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# The Addition Rule

## • Describe events as subsets of a sample space (the set of outcomes) using characteristics of the outcomes or as unions, intersections, or complements of other events (“or”, “and,” “not”) (S-CP.A.1). • Develop the Addition Rule to compute probabilities of compound events in a uniform probability model (S-CP.B.7). • Apply the Addition Rule in a uniform probability model and interpret the answer in terms of the model (S-CP.B.7).

In grade 7, students described sample spaces by creating lists, tables, and diagrams. They determined P(A or B) by counting occurrences of simple events in A ∪ B. In this section, students learn to calculate P(A or B) in terms of P(A), P(B), and P(A and B). They begin by considering compound events as subsets of sample spaces, noting that in the sample space: the event “not A” is the complement of A; that the event “A and B” is the intersection of sets A and B; and that the event “A or B” is the union of A and B. The section culminates in a task where the Addition Rule is used to compute a probability.

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## Tasks

1 Describing Events

WHAT: Students list events in a sample space and describe unions, intersections, or complements of given compound events.

WHY: Being able to view events as unions, intersections, or complements of given compound events is a preliminary for calculation of the probability of one event in terms of probabilities of other events.

2 The Addition Rule

WHAT: This task describes how to use a diagram to represent compound events in a sample space where each simple event is assumed to be equally likely (thus generating a uniform probability model). This is used to develop the Addition Rule for calculating the probability of the union of two events.

WHY: This task gives students the opportunity to build on the understanding of compound events in a sample space from the task "Describing Events" (https://www.illustrativemathematics.org/illustrations/1884) to develop and apply the Addition Rule.

3 Coffee at Mom's Diner

WHAT: Students are told numerical values for P(A), P(B), and P(A or B), and are asked to find P(A and B) and interpret its value in the context from which the values arose.

WHY: In the previous task, students developed the Addition Rule. This task asks them to use it in a context that differs at least three ways: diner vs student council; written descriptors vs Venn diagram; and estimated frequencies vs uniform probability model.