Download Standard 3.3 Probability

Curriculum Guide
High School – Math Standard 3.3
Math.3.3 - Probability models outcomes for situations in which there is inherent randomness.
Related Colorado Department of Education Sample Units:


Algebra II - Independently Lucky (Concepts: two-way frequency tables, associations, conclusions, categorical variables, unions, intersections,
complements, events, subsets, sample space, independence, probabilities, products, conditional probability, given)
Integrated Math 2 - Independently Lucky
Essential Questions - 21st Century Skills and Readiness Competencies (District):
1a. In statistics, what is an event? What is the complement of an event?
1a. What is the sample space of a probability experiment? How can you describe
events as subsets of a sample space by using characteristics of the outcome?
1a. What is a union? What is an intersection? What are specific examples of each?
1b. What is independent probability? What is conditional probability? How can you
differentiate between the two?
1b. How can you determine if two events are independent by looking at their individual
probabilities and the probability of the two events occurring together?
1c. How can you use the concept of conditional probabilities to explain independent
events?
1d. How can you construct a sample space for two-event activity (e.g., roll of two fair
dice) and use it to calculate conditional and compound probabilities?
1e. How can you use probability models to disprove the idea of a “winning streak”?
2. How can you use the concept of equal probability outcomes to determine probability
of compound events?
2a. How do Venn diagrams play a role in determining conditional probabilities?
2a. What is the difference between a combination and a permutation?
2a. How do you decide when to use combinations and permutations?
2a. In what types of situations would compound probability be used?
2b. What is the "Addition Rule" in statistics?
3. How does probability relate to obtaining insurance?
Evidence Outcomes (District):
1. Understand independence and conditional probability and use them to interpret
data.
a. Describe events as subsets of a sample space using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of
other events.
b. Explain 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.
c. Using the conditional probability of A given B as P(A and B)/P(B), interpret
the independence of A and B as saying that the conditional probability of A
given B is the same as the probability of A, and the conditional probability of B
given A is the same as the probability of B.
d. 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.
e. Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations.
2. Use the rules of probability to compute probabilities of compound events in a
uniform probability model.
a. 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.
b. 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.
3. Analyze the cost of insurance as a method to offset the risk of a situation.
Academic Vocabulary (District):
Addition Rule of Probability
complement
compound events
conditional probability
dependent events
element
Assessment (District):
Informal Checks for Understanding: Accuracy and thoroughness of student notes,
study guides, and graphic organizers will demonstrate informal evidence of concept
processing.
Observation/Dialogue: Teacher observation and monitoring of the frequency and
quality of student contributions to discussion and the sophistication of student
event
experimental probability
frequency table
independent events
intersection of sets
mutually-exclusive events
null set
outcomes
probability
sample space
set
simple events
subset
theoretical probability
two-way frequency table
union of sets
Venn diagram
responses to critical questioning will serve as informal evidence of concept processing
and skill development.
Suggested Activities/Strategies (District):
1a. The Marble Game task challenges students to demonstrate an understanding of
theoretical and empirical probability and to be able to represent the sample space for a
given event in an organized manner. Students must be able to represent and compare
probabilities as ratios, proportions, decimals or percent. Finally, students must
determine the game with the best probability of winning.

1a. Marble Game
1a. The Probability of Combined Events lesson teaches new concepts in probability,
and provides examples and practice for students. During the lesson, students
determine the probability of an event not occurring, calculate the intersection of two
events, determine when two events are independent, and calculate the union of two
events.

1a. Probability of Combined Events
1a. Designing and Examining Random Samples is a lesson wherein students examine
different sampling techniques and determine bias within them. Students should grasp
the importance of random sampling and using appropriate sample sizes for a given
population.

1a. Designing and Examining Random Samples
1a. As students work through Simulations Using the Random Number Table,
they perform a simulation in the five-step procedure, demonstrate the appropriate use
of the Random Number Table in order to complete the
simulations, analyze the data, interpret the results, and communicate their results both
in writing and orally.

1a. Simulations Using the Random Number Table
Resources/Technology (District):

1a. Notes on Concepts of Probability

1a. Union and Intersection of Events, Complement of an Event, and Odds

1b. Probability of Independent and Dependent Events Textbook Practice

Math Activity Buffon's Needle

Probability PowerPoint

Questions and Answers on Understanding Probability

Probability Notes, Definitions, Worked Examples

Probability Examples

http://www.insidemathematics.org/
o
Problems of the Month Resources
1a. In the Card Game task, students use probability to make predictions.

1a. Card Game
1a, 1b. The task Return to Fred's Fun Factory addresses standards regarding sample
space, independence, probability distributions, and permutations/combinations.

1a-b. Return to Fred's Fun Factory
1a, 1d, 2a. (The Titanic Tasks 1, 2, 3 all ask related questions, but are scaffold.) Titanic
1 Task guides students by asking the series of specific questions and lets them explore
the concepts of probability as a fraction of outcomes, and using two-way tables of data.
The emphasis is on developing understanding of conditional probability.

