
Date________________________
Experiment using idea of Bernoulli
Opening Task: An Argument about Probability
I flipped a coin ten times and got a head 4 times. My friend Eduardo flipped the
coin 100 times got a head 56 times. We had an argument about who got closer to
the expected probability. Who is right? Explain your reasoning.
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Jacob Bernoulli, a mathematician who lived in the 17th century, stated that when you look at
probability it is important to understand that the greater amount of trials you do the closer
you will get to the expected probability.
What does that mean? Do you think it is true?
Today you will be doing an experiment that will test the Bernoulli idea.
Performance Task: An Experiment about Bernoulli’s Law: Is it true?
Step 1: You are going to use the calculator to randomly pick a number from 1 to 10.
How do you do it?
1) Go to MATH then PRESS ENTER
2) Move > THREE PLACES to PRB
3) PRESS 5
4) You will see randInt(
5) Type 1,10)
If you press enter the calculator will randomly pick a number from 1 to 10. TRY IT.
Step 2: Now you are going to decide what outcome you are looking for. You might want the
number six, or a number less than 4. Throughout the experiment you will always work wit the
same outcome regardless of the number of trials.
What outcome do you want?
What is the probability of your chosen outcome?
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Step 3: You are going to begin your experiment. You will use the following table to list your
results.
# of
trials
What is the
probability?
(as a fraction)
What is the
probability?
(as a
percent)
How many
favorable
outcomes
are
expected?
How many
favorable
outcomes
really
happened?
Percent of
real
favorable
outcomes
Difference between
expected
and real
outcomes
10
20
50
100
200
How are you going to do this? Let me show you an example.
1) I decide I’m looking for a two to come up.
2) The probability of getting a 2 is 1/10.
3)1/10 = 10%
4) I’m going to have 10 trials so since the probability is 1/10 I expect 1 out of 10 to be a two.
5) I randomly pick ten numbers. How many twos came up? (Let’s say I got two 2’s.)
Now I fill out the chart
10
1/10
10%
1
2
20%
10%
Step 4: You are now going to run this experiment for the different amount of trials given to
you.
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AFTER YOU HAVE COMPLETED THIS EXPERIMENT ANSWER THE FOLLOWING QUESTIONS.
1) Did Bernoulli’s idea hold true for your experiment? Explain in detail using the results of
your experiment
2) What do you think would happen if you ran the experiment 1000 times?
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A Model Project
Your written piece should be broken up into three parts. The three parts are; game description
and instructions, explanation of probability, and fairness of game
1) Game Description and Instructions:
In this section you will give a detailed description of the game including all the necessary rules
and instructions, describing how a person wins the game. You should describe what materials
are needed to play the game.
2) Explanation of Probability
This is the most important section. You need to show through using the various probability
concepts we learned how you came up with your game. You should include wherever possible
the counting principle, permutation, combination, simple probability, compound probability,
showing all possible outcomes using a sample space and tree diagram.
3) Is the game fair?
Show through various rules and the probability involved if the game is fair to play.
For example if one outcome is three times harder to get than a second outcome then a rule
involved with those two outcomes will show that.
Example Project
The Coin Game
I) Game Description
The Coin Game can be played by two or more players. Each player is given three pennies and a
little man. Each player places their little man at GO. The goal of the game is to be the first
player to move around the board and reach END.
How do you play the game?
 Each player flips his/her coins. The person with the most heads goes first.
 Player 1 takes his/her three coins and has to predict how the coins will land before
he/she throws the coins.
 Player 1 flips the coins and counts the number of heads and tails.
 Use the following chart to determine the movement of your little man.
Number of
Tails
Number of
Heads
Prediction is
correct
Prediction is
wrong
3
2
1
0
0
1
2
3
6 steps forward
2 steps forward
2 steps forward
6 steps forward
1 step back
2 step back
2 step back
1 step back
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




When you land on ? You must answer one of the questions from the deck of questions.
Another player reads the question. The answer is at the bottom of the card. If you are
wrong you go to TUTORING.
To get out of TUTORING you must answer a question or flip three tails .If you are
unsuccessful you lose a turn.
You can not move backwards from GO.
Every player gets one turn then the next player goes. After Player 1 gets his/her turn
Player 2 goes.
The first player to reach END wins the game.
2) Explanation of Probability
This game uses three coins. Each coin has two possible outcomes; head or tail. If you flip
three coins there are going to be eight possible outcomes. Why?
Using the counting principle we know that there are 2 possible outcomes on each coin. We
multiply the possible outcomes on each coin and we get 2*2*2 or 8 possible outcomes. Let
me prove it to you using a tree diagram and sample space.
Here is the tree diagram showing 8 possible outcomes:
Here is the sample space showing eight possible outcomes:
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
Here is a chart showing the probability of the different outcomes:
Number of
Tails
Number of
Heads
Probability
3
2
1
0
0
1
2
3
1/8
3/8
3/8
3/8
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You should notice that the probabilities add up to 8/8 or 1 whole. This will always be true when
you are up all the probabilities of an experiment.
When you look at the chart you will notice that it is three times more likely to get a head
and two tails or a tail and two heads than getting three heads or three tails. This fact is
reflected in the rules of the game. Notice that if you predict getting three heads or three
tails and this event occurs then you move six places forward. If you predict two heads and a
tail or two tails and a head and this event occurs you only move two places forward.
The questions on the cards in the deck are all questions from probability covering all the
topics we studied.
For example:
How many three person groups can you form from 5 people?
Answer: 5C3 or 10 different groups.
3) Is the game fair?
The game is definitely fair. It doesn’t favor the person who predicts the less likely
probability or the person who predicts the more likely probability. We can see that in the
following statements.
If you predict three heads with a probability of 1/8 you move forward six places if you are
correct and move back one place if you are wrong.
If you predict two heads and one tail with a probability of 3/8 you move forward two
places if you are correct and move back two places if you are wrong.
Will the person who goes for the long shot or the person who goes for the
sure thing win? It is hard to tell. That is why the game is fair.
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