 # Standard 1 : Calculate expected values and use them to solve problems

This document was generated on CPALMS - www.cpalms.org

### General Information

Number: MAFS.912.S-MD.1
Title: Calculate expected values and use them to solve problems
Type: Cluster
Subject: Mathematics
Domain-Subdomain: Statistics & Probability: Using Probability to Make Decisions

#### Related Standards

This cluster includes the following benchmarks
 Code Description MAFS.912.S-MD.1.1: 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. ★ MAFS.912.S-MD.1.2: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. ★ MAFS.912.S-MD.1.3: 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. ★ MAFS.912.S-MD.1.4: 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? ★

#### Related Access Points

This cluster includes the following access points.

#### Access Points

 Access Point Number Access Point Title MAFS.912.S-MD.1.AP.3a: Determine the theoretical probability of multistage probability experiments (e.g., draw or select a representation of the theoretical probability for a sample space). MAFS.912.S-MD.1.AP.3b: Collect data from multistage probability experiments. MAFS.912.S-MD.1.AP.3c: Compare actual results of multi-stage experiment with theoretical probabilities (e.g., make a statement that describes the relationship between the actual results of a multistage experiment with its theoretical probabilities [ex., more, less, same, different, equal]).

#### Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

#### Educational Software / Tool

 Name Description Free Graph Paper: A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

#### Perspectives Video: Experts

 Name Description How Math Models Help Insurance Companies After a Hurricane Hits: Hurricanes can hit at any time! How do insurance companies use math and weather data to help to restore the community? Download the CPALMS Perspectives video student note taking guide. Probabilistic Weather Modeling: Meteorologist from Risk Management discusses the use of probability in predicting hurricane tracks. Download the CPALMS Perspectives video student note taking guide. History of Probability and the Problem of Points: What was the first question that started probability theory? Download the CPALMS Perspectives video student note taking guide.

 Name Description Bob's Bagel Shop: The purpose of this task is to assess a student's ability to compute and interpret the expected value of a random variable.

#### Teaching Idea

 Name Description Conditional Probability and Probability of Simultaneous Events: This lesson is designed to further students' practice with probability as well as introduce them to conditional probability and probabilities of simultaneous independent events. The lesson provides links to discussions and activities related to conditional and simultaneous probabilities as well as suggested ways to integrate them into the lesson. Finally, this lesson provides links to follow-up lessons designed for use in succession with this one.

#### Video/Audio/Animation

 Name Description MIT BLOSSOMS - 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. During this lesson, students will see and use both deterministic and probabilistic models and will learn by doing through role-playing exercises. There are no formal prerequisites, as students in any high school or even middle school math class could enjoy this learning video. But more advanced classes can go into the optional applied probability modeling that accompanies the module in a downloadable pdf file. The primary exercises between video segments of this lesson are class-intensive simulation games in which members of the class 'infect' each other under alternative math modeling assumptions about disease progression. Also there is an occasional class discussion and local discussion with nearby classmates.

#### Virtual Manipulatives

 Name Description Advanced Fire Simulator - Shodor: In this online activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree to the other. This simulation also records data for each trial including the burn probability, where the fire started, the percent of trees burned, and how long the fire lasted. This activity allows students to explore the idea of chaos in a simulation of a realistic scenario. Supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet are linked to the applet. Interactive Marbles: This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet. Plinko Probability: The students will play a classic game from a popular show. Through this they can explore the probability that the ball will land on each of the numbers and discover that more accurate results coming from repeated testing. The simulation can be adjusted to influence fairness and randomness of the results.

#### Worksheet

 Name Description Replacement and Probability: This lesson is designed to develop students' understanding of sampling with and without replacement and its effects on the probability of drawing a desired object. The lesson provides links to discussions and activities related to replacement and probability as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons that are designed to be used in succession with the current one.

#### Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

 Title Description Bob's Bagel Shop: The purpose of this task is to assess a student's ability to compute and interpret the expected value of a random variable.

#### Virtual Manipulatives

 Title Description Advanced Fire Simulator - Shodor: In this online activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree to the other. This simulation also records data for each trial including the burn probability, where the fire started, the percent of trees burned, and how long the fire lasted. This activity allows students to explore the idea of chaos in a simulation of a realistic scenario. Supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet are linked to the applet. Interactive Marbles: This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet. Plinko Probability: The students will play a classic game from a popular show. Through this they can explore the probability that the ball will land on each of the numbers and discover that more accurate results coming from repeated testing. The simulation can be adjusted to influence fairness and randomness of the results.

#### Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.