1a, 1d, 2a. The Titanic 1
Quiz/Test: Formally measured evidence of outcomes and overall standard
achievement will be established via quizzes, unit tests, and benchmark tests.
Performance Tasks/Projects: Student self-evaluation and peer evaluation will be
utilized for informal feedback on at least one activity per standard. Performance tasks
and projects will be formally assessed by the teacher using rubrics for holistic
evaluation.

Probability Test
1a, 1b, 1d, 1e, 2a. In the Titanic 2 Task, students explore the concepts of probability as
a fraction of outcomes and use two-way tables of data. The special emphasis is on
developing understanding of conditional probability and independence. Students use
conditional probability to decide if events A and B are independent (i.e., They check if
P(A|B)=P(A), or if P(B|A)=P(B).) In the last part of the task, students decide which
probabilities would be useful to answer the given question and calculate those
probabilities.

1a, 1b, 1d, 1e, 2a. The Titanic 2
1d, 1e, 2a. The Titanic 3 Task is an open-ended task. It poses a question, then the
students have to formulate a plan to answer it and use a two-way table of data to find
all the necessary probabilities. The special emphasis is on developing understanding
of conditional probability and independence. This task could be used as a group
activity where students cooperate to formulate a plan for answering the question and
calculating the appropriate probabilities. The task could lead to extended class
discussions about the different ways of using probability to justify general claims (e.g.,
Can we really say that first class passengers had a larger chance of being rescued?
Why or why not? What was the role of gender in the rescue procedures?)

1d, 1e, 2a. The Titanic 3
1e, 2a. Use the "Life is like a box of chocolates" example as follows. Suppose your box
of 36 chocolates have some dark and some milk chocolate, divided into cream or nutty
centers. Out of the dark chocolates, 8 have nutty centers. Out of the milk chocolates, 6
have nutty centers. One-third of the chocolates are dark chocolate. What is
the probability that you randomly select a chocolate with a nutty center? Given that it
has a nutty center, what is the probability you chose a dark chocolate? Show how you
determined your answers.
1a, 1b, 2a, 2b. Got Your Number provides students with leveled activities involving
number games. Students use number properties, number operations, organized lists,
systematic charting, counting methods/principles, and probability to solve the
problems.

1a, 1b, 2a, 2b. Got Your Number
1b, 1c, 1e, 2b. This task uses the same situation to explore different concepts of
probability theory. Part A explores the idea of independence of events. Part B lets
students explore the idea of conditional probability.

1b, 1c, 1e, 2b. Rain and Lightning
In Part C, students practice using the Addition Rule, P(A or B)=P(A)+P(B)−P(A and
B). Part D is a variant of Part C. Students must apply the Addition Rule and solve the
equation for the unknown quantity.
1b, 1c. Following videos provides practice in the probabilities of independent and
dependent events.

1b, 1c. Khan Academy Probability of Independent and Dependent Events
1d, 2a. How Do You Get to School? is a task designed as an assessment item. It
requires students to use information in a two-way table to calculate a probability and a
conditional probability.

1d, 2a. How Do You Get to School?
1e. The purpose of this task is to assess student ability to explain the meaning of
independence in a simple context. Consider expanding this task by asking students to
also explain what it would mean to say that the two events are not
independent. Provide probability values such as "On days when Janelle eats breakfast,
she is late to school about 20% of the time and on days when she does not eat
breakfast, she is late to school about 10% of the time." Then ask if this indicates that
the two events of "eats breakfast" and "late to school" are independent or not
independent.

1e. Breakfast Before School
2a. This worksheet (with teacher answer key) allows students to practice the concepts
of permutations and combinations.

2a. Permutations and Combinations Worksheet with Solutions
2b. Coffee at Mom's Diner is a task assessing a student's ability to use the addition
rule to compute a probability and to interpret a probability in context.

2b. Coffee at Mom's Diner
3. The articles "How Do Insurance Companies Determine Rates" and "How Do
Companies Determine Life Insurance Rates" provide students with background
information that can be incorporated into a classroom discussion or project. Students
should recognize how mathematics (especially probability and statistics) is a key
component in the determinations of rates made by insurance companies.

3. How Do Insurance Companies Determine Rates?

3. How Do Companies Determine Life Insurance Rates?
Cumulative Activity: Make a “Human Venn Diagram” where the sample space is all the
students in the class. Use lengths of rope to create three overlapping circles. Assign an
event to each of the three circles, such as: ate breakfast, brought a cell phone to
school, and got at least seven hours of sleep. Have students place themselves in the
appropriate locations. Using correct probability notation, identify each of the spaces in
the Venn diagram (don’t forget to include the space outside the circles). Analyze,
explore and record the results in terms of conditional probabilities.

