### HS-Stats-Prob-ppt-fo..

```+
NEKSDC CCSSM HS Statistics and Probability
Elaine Watson, Ed.D.
March 13, 2013
+
Arthur Benjamin TED Talk: Teach
Statistics Before Calculus
+
Essential Questions
 How
do we organize data so we can describe
it?
 What is an adequate description of data?
 How do we interpret and analyze data?
 How is statistics different from mathematics?
 Why is probability so closely linked to
statistics?
+
Structure of Presentation

General Overview of Statistics and Probability




GAISE Report
 Levels A, B, C
 Components of Statistical Problem Solving
Statistics Framework:
 Population versus Sample
 Descriptive Statistics and Inferential Statistics
Common Core Statistics and Probability
K – 5 Data (Categorical and Measurement)
 6 – 8 Statistics
 HS Statistics
Time to Work Problems
 Illustrative Mathematics
 Other Text Resources


+
Activity:





What CCSSM Math Practice Standards did you use?
How does this activity relate to Probability and Statistics?

What are all three-digit numbers that you can make using each of the
digits 1, 2, and 3, and using each digit only once?

Angel, Barbara, and Clara run a race. Assuming there is no tie, what are
all the possible outcomes of the race (first, second, third)?

You are watching Angel, Barbara, and Clara playing on a merry-goround. As the merry-go-round spins, what are all the different ways that
you see them from left to right?

You want to form two partners from among Angel, Barbara, and Clara,
by the following procedure: You choose one of them, and then let that
child choose her partner. What are all the possible outcomes of this
process?

In a 3 x 3 grid square, color three of the (unit) squares blue, in such a
way that there is at most one blue square in each row and in each
column. What are all the ways of doing this?

Find all of the symmetries of an equilateral triangle.
https://www.math.purdue.edu/~goldberg/Math453/eqi-slides-web.pdf
+
The Difference between Statistics
and Mathematics
“Statistics is a methodological discipline. It exist not for itself,
but rather to offer to other fields of study a coherent set of
ideas and tools for dealing with data. The need for such a
discipline arises from the omnipresence of variability.”
Moore and Cobb, 1997
Statistical problem solving and decision making depend on
understanding, explaining, and quantifying the variability in
the data.
It is this focus on variability in data that sets apart statistics from
mathematics.
GAISE Report
+
The Nature of Variability

Measurement Variability



Natural Variability


Individuals differ in size, aptitudes, abilities, opinions, emotional
responses
Induced Variability



Measuring devices can produce unreliable results
Changes in the system being measured (blood pressure from one
moment to the next)
Planting seeds in two different locations with different conditions
A carefully designed experiment can help determine the effects of
different factors
Sampling Variability

We use a sample of a population to make an estimate of a whole
population. However, it is rare for two samples to have identical
results. Proper sampling techniques and adequate sampling size
help to lower sampling variation.
GAISE Report
+
The Role of Context
“The focus on variability naturally gives statistics a
particular content that sets it apart from
mathematics, itself, and from other mathematical
sciences but there is more than just content that
distinguishes statistical thinking from mathematics.
Statistics requires a different kind of thinking,
because data are not just numbers, they are numbers
with a context. In mathematics, context obscures
structure. In data analysis, context provides
meaning.”
Moore and Cobb, 1997
GAISE Report
+
What is the role of context?
Mexico 2000
Male
Female
Population (in millions)
United States 2000
Female
Male
Population (in millions)
Statistics in Action (Watkins, Scheaffer, Cobb) Key Curriculum Press, 2004
+ Components of Statistical Problem Solving
I.
Formulate Questions
II.
Collect Data
III.
Analyze Data
IV.
Interpret Results
C
Compare to CCSSM
Modeling Cycle
+
Statistics is Modeling ★
In the HS CCSSM Standards, Modeling is considered not only a
Practice Standard, but also one of the six Conceptual
Categories (Numbers & Quantity, Algebra, Functions,
Modeling, Geometry, Statistics & Probability)
The Modeling standards are interspersed throughout the
Numbers & Quantity, Algebra, Functions, and Geometry
Conceptual Categories and are indicated with a ★.
However, ALL standards in the the Statistics Conceptual
Category are considered modeling standards. As a result, the
Category is marked with a ★, but the individual standards are
not.
+
Statistical Education:
A Developmental Process
GAISE Report: 3 Levels of Statistical Understanding
Level A
Students develop “data sense” – an understanding that data are
more than just numbers. Statistics changes numbers into
information.
