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High School – Conditional Probability and the Rules of Probability
Essential Questions:
1. How can we gather, organize and display data to communicate and justify results in the real world?
2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies?
Essential Vocabulary : independent events, conditional probability, two-way frequency table, independence, addition rule for probability, general
multiplication rule for probability, permutation, combination
HS.S-CP.2: Describe events as subsets of a sample space (the set of outcomes) using the characteristics (or categories) of the outcomes, or as
unions, intersections, or complements of other events(“or,” “and,” “not”).
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. that events can be described as subsets of
1. use characteristics to describe events.
sample space.
HS.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. that two events A and B are independent if 1. justify that two events are independent
1. independent events.
through the use of probability rules.
the probability of A and B occurring
together is the product of their
probabilities.
HS.S-CP.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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. determine the conditional probability of an
1. that the conditional probability of A given
1. conditional probability
event given that another event has already
B is the same as the probability of A, and
occurred.
the conditional probability of B given A is
the same as the probability of B.
HS.S-CP.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.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. construct a two-way frequency table from
1. two-way frequency tables
1. that frequency tables can be used to
a set of data and calculate conditional
determine conditional probabilities and
probabilities.
independence.
HS.S-CP.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.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. conditional probability and independence
1. that conditional probability and
1. recognize conditional probability and
independence occur in many everyday
independence in everyday situations.
situations and will be able to discuss the
role they play in these situations.
HS.S-CP.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.
Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that conditional probability can be
1. construct and appropriate model that
1. conditional probability.
modeled in a variety of ways.
displays the information needed to
calculate the conditional probability.
HS.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.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. Addition Rule for probability
1. that when using the Addition Rule for
1. compute the probability of A or B using
probability that the probability of A and B
the Addition Rule for probability.
should not be counted twice.
HS.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.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. General Multiplication Rule for probability 1. that when using the General Multiplication 1. compute the probability of A and B using
the General Multiplication Rule.
Rule that if events are dependent
conditional probabilities used.
HS S-CP 9: (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. permutation and a combination
1. that there is a difference between a
1. compute probabilities of compound events
permutation.
using permutations and combinations
High School – Making Inferences and Justifying Conclusions
Essential Questions:
1. How can we gather, organize and display data to communicate and justify results in the real world?
2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies?
Essential Vocabulary : random sampling, statistic, parameter, population, sample, theoretical and experimental statistics, an experiment and an
observational study, mean, proportion and margin of error, significance
HS.S-IC.1: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. make inferences about a population from a
1. that statistics is a process for making
1. random sample
random sample.
inferences about population parameters
2. statistic
based on a random sample from that
3. parameter
population.
4. population
5. sample
HS.S-IC.2: 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 5tails in a row cause you to question the model?
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. theoretical and experimental statistics
1. that consistent results with the given event 1. design a simulation that models a desired
are important.
event.
HS.S-IC.3: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how
randomization relates to each.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. recognize a survey, an experiment and an
1. the components of a survey
1. that randomization relates to sample
observational study.
2. an experiment and an observational study
surveys, experiments, and observational
2. explain the differences between a survey,
studies.
experiment, and observational study.
HS.S-IC.4: 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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. estimate a population mean or proportion
1. that repeated simulation is a way of
1. mean
and develop a margin of error by the use of
estimating a population mean or
2. proportion
simulation.
proportion and can be used to develop an
3. margin of error
estimate of a margin of error.
HS.S-IC.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are
significant.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. significance
1. that statistical significance is important.
1. decide if the differences between
parameters are significant.
HS.S-IC.6: Evaluate reports based on data.
Enduring Understandings
Students will understand…
1. that statistics are important in evaluating
the options presented in a report
Students will know…
1. how to read a report
Students will be able to…
1. evaluate reports based on data.
High School – Interpreting Categorical and Quantitative Data
Essential Questions:
1. How can we gather, organize and display data to communicate and justify results in the real world?
2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies?
Essential Vocabulary: categorical data, center, shape and spread, extreme data points, joint, marginal, and conditional relative frequencies, slope,
intercept, correlation coefficient, correlation, causation
HS.S-ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).
GFPS will also address Categorical Data in this Benchmark
High School Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. categorical data
1. that quantitative and categorical single
1. construct graphical representations of
variable data can be represented
categorical and quantitative data (dot plots,
graphically.
histograms, box plots, bar graphs, pie
graphs, stem and leaf plots . . .)
