Abstract

The activity and two discussions that make up this lesson introduce ideas that are the basis of probability theory. By using everyday experiences and intuitive understanding, this lesson gives students a gradual introduction to probability.

Objectives

Upon completion of this lesson, students will:

• have been introduced to the concept of probability
• have worked with random number generators
• have learned what it means for a game to be fair

Activities

• The Racing Game with One Die

Standards

The activities and discussions in this lesson address the following Standard:

• Probability

Key Terms

• probability
• theoretical probability
• experimental probability
• equally likely
• random number generators
• rule of probabilities
• frequency view
• personal view

Student Prerequisites

• Arithmetic: Students must be able to:
  • use addition when working with dice
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag
  • use a browser such as Netscape for experimenting with the activities

Teacher Preparation

Students will need:

• access to a browser
• copies of the supplemental materials:
  • A printed copy of the Racing Game Field if they play the Racing Game with One Die without using the computer
  • Racing Game Worksheet
• The following objects or their pictures may be used in activities or to illustrate the discussions.
  • 1. Dice with various numbers of sides.
    
  • 2. Spinners. 3. Bag of lotto pieces with numbers. 4. Lottery machine.
  

  Lesson Outline

  1. The lesson can start by playing The Racing Game with One Die using computers decreasing the number of steps to the finish line to 1. As the game is used for illustration only, it can be played by each student individually, by groups of students, or by one person who broadcasts it for everybody else to see. Each group of students can come up with their own way of randomly choosing which players move on which rolls.
  2. Lead a discussion about fair choice. Students can use The Racing Game with One Die as an example of a game that is either fair or not. Next run The Racing Game with One Die with multiple steps to discover how this changes a players chances of winning.
  3. Students can start a discussion with each other or with mentor(s) by trying to answer the following questions:
     • What does it mean for a game to be fair?

     • What are some examples of fair games that use dice? What are some examples of unfair games?

     • Is it always easy to tell if the game is fair or not? Is it possible to make some experiments to find that out?

  4. Lead a discussion about random number generators. Everybody has some expertise with random choices. This fact allows the following questions to lead to spark a discussion:

     • How can you randomly choose between any given numbers? Can you use some devices to help you with that? What devices?

     • How do you know if the choice is truly random? How do you know if it is fair?

     Each group of students can come up with their own device for randomly choosing numbers. Different groups can discuss each other's devices and answer the above questions about them.

Alternate Outlines

This lesson can be rearranged in several ways.

• If computers are not available, after describing the game as it is in The Racing Game with One Die, students can use dice or spinners and a printed copy of the Racing Game Field to record their moves.
• To go into more depth, use the Racing Game with one Die activity to show that multiple steps increase whatever advantage a player has of winning, and then lead a discussion about the Probability of simultaneous events to reinforce the idea.
• If not used earlier, use the Probability vs. Statistics discussion to demonstrate the difference between these two concepts.
• Combine this lesson with the Introduction to the Concept of Probability lesson to give students an understanding of outcomes, events, and set operations along with the concepts of randomness and fair choice that are part of this lesson.

Suggested Follow-Up

After these discussions and activities, the students will have the beginnings of an understanding of probability, randomness and fair choice. The next lesson, Unexpected Answers, continues the initial exploration of probability and presents some unusual examples of games that require close examination to determine if they are fair.