Level B
Students see statistical reasoning as a process for solving
problems through data and quantitative reasoning
Level C
Students extend concepts learned in Levels A and B to cover a
wider scope of investigatory issues, and develop a deeper
understanding of inferential reasoning and its connection to
probability. Students also should have an increased ability to
explain statistical reasoning to others.
+
Statistical Education:
A Developmental Process
Although these three levels may parallel grade levels,
they are based on development in statistical literacy, not
age.
Thus, a student who has no prior experience with
statistics will need to begin with Level A concepts and
activities before moving to Level B and Level C.
For this reason, we will spend time looking at the 6 – 8
Standards, since many students currently in HS may be at
Level A or B.
The learning is more teacher driven at Level A, but
becomes more student-driven at Levels B and C.
+
GAISE Framework
Levels A,B,C in the Statistical Modeling Process
As we go through the next few slides on the Statistical
Modeling Process…
Refer to the one-page handout from the GAISE Document that
has a Table with Column Headings

Level A, Level B, Level C

I. Formulate Questions, II. Collect Data, III. Analyze Data, and
IV. Interpret Results
+ Components of Statistical Problem Solving
I. Formulate Questions
 Clarify
the problem at hand
 Formulate
one (or more) questions that can be
 The
question should anticipate variability
Which of these questions anticipates variability?
 How
tall am I?
 How tall are adult men in the USA?
Discuss the horizontal progression in the
Formulate Question row across Levels A, B, C
+ Components of Statistical Problem Solving
II. Collect Data
 Design
 Employ
a plan to collect appropriate data
the plan to collect the data
 Data
collection designs must acknowledge
variability in data, and frequently are intended to
reduce variability (random sampling)
 The
sample size influences the effect of sampling
variation (error)
Discuss the horizontal progression in the
Collect Data row across Levels A, B, C
GAISE Report
+ Components of Statistical Problem Solving
III. Analyze Data
 Select
 Use
appropriate graphical and numerical methods
these methods to analyze the data
 The
main purpose of statistical analysis is to give an
accounting of the variability in the data
 42% of those polled support the candidate with a
margin of error +/- 3% at the 95% confidence level
 Test scores are described as “normally distributed
with mean 450 and standard deviation 100
Discuss the horizontal progression in the
Analyze Data row across Levels A, B, C
GAISE Report
+ Components of Statistical Problem Solving
IV. Interpret Results
 Interpret
the analysis
 Relate
the interpretation to the original
question
 Statistical
presence of variability and must allow for it.
 Looking
beyond the data to make
generalizations must allow for variability in the
data.
GAISE Report
Discuss the horizontal progression in the
Interpret Results row across Levels A, B, C
+ At the point of question formulation, the statistician anticipates
the data collection, the nature of the analysis, and the possible
interpretations – all of which involve possible sources of
variability.
1.
Formulate
Question
4.
Interpret
Results
Variability
2.
Collect
Data
Variability
3.
Analyze
Data
Variability
+ In the end, the mature practitioner reflects upon all aspects of
data collection and analysis as well as the question itself when
interpreting results.
1.
Formulate
Question
4.
Interpret
Results
2.
Collect
Data
3.
Analyze
Data
+ Likewise, he or she links data collection and analysis to each
other and the other two components.
1.
Formulate
Question
4.
Interpret
Results
2.
Collect
Data
3.
Analyze
Data
+
Statistical Education:
A Developmental Process
The mature practitioner understands the role of
variability in the statistical problem-solving process.
Beginning students cannot be expected to make all
of these linkages. They require years of experience
and training.
The GAISE Report, therefore, provides a framework
for statistical education over three levels for K – 12.
A mature practicing statistician would go beyond
these three levels.
+
Resources for Deeper Study of
Statistics and Probability Standards
See handout for links to the following resources:

The GAISE Report

Progressions Documents

6 – 8 Statistics and Probability

High School Statistics and Probability
+
What is Meant by Statistics and
Probability?
Statistics
is the study of what has
structure.
Probability
is using the structure to
predict the future.
+
Probability:
An essential tool in mathematical
modeling and in statistics
 The
use of probability as a mathematical model
and the use of probability as a tool in statistics
employ not only different approaches, but also
different kinds of reasoning.
 Two
problems and the nature of the solutions will
illustrate the difference…
GAISE Report
+
Probability:
An essential tool in mathematical
modeling and in statistics

Problem 1:
Assume a coin is “fair.”
Question: If we toss the coin five times, how many heads will we get?

Problem 2:
You pick up a coin.