HS.S-ID.2: 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.
High School Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. use the appropriate measure of center and
1. how to recognize the differences in shapes 1. that based on the shape of a distribution
spread to describe a distribution.
that different measure of center and spread
of different distributions.
are used to describe the distribution.
HS.S-ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points
(outliers).
High School Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. center
1. that data sets can be compared and
1. describe the center, shape and spread of a
2. shape
contrasted using center, shape and spread.
data set.
3. spread
2. identify any extreme data points.
4. extreme data points
HS.S-ID.4: 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, and Montana Indian data sources
to estimate areas under the normal curve.
Students will know…
1. the properties of a normal distribution
Enduring Understandings
Students will understand…
1. that mean and standard deviation of a data
set can be used to estimate population
percentages.
Students will be able to…
1. use calculators, spreadsheets, and tables to
estimate areas under the normal curve.
2. calculate mean and standard deviation
HS.S-ID.5: 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.
Enduring Understandings
Students will understand…
Students will be able to…
Students will know…
1. that possible associations and trends may
1. construct a two way frequency table
1. joint
occur in data.
2. calculate the various joint, marginal and
2. marginal
conditional relative frequencies.
3. conditional relative frequencies
HS.S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. 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.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. represent two quantitative variables on a
1. that the appropriate selection of a
scatter plot and calculate an appropriate
regression model begins with a scatter plot
regression model for the data (using
and is then further analyzed by the use of
appropriate technology).
residuals.
2. that two quantitative variables can be
displayed graphically on a scatter plot and
there are various types of regression
models that can used to model the data
HS.S-ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. slope (rate of change) and intercept
1. that slope as a rate of change in the context 1. given a linear model, identify the slope and
of the data and intercept in the context of
the intercept.
the data needs to be interpreted.
HS.S-ID.8: Compute (using technology) and interpret the correlation coefficient of a linear fit. Benchmark
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. correlation coefficient
1. that technology can compute the linear
1. use technology to compute the linear
model and the correlation coefficient
model and the correlation coefficient.
HS.S-ID.9: Distinguish between correlation and causation.
Enduring Understandings
Students will know…
Students will understand…
1. correlation and causation
1. that correlation does not imply causation.
Students will be able to…
1. distinguish between correlation and
causation.
High School – Using Probability to Make Decisions
Essential Questions:
1. How can we gather, organize and display data to communicate and justify results in the real world?
2. How can we analyze data to make inferences and/or predictions, based on surveys, experiments, probability and observational studies?
Essential Vocabulary : random variable, expected value, theoretical probability, empirical probability, fair decision
HS.S-MD.1: (+) 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.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. assign probabilities to random variables
1. that a random variable for a quantity of
1. random variable
and graph the results.
interest can be defined by assigning a
numerical value to each event in a sample
space.
HS.S-MD.2: (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. expected value
1. that the expected value in a given situation 1. calculate expected values.
can be interpreted as the mean of the
probability distribution.
HS.S-MD.3: (+) 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 multiple-choice test where each question has four choices, and find the expected grade under various grading
schemes.
Enduring Understandings
Students will be able to…
Students will understand…
Students will know…
1. calculate the theoretical probabilities for a
1. that in a given situation the theoretical
1. theoretical probability
given distribution.
probabilities can be calculated to find the
expected value of a certain outcome.
HS.S-MD.4: (+) 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?
Enduring Understandings
Students will be able to…
Students will know…
Students will understand…
1. calculate the empirical probabilities for a
1. empirical probability
1. that in a given situation the empirical
given distribution.
probabilities can be calculated to find the
expected value of a certain outcome.
HS.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.
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. expected value.
1. that there is a difference between the
1. calculate expected values of various events
expected value and the payoff of various
and make decisions based upon these
events.
values.
HS.S-MD.6: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. fair decision
1. that probability is used in decision making. 1. calculate the probability of an outcome of
a given situation and determine if it is fair
or not by the definition of fair for that
event.
HS.S-MD.7: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the
end of a game).
Enduring Understandings
Students will know…
Students will understand…
Students will be able to…
1. probability’s role
1. that probability is used in decision making. 1. use probability to make a decision giving
the best chance of a desired outcome.