Common Core High School Statistics & Probability Teaching and Learning Resources
Domain: CP: Conditional Probability and the Rules of Probability
Select the cluster for resources on this page:
CP-A: Understand independence and conditional probability
and use them to interpret data
Standards:
- CP-A.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”).
- CP-A.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.
- CP-A.3. Understand the conditional
probability of A given B as P(A and B)/P(B),
and interpret independence of A and B as saying that
the conditional probability of A given B is the same
as the probability of A, and the conditional probability of
B given A is the same as the probability of B.
- CP-A.4. Construct and interpret two-way
frequency tables of data when two categories are associated with
each object being classified. Use the two-way table as a sample
space to decide if events are independent and to approximate
conditional probabilities. For example, collect data from a
random sample of students in your school on their favorite subject
among math, science, and English. Estimate the probability that a
randomly selected student from your school will favor science given
that the student is in tenth grade. Do the same for other subjects
and compare the results.
- CP-A.5. Recognize and explain the concepts
of conditional probability and independence in everyday language and
everyday situations. For example, compare the chance of having
lung cancer if you are a smoker with the chance of being a smoker if
you have lung cancer.
Technology-enhanced investigations:
Stattrek.com Statistics Dictionary contains the following key vocabulary for this domain.
Select the term from the Stattrek pull-down menu. Use with CP-A,
CP-B:
|
addition rule |
complement |
independent |
sample space |
|
Bayes' theorem/rule |
conditional probablity |
intersection |
event |
|
combination; permutation |
two-way table (frequency) |
multiplication rule |
union |
Art of Problem Solving:
Counting and Probability Videos: These are the videos that go with
each of the sections within the 14-chapter textbook,
Introduction to Counting & Probability.
Interactive Mathematics:
Counting and Probability - Introduction delves into factorial
notation, basic principles of counting, permutations, and combinations.
Probabiltiy theory and probability of an event are introduced.
Probabilities of two or more events addresses conditional probability,
independent and dependent events, mutually exclusive events, and Bayes'
Theorem. Concepts of probability distributions are presented along
with binomial probability distributions, Poisson probability
distributions, and normal probability distributions. An
interactive for a normal probability distribution is included. Use with
CP-A and CP-B.
Imagine Learning Classroom (formerly LearnZillion) subscription
needed:
- Lesson set:
Describe events as subsets of a sample space:
Five video lessons: Organize sample space information, identify the
intersection and union of two events, identify the complement of
an event, determine quantities belonging to subsets of a sample
space. Aligns with CP-A.1.
- Lesson set:
Describe events as subsets of a sample space, or as unions,
intersections, or complements of other events: Five video
lessons: Describe a sample space by organizing the set of all
possible outcomes, describe an event and its complement,
describe the intersection of two events, describe the union of
two or more events, describe the complement, union, and
intersection of events in a word problem. Aligns with CP-A.1.
- Lesson set:
Describe events as subsets of a sample space, or as unions,
intersections, or complements of other events: Five video
lessons: Describe events as subsets of a sample space, determine
the probability of an event and its complement; determine the
probability of union and intersection of several events, compare
the probability of union and intersection of events.
Aligns with CP-A.1.
- Lesson set:
Understand independence and determine whether events are
independent: Three video lessons: Determine whether events
are independent, determine independence using a formula,
calculate the probability of independent events. Aligns
with CP-A.2.
- Lesson set:
Understand and determine independence: Four video lessons:
Determine whether events are independent or associated,
determine whether events are independent within a two-way table,
determine independence using products of probabilities,
understand selecting without replacement. Aligns with
CP-A.2.
- Lesson set:
Understand conditional probability and use it to determine
independence: Three video lessons: Understand conditional
probability, compute conditional probability using a formula,
determine independence using conditional probability.
Aligns with CP-A.3.
- Lesson set:
Understand the conditional probability of A given B as P(A and
B)/P(B): Four video lessons: Interpret conditional
probabilities; identify conditional probability in a two-way
table and in a tree-diagram; interpret independent variables
using conditional probabilities. Aligns with CP-A.3.
- Lesson set:
Recognize and explain the concepts of conditional probability
and independence: Four video
lessons: Understand conditional probability using scenarios,
calculate conditional probabilities, determine dependent or
independent by comparing scenarios, prove independence
numerically by using the independence formula. Aligns with CP-A.5.
Purple Math: Venn Diagrams: Introduction, Logic and set notation, Set notation
exercises, Venn word problems. Use with CP-A.1.
MIT BLOSSOMS: Video lessons with additional teacher and learner
resources. Descriptions are from the video summaries.
- Flu Math Games:
"This video lesson shows students that math can play a role in
understanding how an infectious disease spreads and how it can be
controlled." Additional simulations are included. Aligns
with Algebra standards SSE-B.3.c and REI-A.1; Function standards
IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability
standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.
- The Monty Hall Problem or How to Outsmart a Game Show and Win a
Car: "This lesson teaches students how to make decisions in
the face of uncertainty by using decision trees. It is aimed for
high school kids with a minimal background in probability; the
students only need to know how to calculate the probability of two
uncorrelated events both occurring (ie flipping 2 heads in a row).