Question: Is this a fair coin?
Problem 1 is a mathematical probability problem.
Problem 2 is a statistics problem that can use the mathematical
probability model determined in Problem 1 as a tool to seek as solution.
GAISE Report
+
Statistics
Probability shows you the likelihood, or chances, for each of
the various future outcomes, based on a set of assumptions

Allows you to handle randomness (uncertainty) in a
consistent, rational manner.

Forms the foundation for statistical inference (drawing
conclusions from data), sampling, linear regression,
forecasting, risk management.
+
Statistics
With Statistics, you go from observed data to generalizing how
the world works.
The 7 hottest years on
record occurred in the
There is global
warming
Perhaps
without
justification
+
Statistics
world works, and then figure out what kind of data you are
likely to see.
Assume no global
warming.
How likely would we
be to get such high
temperatures as we
have been having?
Probability provides the justification for statistics.
Probability is the only scientific basis for decision-making in
the face of uncertainty.
+
Looking at the World through a
Statistical Lens
From a statistics lens, if
you are given a jar of
different colored jelly
beans (the world), you
won’t be able to see
what’s in the jar.
In the world of
jelly beans in
this jar
You will use a sampling
method to collect
information to infer the
percentage of each
color of jelly beans. Julie Conrad
+
Looking at the World from a
Probability Lens
From a probability lens,
you know the percentage
of each color of jelly
beans. (how this world
works)
In the world of
jelly beans in this
jar, I know that
20% are red
You predict what’s going
to happen when you
choose one at random.
+
A Framework for Studying
Statistics
The Practice of Statistics
Descriptive Statistics
Inferential Statistics
Statistics for K-8 Educators by Robert Rosenfeld
+
A Framework for Studying
Statistics
Descriptive Statistics
Graphs
Measures
of Center
Measures of
Variability
Measures of
Relationship
Placement of Individuals
Normal Curve
Statistics for K-8 Educators by Robert Rosenfeld
+
A Framework for Studying
Statistics
Inferential Statistics
Connections between
statistics and probability
Correlation
Inference and
Margin of
Error
Confidence
Intervals
Statistical
Significance
Statistics for K-8 Educators by Robert Rosenfeld
+
Population versus Sample

A population is the total set of individuals, groups, objects, or
events that the researcher is studying. For example, if we
were studying employment patterns of recent U.S. college
graduates, our population would likely be defined as every
college student who graduated within the past one year from
any college across the United States.

A sample is a relatively small subset of people, objects,
groups, or events, that is selected from the population.
United States, which would cost a great deal of time and
which would then be used to generalize the findings to the
larger population.

+
Descriptive Statistics

Descriptive statistics includes statistical procedures that we
use to describe the population we are studying. The data
could be collected from either a sample or a population, but
the results help us organize and describe data. Descriptive
statistics can only be used to describe the group that is being
studying. That is, the results cannot be generalized to any
larger group.

Descriptive statistics are useful and serviceable if you do not
need to extend your results to any larger group. However,
much of social sciences tend to include studies that give us
“universal” truths about segments of the population, such as
all parents, all women, all victims, etc.
+
Inferential Statistics
 Inferential
statistics is concerned with making
predictions or inferences about a population from
observations and analyses of a sample. That is, we
can take the results of an analysis using a sample
and can generalize it to the larger population that
the sample represents. In order to do this, however,
it is imperative that the sample is representative of
the group to which it is being generalized.
+
Compare and Contrast Different
Representations of the Same Data

Activity:

Look at the four graphical representations of the same data set on
page 42

Compare and contrast the graphs

What does each communicate?

Which do you think is the best representation of the data? Justify
+
K – 5 Foundation for Statistics
Two paths for K – 5 Data Standards
Categorical Data
Measurement Data
Sorting
Measuring
Representing on
Bar Graphs
Representing on
Line Plots
Supports
later work
on
bivariate
data and
two-way
tables in
Supports
later work
on
histograms
and box
plots in MS
+
Statistics Begins
In Grades K – 5, students have learned to represent and
interpret data using line plots and bar graphs.
6.SP.2 Understand that a set of data collected to answer a
statistical question has a distribution which can be described
by its center, spread, and overall shape.
Students learn to represent data using
histograms and box plots.
+
Common Core
Grade 6 Develop understanding of statistical variability.

Understand that a set of data collected to answer a statistical
question has a distribution which can be described by its

Recognize that a measure of center for a numerical data set
summarizes all of its values with a single number (mean,
median, mode), while a measure of variation describes how
its values vary with a single number (mean absolute
deviation or interquartile range).
