Standard 1 : Calculate expected values and use them to solve problems (Archived)



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 - Archived
Grade: 912
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.

Lesson Plans

Name Description
Modeling Conditional Probabilities 2: This lesson unit is intended to help you assess how well students understand conditional probability, and, in particular, to help you identify and assist students who have the following difficulties representing events as a subset of a sample space using tables and tree diagrams and understanding when conditional probabilities are equal for particular and general situations.
Winner! Winner! - Expected Values:

County Fairs and Carnivals are wonderful. The smell of the food, the thrill of the rides, and the chance to win prizes make for a perfect combination.

In this Model Eliciting Activity, MEA Winner! Winner! is an activity about the Carnival coming to town with seven (7) exciting games: Rolling Dice, Bottle Bowling, Fishbowl, Weigh in on That! Pin the Wing, Colors, and Spin It! Student groups must look at the cost, rules, and awarded prizes in order to give the best rank order for these games. The groups will be expected to give details on the procedures they used to develop their ranking order. Furthermore, students will be expected to calculate the expected values of the Rolling Dice game and interpret the results in context.

All of a sudden more information is collected about these Carnival Games and now the savvy students must use this information to either revise or rewrite their procedures. Then, when the students are ready to attack this new dilemma, they will encounter Andy, a Carnival Lover, who has had a thrilling time but now discovers his funds are running low. Which games will the students select for Andy and why?

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics to solve an open-ended problem, while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought process. MEAs follow a problem-based, student centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEA’s visit: https://www.cpalms.org/cpalms/mea.aspx.

Probability:

This lesson is designed to develop students' understanding of probability in real life situations. Students will also be introduced to running experiments, experimental probability, and theoretical probability. This lesson provides links to discussions and activities related to probability as well as suggested ways to integrate them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

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.

Problem-Solving Task

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.

Problem-Solving Task

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.

Problem-Solving Task

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.