Abstract

Students are introduced to the concepts of probability and the basic set operations which are useful in solving probability problems that involve counting outcomes.

Objectives

Upon completion of this lesson, students will:

• have clarified the definition of probability
• have learned about outcomes in probability
• know how to calculate experimental probability

Activities

• Crazy Choices Game

Standards

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

• Probability

Key Terms

• probability
• outcome
• theoretical probability
• experimental probability
• union
• intersection

Student Prerequisites

• Arithmetic: Students must be able to:
  • use addition, subtraction, multiplication and division to solve set operations problems
  • calculate experimental probability when given the formula
  • keep simple records of data
• 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

• access to a browser
• pencil and paper
• Crazy Choices Worksheet
• In the Crazy Choices Game if the game is simulated using different random number-generating devices, some of the following will be needed:
  • dice with various numbers of sides
  • spinners
  • bags of numbered lotto chips, or chips of several colors, or marbles of several colors
  • coins
• The Crazy Choices Game Tally Table can be printed for each student or group of students to keep track of their data in the Crazy Choices Game.
• The Events and Set Operations discussion is best illustrated with color diagrams. Pens, pencils or crayons of 3-5 different colors (a set for each student or each group of students working independently) will help to visualize the ideas and to make problem solving more fun.

Lesson Outline

1. The lesson can begin by introducing the Crazy Choices Game to show how probabilities can be compared experimentally, and to lead students to the definition of probability.
2. Students can play the game in groups (2-10 people per group) using computer(s) or various random number generating devices (dice, spinners, etc.). The software keeps the necessary statistics:
   1. number of games played
   2. number of times each player won
   3. experimental probability of winning
3. If students play the game using hands-on materials, they may want to keep track of this data using the Crazy Choices Game Tally Table that can be reproduced for each group of students. Students should play a lot of games (50-100) if they want to obtain reliable statistics. The goal of the game is to determine which player has better chances of winning if players use different devices to determine whether they win. For example, to compare the chances of the player who flips a coin (winning in 1 out of 2 possible outcomes) and the chances of the player who rolls a six-sided die (winning if it rolls a 1 or 2, or in 2 out of 6 possible outcomes).
4. The advantage of the software is that it can simulate many games in a single run. This saves time, and helps students see how experimental probabilities get closer and closer to theoretical probabilities (the Law of Large Numbers).
5. Lead a discussion, based on the Crazy Choices Game, of Outcomes and Probability to introduce the ideas of "outcome" and "probability."
6. Students can try to answer the following questions individually, in group discussions or in discussion with the mentor. Each group of students can answer the whole set of questions, later sharing their answers and discussing them with other groups in order to refine the definitions and understanding.
   1. In the Crazy Choices Game, each player won in so many outcomes out of so many total outcomes. How can we define an outcome?
   2. If the total number of outcomes is the same for all players, it is easy to compare their chances. For example, the player who has four winning numbers on a six-sided die will win twice as often as the player who has two winning numbers. How do we compare the chances of the players if the total number of outcomes is different? Can we do it with experiments? Can we predict the results of the experiments approximately?
   3. What happens to experimental probabilities when we collect more and more data on the same game?
7. Next, initiate a discussion about Events and Set Operations.
8. Discussions of sets work best when they are based on problems, and when students can draw or use manipulatives to work on the problems. Students can work in small (2-4 people) groups, each group discussing a few problems and trying to answer the following questions in the process. Each group can draw a problem from Sample Problems on Set Operations and then come up with several more of their own problems of the same sort.
   1. What is the union of sets? Can you find out how many elements are in the union if you know how many are in each set? What other things do you need to know to answer that question?
   2. What is the intersection of sets?
   3. If each set describes an event, what events are described by the union and the intersection?
9. Students will minimize confusion if they solve a problem or two before attempting to answer the questions. They can start by answering the questions about the problems they solved, and then trying to generalize the answer. After each group works on the questions for a while (with the mentor helping each group as needed), all students can share and discuss their answers to the questions.

Alternate Outlines

This lesson can be rearranged in several ways.

• Combine this lesson with the Ideas That Lead to Probability lesson to give students an understanding of randomness and fair choice along with the concepts introduced here in one single lesson.
• Have the students first try playing the Crazy Choices Game using random number generators and recording their data on the Crazy Choices Game Tally Table, and then show them how quickly the computer can run the experiments for them. Point out how the more times the game is run, the closer the results get to the theoretical probability.
• Encourage the students to use colored pencils or pens to illustrate the solutions to the Sample Problems on Set Operations.
• If not used earlier, use the Probability vs. Statistics discussion to demonstrate the difference between these two concepts.

Suggested Follow-Up

After these discussions and activities, the students will have a clearer understanding of probability, outcomes, and set operations. If students have not yet seen Unexpected Answers, have them continue their exploration of probability and observe some unusual examples of probability games. After that, continue with Probability and Geometry, which brings to light the subtle difference between defining probability by counting outcomes and defining probability by measuring proportions of geometrical characteristics.