+
Distributions
Describe Data by Measures of Center: Mean, Median, Mode
Big Idea: You should not decide which measure or measures of
center to use until you know the reason you are doing it. Pick
one that helps you tell the story of your data. If you have a very
small set of data, you may prefer not to do any of them, but just
to show all the data
Statistics for K-8 Educators by Robert Rosenfeld
+
Distributions
Describe Data by Measures of Variation: What is the spread of
the data?
Range:
20 – 13 = 7
Interquartile Range:
18 – 16 = 2
+
Distributions
Describe Data by Measures of Variation: How do numbers tend
to spread out from the center?
X: 4, 5, 7, 12
Mean = 7
Mean deviations: -3, -2, 0, 5
Absolute values of deviations: 3, 2, 0, 5
Mean absolute deviation: (3 + 2 + 0 + 5)/4 = 10/4 = 2.5
Mean absolute deviation is introduce in Grade 6
+
Standard Deviation is not
introduced until HS
Another Measures of Variation: How do numbers tend to spread out from
the center?
Standard Deviation – also summarizes how the individual numbers in a set
differ from the mean, but it is based on the squares of the deviations rather
than their absolute values.
X: 4, 5, 7, 12
Core
Standard Deviation is introduced in HS in Common
Mean deviations: -3, -2, 0, 5
Squares of the deviations: 9, 4, 0, 25
Mean of the squares: (9 + 4 + 0 + 25)/4 = 38/4 = 9.5
SD = square root (9.5) = 3.08
9.5 (the square of the SD) is called the
variance and is used as a measure of
+
Measures of Variation: Which do I use?
The mean absolute deviation is gaining popularity as the best
way to introduce measuring variability in grades K – 12, saving
standard deviation for more advanced work.
In research, the mean absolute deviation is often chosen as the
measure of variability when the median is used as the measure
of center, while the standard deviation is used when the mean
is the measure of center.
Statistics for K-8 Educators by Robert Rosenfeld
+
Distributions
Describe Data by Overall shape of data distribution:
Normal Distribution
Bimodal Distribution
Uniform Distribution
Skewed Left
Skewed Right
+
Distributions

Display numerical data in plots on a number line, including
dot plots, histograms, and box plots.
+
Distributions
Summarize numerical data sets in relation to their context, such
as by:

Reporting the number of observations.

Describing the nature of the attribute under investigation,
including how it was measured
and its units of measurement.
+
Distributions
Summarize numerical data sets in relation to their context, such
as by:

Giving quantitative measures of center (median and/or
mean) and variability (interquartile range and/or mean
absolute deviation), as well as describing any overall pattern
and any striking deviations from the overall pattern with
reference to the context in which the data were gathered.
+
Distributions
Summarize numerical data sets in relation to their context, such
as by:

Relating the choice of measures of center and variability to
the shape of the data distribution and the context in which
the data were gathered.

Which is the best measure of center for this data set?
+
Work and Share
Go to Illustrative Mathematics and check out the illustrations
for Grade 6 Statistics and Probability.
+
Understand that:
 Statistics can be used to gain information about a
population by examining a sample of the
population;
 Generalizations about a population from a
sample are valid only if the sample is
representative of that population.
 Random sampling tends to produce
representative samples and support valid
inferences.
+

Use data from a random sample to draw inferences about a
population with an unknown characteristic of interest.

Generate multiple samples (or simulated samples) of the
same size to gauge the variation in estimates or predictions.
For example:

Estimate the mean word length in a book by randomly
sampling words from the book;

Predict the winner of a school election based on randomly
sampled survey data. Gauge how far off the estimate or
prediction might be.
+
Draw informal comparative
populations.

Informally assess the degree of visual overlap of two
numerical data distributions with similar variabilities,
measuring the difference between the centers by expressing
it as a multiple of a measure of variability.

For example, the mean height of players on the basketball
team is 10 cm greater than the mean height of players on the
soccer team, about twice the variability (mean absolute
deviation) on either team; on a dot plot, the separation
between the two distributions of heights is noticeable.
+
Draw informal comparative
populations.

Use measures of center and measures of variability for
numerical data from random samples to draw informal

For example, decide whether the words in a chapter of a
seventh-grade science book are generally longer than the
words in a chapter of a fourth-grade science book.
+
Probability of a Chance Event is
a Number Between 0 and 1

Understand that the probability of a chance event is a
number between 0 and 1 that expresses the likelihood of the
event occurring.

