Course 2 Unit 8 - Probability Distributions

In Course 1, students first studied probability in Unit 8, Patterns in Chance. This unit in Course 2 develops student ability to solve problems involving chance by constructing sample spaces of equally likely outcomes or geometric models and to approximate solutions to more complex probability problems by using simulation. (See the descriptions of Course 1 Units and Course 2 Units.)

Unit Overview
This unit continues to address one of the main goals in teaching probability—for students to understand that while they may not know the next outcome in a random process, they often have a very good idea of what the distribution of the next ten thousand outcomes will look like. Further, students should understand that the distributions for related random processes have characteristic shapes. Waiting-time distributions are skewed right. Binomial distributions are symmetric when p = 0.5, skewed right when p < 0.5, and skewed left when p > 0.5. For each of these distributions, you can compute a measure of center, typically the mean (expected value), and a measure of spread, typically the standard deviation. These distributions can be constructed approximately using simulation or constructed exactly using mathematical theory.
Another important goal of this unit is for students to understand why, when events A and B are independent, we multiply to find the probability of A and B both occurring. This idea was introduced in Course 1 in Patterns in Chance and will be developed further through area models and through the intuitive idea that in a waiting-time situation, the proportion of people who have their first success on the xth trial is p times the proportion of people who were left after the previous trial.

Objectives of the Unit
  • Interpret and compute conditional probabilities
  • Use the Multiplication Rule to find P(A and B), when events A and B are independent and when they are not independent
  • Compute the expected value (mean) of a probability distribution
  • Identify waiting-time situations and construct waiting-time distributions

Sample Overview
The sample material for this unit is Investigation 3 of Lesson 2. The Connections and Reflections tasks from the homework section for Lesson 2 are included to provide you samples of tasks written to help students build links between mathematical topics they have studied in the lesson and to connect those topics with other mathematics that they know (Connections tasks) and tasks that provide opportunities for students to re-examine their thinking about ideas in the lesson (Reflections tasks).

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 four-phase cycle of classroom activities—Launch, Explore, Share and Summarize, and Apply. This instructional model is elaborated under Instructional Design.

View Sample Material
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How the Statistics and Probability Strand Continues
Course 3 Unit 1, Reasoning and Proof, develops student understanding of formal reasoning in geometric, algebraic, and statistical contexts and of basic principles that underlie those reasoning strategies, design of experiments including the role of randomization, control groups, and blinding; sampling distribution; randomization test; and statistical significance.
     Course 3 Unit 4, Samples in Variation, extends student understanding of the measurement of variability, develops student ability to use the normal distribution as a model of variation, introduces students to the binomial distribution and its use in decision making, and introduces students to the probability and statistical inference involved in control charts used in industry for statistical process control. Topics studied include normal distributions, standardized scores, binomial distributions (shape, expected value, standard deviation), the normal approximation to a binomial distribution, odds, statistical process control, control charts, and the Central Limit Theorem. (See the CPMP Courses 1-4 descriptions.)

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