Conditional Probability and the Rules of Probability
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”).
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Events
Characteristics
Unions
Intersections
Complements
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3 - Describe (events as subsets of a sample space based on characteristics of outcomes)
3 - Describe (events as unions, intersections, or complements)
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3 - I can describe events as subsets of a sample space based on characteristics of outcomes.
3 - I can describe events as unions, intersection, or complements based on characteristic of outcomes)
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A. An event can be described as a subset of a sample space using characteristic terms.
B. The terms union, intersection, and complement can be used to describe events based on characteristics of outcomes of a sample space.
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A.1 What is a sample space?
A.2 What is a subset?
A.3 How can an event be described as a subset of a sample space?
B.1 What is the relationship among sets in the union, intersection, and complements of the sets?
B.2 How do the terms or, and, and not relate to union, intersection, and complement?
B.3 What notation is used to describe subsets, unions, intersections, and complements of sets?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Independent Events
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2 - Understand (two events A and B are independent if the probability of A and B occurring together is the product of their probabilities)
4 - Determine (if two events are independent by using the characterization of the product of their probabilities)
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2 - I can understand that two events A and B are independent if the probability of A and B occurring together is the product of their individual probabilities.
4 - I can determine if two events are independent by using the product of their probabilities.
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A. Two events A and B are independent if the probability of A and B occurring together is the product of their individual probabilities.
B. It is possible to determine if two events are independent by using the characteristic of the product of their probabilities.
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A.1 Which probability characteristic reveals that two events A and B are independent?
B.1 How can independence between two events be determined by using the characteristic of the product of individual probabilities?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Conditional Probability
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2 - Understand (conditional probability of A given B in notation as P[A and B]/P[B])
4 - Interpret (independence of A and B as saying that the conditional probability of A given B is the same as the probability of A)
4 - Interpret (independence of A and B as saying that the conditional probability of B given A is the same as the probability of B)
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2 - I can understand that the conditional probability of A given B is expressed by the notation P[A and B]/P[B])
4 - I can 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.
4 - I can interpret the independence of A and B as saying that the conditional probability of B given A is the same as the probability of B.
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A. The conditional probability A given B is expressed in notation by P (A and B)/P(B).
B. The independence of two events A and B is the same as saying the conditional probability of A given B is the same as the probability of A.
C. The independence of two events A and B is the same as saying the conditional probability of B given A is the same as the probability of B.
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A.1 How is the conditional probability of A given B expressed in notation?
B.1 Why is the independence of two events A and B the same as saying the conditional probability of A given B is the same as the probability of A?
C.1 Why is the independence of two events A and B the same as saying the conditional probability of B given A is the same as the probability of B?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Two-way table
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3 - Construct (two-way frequency tables)
4 - Interpret (two-way frequency tables when two categories are associated with each classified object)
3 - Use (two way table as sample space)
5 - Decide (if events are independent)
5 - Approximate (conditional probabilities)
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3 - I can construct two-way frequency tables.
4 - I can interpret two-way frequency tables when two categories are associated with each object being classified.
3 - I can use the two-way table as a sample space.
5 - I can decide if events are independent.
5 - I can approximate conditional probabilities by using the two-way table as a sample space.
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A. Data from two categories associated with classified objects can be used to construct two-way frequency tables.
B. Two-way frequency tables can represent a sample space that can be interpreted to decide if events are independent.
C. Two-way frequency tables can represent a sample space that can be used to approximate conditional probabilities.
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A.1 When two categories are associated with classified objects, how can the data be used to construct a two-way frequency table?
B.1 How can a two-way frequency table’s sample space be used to decide if events are independent?
C.1 How can a two-way frequency table’s sample space be used to approximate conditional probabilities?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Concepts
Language
Situations
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2 - Recognize (concepts of conditional probability and independence in everyday situations)
5 - Explain (concepts of conditional probability and independence in everyday language)
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2 - I can recognize conditional probabilities and independence when they occur in everyday situations.
5 - I can explain conditional probabilities and independence with informal language as they relate to everyday situations.
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A. Conditional probability can be recognized in a variety of everyday situations, and can be described with informal language.
B. Independence of two events is demonstrated by a variety of everyday situations, and can be described by informal language.
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A.1 What are some examples of everyday events that demonstrate conditional probability?
A.2 How can conditional probability be explained with informal language?
B.1 What are some examples of everyday events that demonstrate independence?
B.2 How can independence of events be explained with everyday language?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Probability
Fraction
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3 - Find (conditional probability of A given B)
4 - Interpret (answer in terms of model)
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3 - I can find the conditional probability of A given B as a fraction of B’s outcomes that also belong to A.
4 - I can interpret my answer to conditional probability in terms of a model.
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A. The conditional probability of A given B can be expressed as the fraction of B’s outcomes that also belong to A.
B. The answer to a conditional probability can be interpreted in terms of a model.
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A.1 Why can the conditional probability of A given B be expressed as the fraction of B’s outcomes that also belong to A?
B.1 How can the answer to a conditional probability be interpreted in terms of a model?
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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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Answer
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3 - Apply (Addition rule P [A or B])
4 - Interpret (answer in terms of model)
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3 - I can apply the Addition Rule (A or B) = P (A) + P (B) - P(A and B).
4 - I can interpret an answer found by applying the Addition Rule in terms of the model.
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A. The Addition Rule P(A or B) = P(A) + P(B) - P(A and B) can be applied to find probabilities associated with a model.
B. An answer found using the Addition Rule is interpreted in terms of the model.
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A.1 How can the Addition Rule be applied to find probability?
B.1 How can an answer found by applying the Addition Rule be interpreted in terms of a model?
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(+) 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.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Answer
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3 - Apply (general Multiplication Rule in uniform probability model, P[A and B])
4 - Interpret (answer in terms of model)
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3 - I can apply the general Multiplication rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B).
4 - I can interpret the answer found by applying the general Multiplication Rule in terms of the model.
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A. The general Multiplication Rule, P(A and B) = P(A)P(B|A) = P(B)P(A|B), in a uniform probability model, can be applied to find probabilities associated with a model.
B. An answer found using the general Multiplication Rule is interpreted in terms of the model.
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A.1 How can the general Multiplication Rule in a uniform probability model be applied to find probabilities?
B.1 How can an answer found by the general Multiplication Rule be interpreted in terms of a model?
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(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
| Content | Skills | Learning Targets | Big Ideas | Essential Questions |
|---|---|---|---|---|
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Compound Events
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3 - Use (permutations and combinations)
3 - Compute (probabilities of compound events)
3 - Solve (problems)
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3 - I can use permutations and combinations.
3 - I can compute probabilities of compound events.
3 - I can solve problems by using permutations and combinations.
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A. Permutations and combinations can be used to compute probabilities of compound events.
B. Permutations and combinations can be used to solve a variety of problems.
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A.1 What is a permutation?
A.2 What is a combination?
A.3 How are permutations and combinations alike or different?
A.4 How can permutations and combinations be used to compute probabilities of compound events?
B.1 How can permutations and combinations be used to solve problems?
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