Over the course of this lesson, students will learn about the role
of uncertainty in decision making, how to make and use a decision
tree, how to use limiting cases to develop an intuition, and how
this applies to everyday life." Aligns with CP-A.2, CP-B.8,
CP-B.9, MD-B.5.a, MD-B.6, and MD-B.7.
Stat Trek: Tutorials:
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
12 manipulatives addressing CP-A.1,
7 manipulatives for CP-A.2,
6 manipulatives for CP-A.3,
1 manipulative for
CP-A.4,
7 manipulatives for CP-A.5. Among those:
-
Monty Hall Paradox: Aligns with CP-A.1, CP-A.2, CP-A.3,
CP-A.5, and MD-B.6.
-
Multiplying Probabilities: Aligns with CP-A.1, CP-B.9.
-
Probabilities of Matching Colors: Involves multiplication of
probabilities. Aligns with CP-A.1, CP-B.9.
-
Conditional Probability: Aligns with CP-A.1, CP-A.2, CP-A.3,
CP-A.5, and CP-B.6.
-
Venn Diagrams: This shows all 127 nonempty unions and intersections of three sets.
You can show or hide the corresponding set notation, given in
terms of union, intersection, and complement. Aligns with CP-A.1, CP-B.6, CP-B.7.
-
Probability of Being Sick After Having Tested Positive for a
Disease (Bayes' Rule): Aligns with CP-A.3, CP-A.4, CP-A.5,
MD-B.7.
Wolfram MathWorld:
YouTube videos:
GCSE Probability
Part 1: This video is from a professor in Great Britain who nicely
explains probability and shows examples: expected frequency, the typical
problems involving picking balls from a bag, mutually exclusive events,
independent events, and drawing diagrams to help identify possibilities.
Probability
Part 2: tree diagrams. The same professor in Great Britain
discusses the balls in a bag problems when making two picks, which
leads to tree diagrams to help determine probability involving
replacing after each pick, and then not replacing after a pick.
Zweigmedia.com: Tutorials and examples using fill-in or
multiple choice to test understanding. Learners can also choose to
do game versions of some topics. Use with CP-A:
Multiple Choice:
CK-12:
Venn Diagrams: Use Venn diagrams to describe situations when problem
solving. Practice questions with review content and video. Aligns
with CP-A.1.
Khan Academy:
Probability unit. Practice questions with videos.
MathsIsFun.com:
-
Probability: Complement Contains a concise explanation of
the complement, outcomes that are "not" the event, with worked
examples. Following the explanation are ten multiple choice
exercises. Use with CP-A.1.
-
Probability: Independent Events Contains a concise
explanation of the independent events with worked examples.
Following the explanation are ten multiple choice exercises.
Use with CP-A.2.
-
Dependent Events: Conditional Probability Contains a
concise explanation of the conditional probability and dependent
events with worked examples. Following the explanation are ten
multiple choice exercises. Use with CP-A.3.
Constructed-response:
Khan Academy:
HSS-S-CP
includes HSS-CP.A.1 through HSS-CP.A.5 multiple choice and
constructed-response Common Core aligned problems.
Also see the
Probability unit with practice questions with videos.
Performance tasks:
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CP-B: Use the rules of probability to compute
probabilities of compound events.
Standards:
- CP-B.6. Find the conditional probability of
A given B as the fraction of B’s outcomes that
also belong to A, and interpret the answer in terms of the model.
- CP-B.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.
- CP-B.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.
- CP-B.9. (+) Use permutations and
combinations to compute probabilities of compound events and solve problems.
Technology-enhanced investigations:
Art of Problem Solving:
Counting and Probability Videos: These are the videos that go with
each of the sections within the 14-chapter textbook,
Introduction to Counting & Probability.
Calculator.net:
Use the following with CP-B. Explanations of theory related to
each calculator are included.
Cut the Knot:
Bayes' Theorem:
Bayes' Theorem is used as a direct application of conditional probability of
A given B: P(A|B) = P(A∩B)/P(B). Three practical examples are provided
with solutions. Use with CP-B.6.
Interactive Mathematics:
Counting and Probability - Introduction delves into factorial
notation, basic principles of counting, permutations, and combinations.
Probabiltiy theory and probability of an event are introduced.
Probabilities of two or more events addresses conditional probability,
independent and dependent events, mutually exclusive events, and Bayes'
Theorem. Concepts of probability distributions are presented along
with binomial probability distributions, Poisson probability
distributions, and normal probability distributions. An
interactive for a normal probability distribution is included. Use with
CP-A and CP-B.
Omnicalculator.com: Statistics calculators include a Probability
Calculator and Bayes' Theorem Calculator. Explanation of theory is
included for each.
Study.com: College Math: Probability
and Statistics: 11 video lessons. Each includes sample problems and a quiz. View
the probability lessons to "learn about types of events that express
probability, including simple, compound, complementary, independent and
dependent. You'll learn what the outcomes of these events are and how
you can calculate said probability. Other topics covered include the 'at
least one' rule of independent events, permutations, math combinations
and more." Use with CP-B.
