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**Standards**:

- MD-A.1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
- MD-A.2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
- MD-A.3. (+) Develop a probability
distribution for a random variable defined for a sample space in
which theoretical probabilities can be calculated; find the expected
value.
*For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.* - MD-A.4. (+) Develop a probability
distribution for a random variable defined for a sample space in
which probabilities are assigned empirically; find the expected
value.
*For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?*

**Technology-enhanced investigations:**

Stattrek.com Statistics Dictionary contains the following key vocabulary for this domain. Select the term from the Stattrek pull-down menu. Use with MD-A, MD-B:

expected value | mean | probability distribution | sample space |

event | probability | random variable | simple random sampliing |

Art of Problem Solving: Counting and Probability Videos: These are the videos that go with each of the sections within the 14-chapter textbook, Introduction to Counting & Probability. Chapter 11 videos are on expected value.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. Flu Math Games: "This video lesson shows students that math can play a role in understanding how an infectious disease spreads and how it can be controlled." Additional simulations are included. Aligns with Algebra standards SSE-B.3.c and REI-A.1; Function standards IF-C.8.b, BF-B.4.a, and LE-A-1.(a, c); and Statistics and Probability standards ID-B.6.a, IC-A.1, IC-B.4, CP-A.2, and MD-A.1.

Shodor Interactivate: Statistics and Probability: Using Probability to Make Decisions: Calculate expected values and use them to solve problems. This is a series of 11 lessons and 18 activities with virtual manipulatives, which are directly aligned to MD-A.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 8 manipulatives addressing MD-A.1, 4 manipulatives for MD-A.2, 11 manipulatives for MD-A.3, 5 manipulatives for MD-A.4. Among those:

- Game of Dice: Aligns with MD-A.1, MD-A.3, MD-B.5, and MD-B.6.
- Game of Dice 2: Aligns with MD-A.1, MD-A.3, MD-B.5, and MD-B.6.
- Dice Probabilities: Aligns with MD-A.1, MD-A.3.
- An Increasing Preference Distribution: Aligns with MD-A.1, MD-A.2, MD-A.3.
- Playing Cards: Aligns with MD-A.3, MD-A.4.
- Dice Roller and Penny Flipper: Aligns with MD-A.3, MD-A.4, MD-B.6.
- Urn Problem: Aligns with MD-A.3, MD-A.4, MD-B.6.

Zweigmedia.com: Tutorials and examples using fill-in or multiple choice to test understanding. Learners can also choose to do game versions of some topics. Use with MD-A:

- Sample spaces and events
- Random variables and distributions
- Bernoulli Trials and Binomial Random Variables: Note that Bernoulli Trials have two outcomes.

**Multiple Choice:**

Glencoe Online Learning Center, Geometry (2010) self-check quizzes: Ch. 13, Lesson 4: Simulations Aligns with CP-B.9, MD-A.2. Learners will also encounter questions on expected value.

OpenEd:

- Defining a random variable with a value: 5 multiple choice questions. Includes related review resources. Aligns with MD-A.1.
- Calculating expected value: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with MD-A.2.
- Probability distribution for a random variable: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with MD-A.3.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- Constructing probability distributions: Aligns with MD-A.1.
- Expected value: Compute given set of outcomes, probabilities, and payoffs. Aligns with MD-A.2, MD-B.5.
- Expected value with calculated probabilities: Aligns with MD-A.3.
- Expected value with empirical probabilities: Aligns with MD-A.4.

**Performance tasks:**

Georgia Standards of Excellence Framework: Pre-Calculus Unit 8: Probability. This unit contains eight tasks, including a culminating task: Design a Lottery Game. Tasks align with CP-B.8, CP-B.9, all standards in MD-A and MD-B, and related standards CP-A.2, CP-A.3, CP-B.6, and CP-B.7.

Illustrative Mathematics: Fred’s Fun Factory: This task aligns with standards MD-A.2, and MD-B.5 and MD-B.7.

Mathematics Assessment Project: Standards: High School: Statistics & Probability: http://map.mathshell.org/stds.php?standardid=1233

- Task 405: Modeling Conditional Probabilities 2
- Task 409: Modeling Conditional Probabilities 1: Lucky Dip

**Standards**:

- MD-B.5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
- MD-B.5.a. Find the expected payoff for a
game of chance.
*For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.* - MD-B.5.b. Evaluate and compare
strategies on the basis of expected values.
*For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.* - MD-B.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
- MD-B.7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

**Technology-enhanced investigations:**

Art of Problem Solving: Counting and Probability Videos: These are the videos that go with each of the sections within the 14-chapter textbook, Introduction to Counting & Probability. Chapter 11 videos are on expected value.

MIT BLOSSOMS: Video lesson with additional teacher and learner resources. Description is from the video summary. The Monty Hall Problem or How to Outsmart a Game Show and Win a Car: "This lesson teaches students how to make decisions in the face of uncertainty by using decision trees. It is aimed for high school kids with a minimal background in probability; the students only need to know how to calculate the probability of two uncorrelated events both occurring (ie flipping 2 heads in a row). Over the course of this lesson, students will learn about the role of uncertainty in decision making, how to make and use a decision tree, how to use limiting cases to develop an intuition, and how this applies to everyday life." Aligns with CP-A.2, CP-B.8, CP-B.9, MD-B.5.a, MD-B.6, and MD-B.7.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 2 manipulatives addressing MD-B.5, 18 manipulatives for MD-B.6, and 8 manipulatives for MD-B.7. Among those:

- Game of Dice: Aligns with MD-A.1, MD-A.3, MD-B.5, and MD-B.6.
- Game of Dice 2: Aligns with MD-A.1, MD-A.3, MD-B.5, and MD-B.6.
- Dice Roller and Penny Flipper: Aligns with MD-A.3, MD-A.4, MD-B.6.
- Urn Problem: Aligns with MD-A.3, MD-A.4, MD-B.6.
- Monty Hall Paradox: Aligns with CP-A.1, CP-A.2, CP-A.3, CP-A.5, and MD-B.6.
- Probability of Being Sick After Having Tested Positive for a Disease (Bayes' Rule): Aligns with CP-A.3, CP-A.4, CP-A.5, MD-B.7.
- Rock-Scissors-Paper: Aligns with MD-B.6, MD-B.7.
- A Probabilistic Forest Fire Model: Aligns with MD-B.7.

**Multiple-choice:**

Khan Academy: Practice questions with videos: Using probability to make fair decisions: Aligns with MD-A.2, MD-B.6, MD-B.7.

OpenEd: Analyze decisions and strategies using probability: 5 questions combine multiple choice and free response. Includes related review resources. Aligns with MD-B.7.

**Constructed-response:**

Khan Academy: Practice questions with videos.

- Expected value: Compute given set of outcomes, probabilities, and payoffs. Aligns with MD-A.2, MD-B.5.
- Making decisions with expected values: Aligns with MD-B.5, MD-B.7.

**Performance tasks:**

Georgia Standards of Excellence Framework: Pre-Calculus Unit 8: Probability. This unit contains eight tasks, including a culminating task: Design a Lottery Game. Tasks align with CP-B.8, CP-B.9, all standards in MD-A and MD-B, and related standards CP-A.2, CP-A.3, CP-B.6, and CP-B.7.

Illustrative Mathematics: Fred’s Fun Factory: This task aligns with standards MD-B.5 and MD-B.7, and MD-A.2.

Mathematics Vision Project, Secondary 2 Student Edition:

- Module 9: Probability: This module contains 6 classroom tasks. Standard MD-B.7 is found within task 1.

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