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Advanced Topics Unit C: Probability and Statistics
Unit Overview
Math Florida Standards
Content Standards
In this unit, students will define random variables for a quantity of interest, calculate the expected value of random
variables and develop and graph probability distributions. They will also use probabilities to weigh possible
outcomes, make fair decision and analyze those decisions and strategies. Finally, they will utilize conditional
probability, the Addition Rule and the Multiplication Rule.
Textbook Resources
Prentice Hall Algebra 2
Pearson Success Net
How to get here:
 Go to “Other resources”
 Click “Teacher resources”
 Click “Algebra 2 Common Core Additional
Lessons”
 Click “Teacher Resources
Sections:
CC-12, CC-13, CC-15
Glencoe McGraw-Hill Pre-calculus textbook McGraw-Hill
Sections:
0-7, 11-2
Mathematics Formative Assessment System Tasks
The system includes tasks or problems that teachers can
implement with their students, and rubrics that help the
teacher interpret students' responses. Teachers using
MFAS ask students to perform mathematical tasks, explain
their reasoning, and justify their solutions. Rubrics for
interpreting and evaluating student responses are included
so that teachers can differentiate instruction based on
students' strategies instead of relying solely on correct or
incorrect answers. The objective is to understand student
thinking so that teaching can be adapted to improve
student achievement of mathematical goals related to the
standards. Like all formative assessment, MFAS is a process
rather than a test. Research suggests that well-designed
and implemented formative assessment is an effective
strategy for enhancing student learning.
MAFS.912.S-MD.1.1
MAFS.912.S-MD.1.2
MAFS.912.S-MD.1.3
MAFS.912.S-MD.1.4
MAFS.912.S-MD.2.5
MAFS.912.S-MD.2.6
MAFS.912.S-MD.2.7
MAFS.912.S-CP.2.6
MAFS.912.S-CP.2.7
MAFS.912.S-CP.2.8
MAFS.912.S-CP.2.9
Standards for
Mathematical Practice
MAFS.K12.MP.1.1
MAFS.K12.MP.4.1
Other Resources
Replacement and Probability
Unexpected Answers
Coffee At Mom's Diner
Alex, Mel and Chelsea Play a Game
Mathematics Formative Assessment System Tasks
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
Unit Scale (Multidimensional) (MDS)
The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for
measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1,
“What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps
students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the
academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and
declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards.
4.0
In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these
standards. Examples:

3.0
Investigate and compare two situations in regards to probability, fair decisions, and expected values. Make and defend a decision about which situation
is more favorable.
The Student will:
 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability
distribution using the same graphical displays as for data distributions. (MAFS.912.S-MD.1.1)
 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. (MAFS.912.S-MD.1.2)
 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected
value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiplechoice test where each question has four choices, and find the expected grade under various grading schemes. (MAFS.912.S-MD.1.3)
 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected
value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of
sets per household. How many TV sets would you expect to find in 100 randomly selected households? (MAFS.912.S-MD.1.4)
 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. (MAFS.912.S-MD.2.5)
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.
 Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (MAFS.912.S-MD.2.6)
 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
(MAFS.912.S-MD.2.7)
 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
(MAFS.912.S-CP.2.6)
 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. (MAFS.912.S-CP.2.7)
 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. (MAFS.912.S-CP.2.8)
 Use permutations and combinations to compute probabilities of compound events and solve problems. (MAFS.912.S-CP.2.9)
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
2.0
The student will recognize or recall specific vocabulary, such as:
 Addition Rule, Multiplication Rule, probability, random variable, probability distribution, conditional probability, outcome, expected value, sample space,
permutation, combination
The student will perform basic processes, such as:
 The Addition Rule and Multiplication Rule
 Find probabilities and conditional probabilities
 Define random variables
1.0
With help, partial success at 2.0 content but not at score 3.0 content
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
Unpacking the Standard: What do we want students to Know, Understand and Do (KUD):
The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question
#1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what
student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional
scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are
essential for mastery.
Domain: Statistics & Probability: Conditional Probability and the Rules of Probability
Cluster: Use the rules of probability to compute probabilities of compound events in a uniform probability model. (Additional)
Standard: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. (MAFS.912.SCP.2.6)
Understand
“Essential understandings,” or generalizations, represent ideas that are transferable to other contexts.
