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K–12 teaching and learning · from the UNC School of Education

Fair or Unfair-Introduction to Probability

This lesson introduces the idea of probability theory. Students will use everyday experiences and intuitive understanding to gain an understanding of probability.

This lesson uses resources of Shodor Education Foundation, Inc. Permission has been granted for the use of the materials as part of the workshop-"Interactivate Your Bored Math Students."

A lesson plan for grade 3 Mathematics

By Bonnie Boaz

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Learning outcomes

Students will:

  • work with random number generators.
  • gain an understanding of the concept of probability
  • learn what it means for a game to be fair

Teacher planning

Time required for lesson

2 Days

Materials/resources

Racing Game Field:
http://www.shodor.org/interactivate/media/worksheets/193.pdf

Racing Game Worksheet:
http://www.shodor.org/interactivate/media/worksheets/102.pdf

The following are optional:

  • Dice with various numbers of sides.
  • Spinners.
  • Bag of lotto pieces with numbers.
  • Lottery machine.

Technology resources

Color monitor with access to the Internet for each student

Either a computer lab with Internet access or a one computer classroom with an Internet connection.

Pre-activities

Introduce the vocabulary words:

  • experimental probabilty
  • probability
  • random number generator
  • theoretical probability

Link to Shodor online dictionary:
http://www.shodor.org/interactivate/dictionary/#exprobability

Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

If I bet you that we could play a game and that I could win every time, would you believe me? This game is a racing game in which we take turns rolling a six sided die and advancing on the numbers that we each are assigned. I bet you I can assign us an equal quantity of numbers that we move on and no matter how many times we play I will always win. Then tell them that the numbers that you assign yourself are 1, 2, 3, 4, 5, and 6, while the numbers you assign the person who takes you up on your bet are 7, 8, 9, 10, 11, and 12. (If you are only playing with one die then it is impossible to roll anything higher than a 6 so the person assigned 6 - 12 will never move.) Who thinks this game is fair?


Adapted from Shodor.org

Activities

Let the students know what it is they will be doing and learning today. Say something like this:

Today, class, we are going to begin learning about random number generators and probability. We are going to use the computers to learn about random number generators and probability, but please do not turn your computers on until I ask you to. I want to show you a little about this activity first.

Teacher Input

  1. Lead a discussion about Fair Choice.
  2. 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?

Guided Practice

Have students play the “Racing Game with One Die” as an example of a game that is either fair or not. Make sure to adjust the settings on the game so that the race is only one step long. Since 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. Have them discuss different ways that they can make the game fair and not fair.
Individual Practice
Give implicit directions on what they are to do. For example “Today we are going to play the racing game. We are going to try to determine the likelihood of the cars winning if they advance when certain numbers are rolled. I want you to figure out how often the red car wins when he advances on one number, two numbers,…, all six numbers on the die…”
Racing Game (One Die) Suggestions

Several games can be based on this applet. Possible math goals of each game are indicated in parenthesis.

Game 1 (introduction of multiple-outcome events; set operations; computing particular probabilities): Set the controls on the applet in such a way that Player 1 wins if the die shows 1 or 2, and Player 2 wins if it shows 3, 4, 5, or 6. Also, set the number of steps to two. The game is not fair. Just how unfair is it? Simulate many games to find out the chances of each racer winning. Change winning numbers for the racers. How often do you expect each player to win? Run the program many times. Are the players winning as often as you expected?

Game 2 (multiple-outcome events; set operations; probability intuition): Set the winning numbers so that Player 1 has 1 and 2, and Player 2 has 3, 4, 5, and 6. Now reset the controls so it takes 3 steps to get to the finish. What happens to the chances of winning? Reset controls to more steps (4, 5, 10). What happens to the chances of winning now? Why?

Closure
You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Adapted from Shodor.org

Assessment

Students will devise a game using one dice and 2 players. Rules will be written for winning and advancing for each of the two players. Then in a separate paragraph the student will explain if the game is fair or not fair and why.

Rubric for scoring:
Game Rubric

  • 5pts-Covers game and rules indepth
  • 3pts-Covers some of the game rules
  • 1pt-Covers little of the game rules

Fair or Unfair Paragraph Rubric

  • 5pts.-Indicates a strong understanding of what makes a game Fair or Unfair
  • 3pts.-Indicates a good understanding with minor ommissions of what makes a game Fair or Unfair
  • 1pt.-Indicates some understanding with major ommissions or flaws of what makes a game Fair or Unfair
  • 0pts.-Does not indicate any understanding or totally flawed understanding of what makes a game Fair or Unfair

Supplemental information

None

Related websites

http://www.shodor.org/interactivate/lessons/IdeasLeadProbability/

http://www.shodor.org/interactivate/dictionary/#exprobability

Comments

Most of this lesson plan comes directly from the Shodor.org website. Also included on the website is the alternative plan:

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.

Here is the site for that discussion:
http://www.shodor.org/interactivate/discussions/ProbabilityOfSimulta/

If not used earlier, use the Probability vs. Statistics discussion to demonstrate the difference between these two concepts.
Here is the site for that discussion:
http://www.shodor.org/interactivate/discussions/ProbabilityVsStatis/

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.
Link to Unexpected Answers follows:
http://www.shodor.org/interactivate/lessons/UnexpectedAnswers/

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.

North Carolina Curriculum Alignment

Mathematics (2004)

Grade 3

  • Goal 4: Data Analysis and Probability - The learner will understand and use data and simple probability concepts.
    • Objective 4.03: Solve probability problems using permutations and combinations.