Imagine Learning Classroom (formerly LearnZillion) subscription
needed:
- Lesson set:
Find the conditional probability of A given B as the fraction of
B's outcomes that also belong to A: Two video lessons:
Understand conditional probability, calculate conditional
probabilities using a two-way table. Aligns with CP-B.6.
- Lesson set:
Apply the addition rule: Two video lessons: Use addition
rule to calculate the probability of disjoint event A or B,
non-disjoint events and the addition rule, solve word problems
using the addition rule, calculate probabilities by using the
complement and addition rule, use conditional probability,
calculate different probabilities by working backward using the
addition rule. Aligns with CP-B.7.
MIT BLOSSOMS: Video lesson with additional teacher and learner
resources. Description is from the video summary.
The Monty Hall Problem or How to Outsmart a Game Show and Win a
Car: "This lesson teaches students how to make decisions in
the face of uncertainty by using decision trees. It is aimed for
high school kids with a minimal background in probability; the
students only need to know how to calculate the probability of two
uncorrelated events both occurring (ie flipping 2 heads in a row).
Over the course of this lesson, students will learn about the role
of uncertainty in decision making, how to make and use a decision
tree, how to use limiting cases to develop an intuition, and how
this applies to everyday life." Aligns with CP-A.2, CP-B.8,
CP-B.9, MD-B.5.a, MD-B.6, and MD-B.7.
Webmath.com: Permutations. Verify
your list of permutations for up to 8 entries.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives. Note: Within the
Wolfram Demonstration Project are
6 manipulatives addressing CP-B.6,
5 manipulatives for CP-B.7,
6 manipulatives for CP-B.8,
23 manipulatives for CP-B.9. Among those:
-
Conditional Probability: Aligns with CP-A.1, CP-A.2, CP-A.3,
CP-A.5, and CP-B.6.
-
Venn Diagrams: This shows all 127 nonempty unions and intersections of three sets.
You can show or hide the corresponding set notation, given in
terms of union, intersection, and complement. Aligns with CP-A.1, CP-B.6, CP-B.7.
-
Venn Diagrams for two sets: Aligns with CP-B.6, CP-B.7.
-
The perfect Venn Diagram: Aligns with CP-B.6, CP-B.7.
-
Urn sampling with or without replacement: Aligns with
CP-B.7, CP-B.8.
-
Probabilities of permutations: To calculate the probability of a selected sequence of results
from a series of independent events, multiply the probabilities
of each selected result occurring together. Aligns with
CP-B.8, CP-B.9.
-
Probabilities of combinations:
Order does not matter. Aligns with CP-B.8, CP-B.9.
-
Permutations: There are n! ways to arrange n
elements. Aligns with CP-B.9.
-
Number of handshakes: Aligns with CP-B.9.
-
Binomial probability distribution: Aligns with CP-B.9.
-
Clinking wineglasses: Aligns with CP-B.9.
-
Expanding a set of possibilities: Uses combinations of
different choices to make ice cream desserts. Aligns with
CP-B.9.
-
Pascal's triangle: Aligns with CP-B.9, APR-C.5.
YouTube videos:
GCSE Probability
Part 1: This video is from a professor in Great Britain who nicely
explains probability and shows examples: expected frequency, the typical
problems involving picking balls from a bag, mutually exclusive events,
independent events, and drawing diagrams to help identify possibilities.
Probability
Part 2: tree diagrams. The same professor in Great Britain
discusses the balls in a bag problems when making two picks, which
leads to tree diagrams to help determine probability involving
replacing after each pick, and then not replacing after a pick.
Zweigmedia.com: Tutorials and examples using fill-in or
multiple choice to test understanding. Learners can also choose to
do game versions of some topics. Use with CP-B:
Multiple Choice:
MathsIsFun.com:
- Mutually Exclusive Events: Contains a concise explanation of
the Addition Rule with examples for events that are mutually
exclusive and not mutually exclusive. Following the
explanation are ten multiple choice exercises. Use with
CP-B.7.
- Probability Tree Diagrams: Contains explanation of the use
of tree diagrams and the multiplication principle to calculate
probabilities of independent and dependent events.
Following the explanation are ten multiple choice exercises.
Use with CP-B.8.
- Permutations and Combinations: Contains a concise
explanation of the concepts and formulas used to calculate
permutations and combinations. Following the explanation
are ten multiple choice exercises. There is a
Permutations and Combinations Calculator. Use with CP-B.9.
Constructed-response:
Khan Academy:
HSS-S-CP
includes HSS-CP.B.6 through HSS-CP.B.9 multiple choice and
constructed-response Common Core aligned problems.
Also see the following practice questions with videos.
L. Spector: The Math Page: Topics in Precalculus: Use with CP-B.9:
-
Factorials: The concept of factorials and working with those is
needed for calculating permutations and combinations.
- Permutations and Combinations: These lessons help learners to
understand the concepts of permutations and combinations and include
practice problems with answers provided.
Performance tasks:
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Common Core Math:
Intro | HS Statistics & Probability Domain:
ID | IC | CP | MD |
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