Probability determines the likelihood of events and explores the ways in which one event impacts the probability of another event occurring.
Know
Declarative knowledge: Facts, vocab., information





Dependent events
Conditional probability
Intersection
Set
Mutually exclusive
Do
Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts.
Comprehension
 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A.
Analysis
 Interpret the conditional probability in terms of the model.
Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard?
Probability of independent events, sample space, outcomes,
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
Uni-Dimensional, Lesson Scale:
The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of
knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown
below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to
further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1,
“Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the
actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level,
not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard.
Standard:
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. (MAFS.912.S-CP.2.6)
Score
Learning Progression
4.0
I can…
 Create a practical application modeling conditional probability.
3.5
I can do everything at a 3.0, and I can demonstrate partial success at score 4.0.
I can…
 Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A
 Interpret the conditional probability in terms of the model
Sample Tasks
Design a situation involving conditional probability. Compute various
probabilities and interpret the results.
Researchers asked shampoo users whether they apply shampoo directly to the
head or indirectly using their hand. The results were recorded in the table.
1.
What is the probability that a
respondent applies shampoo
directly to the head given the
respondent is a female? Is
male?
2.
Interpret the results.
3.0
2.5
I can do everything at a 2.0, and I can demonstrate partial success at score 3.0.
I can…
 Find the probability of independent events
 Identify if conditional probability exists in terms of the model
2.0
1.0
A utility company asked 50 of its customers
whether they pay their bills online or by mail.
Results were recorded in the table.
1.
What is the probability the bill is paid
online? Is this an example of conditional
probability?
I need prompting and/or support to complete 2.0 tasks.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
Sample High Cognitive Demand Tasks:
These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply
demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them
as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and
common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?”
and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New
Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors).
MAFS Mathematical Content Standard(s)
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. (MAFS.912.SCP.2.7)
Design Question 1; Element 1
MAFS Mathematical Practice(s)
Make sense of problems and persevere in solving them. (MAFS.K12.MP.1.1)
Design Question 1; Element 1
Marzano’s Taxonomy
Analysis
This task uses the same situation to explore different concepts of probability theory.
Part (a) explores the idea of independence of events. Students must use the fact that two events A and B are independent
if P(A and B) = P(A)⋅ P(B).
Teacher Notes
Questions to develop mathematical thinking,
possible misconceptions/misunderstandings,
how to differentiate/scaffold instruction,
anticipate student problem solving strategies
Part (b) lets students explore the idea of conditional probability. Students should understand the difference between
P(A and B) and P(A|B). They should also understand that P(A and B) = P(B and A). In this part, they are given three
probabilities, but only two of them are needed to answer the question. In general, students very often assume that every
piece of given information must be used in the solution, so this is a good way to realize that sometimes we have more data
than we really need, and have to choose which information is useful.
In part c) students practice using the Addition Rule, P(A or B) = P(A) + P(B) − P(A and B). Teachers should make students
aware of the fact, that in mathematics the expression “or” is almost always used in the inclusive sense, that is “A or B”
really means “either A or B or both”. Students must also understand that P(A or B) = P(B or A) and that P(A and B) =
P(B and A).
Part (d) is a variant of part c). Students must apply the Addition Rule, P(A or B) = P(A) + P(B) − P(A and B), and solve the
equation for the unknown quantity.
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015
Advanced Topics Unit C: Probability and Statistics
Rain and Lightning
a.
Task
*These tasks can either be teacher created or b.
modified from a resource to promote higher
order thinking skills. Please cite the source for
c.
any tasks.
d.
Today there is a 55% chance of rain, a 20% chance of lightning, and a 15% chance of lightning and rain together. Are
the two events “rain today” and ”lightning today” independent events? Justify your answer.
Now suppose that today there is a 60% chance of rain, a 15% chance of lightning, and a 20% chance of lightning if it’s
raining. What is the chance of both rain and lightning today? Justify your answer.
Now suppose that today there is a 55% chance of rain, a 20% chance of lightning, and a 15% chance of lightning and
rain. What is the chance that we will have rain or lightning today? Justify your answer.
Now suppose that today there is a 50% chance of rain, a 60% chance of rain or lightning, and a 15% chance of rain and
lightning. What is the chance that we will have lightning today? Justify your answer.
Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License
This a working document that will continue to be revised and improved taking your feedback into consideration.
Pasco County Schools, 2014-2015