This lesson unit is intended to help teachers assess how well students are able to: make sense of a real life situation and decide what math to apply to the problem; understand and calculate the conditional probability of an event A, given an event B, and interpret the answer in terms of a model; represent events as a subset of a sample space using tables, tree diagrams, and Venn diagrams; and interpret the results and communicate their reasoning clearly.
1. Write a function that describes a relationship between two quantities ?
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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").
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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.
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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.
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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.
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5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.
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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.
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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.
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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.
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9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
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Make sense of problems and persevere in solving them.
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Reason abstractly and quantitatively.
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Construct viable arguments and critique the reasoning of others.
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Model with mathematics.
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