Larger numbers indicate greater likelihood.

A probability near 0 indicates an unlikely event, a probability
around 1/2 indicates an event that is neither unlikely nor
likely, and a probability near 1 indicates a likely event.
+
Approximate Probability by
Collecting Data, Observing, and
Predicting

Approximate the probability of a chance event by collecting
data on the chance process that produces it and observing its
long-run relative frequency, and predict the approximate
relative frequency given the probability.

For example, when rolling a number cube 600 times, predict
that a 3 or 6 would be rolled roughly 200 times, but probably
not exactly 200 times.
+
Develop, Use, and Evaluate
Probability Models

Develop a probability model and use it to find probabilities
of events. Compare probabilities from a model to observed
frequencies; if the agreement is not good, explain possible
sources of the discrepancy.

Develop a uniform probability model by assigning equal
probability to all outcomes, and use the model to determine
probabilities of events.

For example, if a student is selected at random from a class,
find the probability that Jane will be selected and the
probability that a girl will be selected.
+
Develop, Use, and Evaluate
Probability Models

Develop a probability model (which may not be uniform) by
observing frequencies in data generated from a chance
process.

For example, find the approximate probability that a spinning
penny will land heads up or that a tossed paper cup will land
open-end down.

Do the outcomes for the spinning penny appear to be
equally likely based on the observed frequencies?
+
Probability of Compound Events
Using Organized Lists, Tables, Tree Diagrams,
Simulation

Understand that, just as with simple events, the probability of a
compound event is the fraction of outcomes in the sample space
for which the compound event occurs.

Represent sample spaces for compound events using methods
such as organized lists, tables and tree diagrams. For an event
described in everyday language (e.g., “rolling double sixes”),
identify the outcomes in the sample space which compose the
event.

Design and use a simulation to generate frequencies for
compound events.

For example, use random digits as a simulation tool to
approximate the answer to the question: If 40% of donors have
type A blood, what is the probability that it will take at least 4
donors to find one with type A blood?
+
Work and Share
Go to Illustrative Mathematics and check out the
illustrations for Grade 7 Statistics and Probability.
+
Investigate patterns of
association in bivariate data.

Construct and interpret scatter plots for bivariate
measurement data to investigate patterns of association
between two quantities.

Describe patterns such as clustering, outliers, positive or
negative association, linear association, and nonlinear
association.
+
Investigate patterns of
association in bivariate data.

Know that straight lines are widely used to model
relationships between two quantitative variables.

For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by
judging the closeness of the data points to the line.
+
Investigate patterns of
association in bivariate data.
Students in Grade 8 study linear functions, which gives them
experience with the concept of m and b in of f(x) = mx + b

Use the equation of a linear model to solve problems in the
context of bivariate measurement data, interpreting the slope
and intercept.
For example, in a linear model for a biology experiment,
interpret a slope of 1.5 cm/hr as meaning that an additional
hour of sunlight each day is associated with an additional 1.5
cm in mature plant height.
+
Investigate patterns of
association in bivariate data.
Students in Grade 8 study linear functions, which gives them
experience with the concept of m and b in of f(x) = mx + b

Use the equation of a linear model to solve problems in the
context of bivariate measurement data, interpreting the slope
and intercept.
For example, in a linear model for a biology experiment,
interpret a slope of 1.5 cm/hr as meaning that an additional
hour of sunlight each day is associated with an additional 1.5
cm in mature plant height.
+
Investigate patterns of
association in bivariate data.

Understand that patterns of association can also be seen in
bivariate categorical data by displaying frequencies and
relative frequencies in a two-way table.

Construct and interpret a two-way table summarizing data on
two categorical variables collected from the same subjects.

Use relative frequencies calculated for rows or columns to
describe possible association between the two variables.
For example, collect data from students in your class on whether
or not they have a curfew on school nights and whether or not
they have assigned chores at home. Is there evidence that those
who have a curfew also tend to have chores?
+
Work and Share
Go to Illustrative Mathematics and check out the
illustrations for Grade 8 Statistics and Probability.
+
HS Statistical Analysis

Teaching Channel Video
Statistical Analysis Lesson to Rank Baseball Players
+
HS Common Core
Statistics and Probability Domains

Interpreting Categorical and Quantitative Data (S-ID)

Making Inferences and Justifying Conclusions (S-IC)

Conditional Probability and the Rules of Probability (S-CP)

Using Probability to Make Decisions (S-MD)
+
Interpreting Categorical and
Quantitative Data (S-ID)
Summarize, represent, and interpret data on a single count
or measurement variable

dot plots, histograms, and box plots

Use statistics appropriate to the shape of the data distribution
to compare center (median, mean) and spread (interquartile
range, standard deviation) of two or more different data sets.

