### Course 2 Unit 7 - Patterns in Chance 1st Edition

Patterns in Chance is the second unit in Course 2 from the statistics and probability strand of the Contemporary Mathematics in Context curriculum. By this time, students will have developed the ability to model a wide variety of situations using concepts and methods from algebra, statistics, probability, geometry, trigonometry, and discrete mathematics. This unit builds on simulation methods for investigating probabilistic situations as developed in the Course 1 unit, Simulation Models. (See the descriptions of Course 2 Units.)

#### Unit Overview

Patterns in Chance introduces students to probability distributions and to the Multiplication Rule for independent events: P(A and B) = P(A)P(B). Students learn how to construct waiting-time distributions both theoretically and by simulation; then they discover how to find the average waiting time. The specific problem that motivates the unit and is studied throughout is that of rolling two dice until doubles appears.

 Unit Objectives To construct a probability distribution by simulation and use it to understand and analyze the probabilistic situation To explore in depth the geometric, or waiting-time, distribution To understand some fundamental ideas of probability: independent events, the Multiplication Rule, the expected value of a probability distribution, and rare events

#### Sample Overview

In the sample material from Lesson 3 of Patterns in Chance, students construct waiting-time or geometric distributions theoretically. In Lessons 1 and 2, students revisit waiting-time distributions which were first developed in Course 1, Unit 7, and develop the Multiplication Rule for Independent Events. Students discover the formula for the expected value of a waiting-time distribution in Lesson 4.

#### Instructional Design

Throughout the curriculum, interesting problem contexts serve as the foundation for instruction. As lessons unfold around these problem situations, classroom instruction tends to follow a common pattern as elaborated under Instructional Design.

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#### How the Statistics and Probability Strand Continues

In Course 3, students will continue their study of this strand in Unit 2, Modeling Public Opinion and Unit 5 Patterns in Variation. In Unit 2, students study survey methods, sampling, sampling distributions, relationships between a sample and a population, confidence intervals, and margin of error. In Unit 5, students extend their understanding of measurement of variation, use the normal distribution as a model of variation, and are introduced to probability and statistical inference involved in the control charts used in industry for statistical process control.

In Course 4, students extend their understanding of the binomial distribution and how the binomial distribution is used in statistical inference to test a single proportion and to compare two treatments in an experiment.

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