Interpret differences in shape, center, and spread in the
context of the data sets, accounting for possible effects of
extreme data points (outliers).
+
Interpreting Categorical and
Quantitative Data (S-ID)
Summarize, represent, and interpret data on a single count
or measurement variable

Use the mean and standard deviation of a data set to fit it to a
normal distribution and to estimate population percentages.

Recognize that there are data sets for which such a procedure
is not appropriate.

Use calculators, spreadsheets, and tables to estimate areas
under the normal curve.
+
Interpreting Categorical and
Quantitative Data (S-ID)
Summarize, represent, and interpret data on two
categorical and quantitative variables

Summarize categorical data for two categories in two-way
frequency tables.

Interpret relative frequencies in the context of the data
(including joint, marginal, and conditional relative
frequencies).

Recognize possible associations and trends in the data.
+
Interpreting Categorical and
Quantitative Data (S-ID)
Summarize, represent, and interpret data on two categorical
and quantitative variables

Represent data on two quantitative variables on a scatter plot,
and describe how the variables are related.

Fit a function to the data; use functions fitted to data to solve
problems in the context of the data.

Use given functions or choose a function suggested by the
context. Emphasize linear, quadratic, and exponential models.


Informally assess the fit of a function by plotting and analyzing
residuals.
Fit a linear function for a scatter plot that suggests a linear
association.
+
Interpreting Categorical and
Quantitative Data (S-ID)
Interpret Linear Models

Interpret the slope (rate of change) and the intercept
(constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation
coefficient of a linear fit.

Distinguish between correlation and causation.
+
Correlation and Causation

TED Talk on Correlation and Causation
+
Making Inferences and Justifying
Conclusions (S-IC)
Understand and evaluate random processes underlying
statistical experiments

Understand statistics as a process for making inferences
about population parameters based on a random sample
from that population.

Decide if a specified model is consistent with results from a
given data-generating process, e.g., using simulation.

For example, a model says a spinning coin falls heads up
with probability 0.5. Would a result of 5 tails in a row cause
you to question the model?
+
Making Inferences and Justifying
Conclusions (S-IC)
Make inferences and justify conclusions from sample
surveys, experiments, and observational studies

Recognize the purposes of and differences among sample
surveys, experiments, and observational studies; explain how
randomization relates to each.

Use data from a sample survey to estimate a population mean
or proportion; develop a margin of error through the use of
simulation models for random sampling.
+
Making Inferences and Justifying
Conclusions (S-IC)
Make inferences and justify conclusions from sample
surveys, experiments, and observational studies

Use data from a randomized experiment to compare two
treatments; use simulations to decide if differences between
parameters are significant.

Evaluate reports based on data.
+
Conditional Probability and Rules
of Probability (S-CP)
Understand independence and conditional probability and
use them to interpret data

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”).

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.
+
Conditional Probability and Rules
of Probability (S-CP)
Understand independence and conditional probability and
use them to interpret data
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.
+
Conditional Probability and Rules
of Probability (S-CP)
Understand independence and conditional probability and use them
to interpret data

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

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?

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?

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?
+
Conditional Probability and Rules
of Probability (S-CP)
Understand independence and conditional probability and
use them to interpret data

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.
+
Conditional Probability and Rules
of Probability (S-CP)
Understand independence and conditional probability and use
them to interpret data

Estimate the probability that a randomly selected student from
your school will favor science given that the student is in tenth

Do the same for other subjects and compare the results.

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.
+
Conditional Probability and Rules
of Probability (S-CP)
Use the rules of probability to compute probabilities of
compound events in a uniform probability model

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.

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.
+
Conditional Probability and Rules
of Probability (S-CP)
Use the rules of probability to compute probabilities of
compound events in a uniform probability model
Honors or Year 4
(+) 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.
(+) Use permutations and combinations to compute
probabilities of compound events and solve problems.
+
Using Probability to Make
Decisions (S-MD)
Calculate Expected Values and Use Them to Solve Problems
All of the Standards in this Domain are for Honors Students or
Year 4 Classes. They will not be assessed by SBAC.
+
Work and Share
Go to Illustrative Mathematics and check out the
illustrations for HS Statistics and Probability.