Ideas that Lead to Probability

Ideas that Lead to Probability

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

Standards

The activities and discussions in this lesson address the following NCTM standards:

Data Analysis and Probability

Understand and apply basic concepts of probability

understand and use appropriate terminology to describe complementary and mutually exclusive events
use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models

Links to other standards.

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.

Key Terms

This lesson introduces students to the following terms through the included discussions:

Lesson Outline

Focus and Review
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?
Objectives
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
- Lead a discussion about Fair Choice
- 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:
  1. How can you randomly choose between any given numbers? Can you use some devices to help you with that? What devices?
  
  2. How do you know if the choice is truly random? How do you know if it is fair?
Guided Practice
- Have students can use 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.
Independent Practice
- Now have the students play The Racing Game with One Die. Each group of students can come up with their own way of randomly choosing which players move on which rolls.
- Also have them adjust the number of steps in the race and observe the affect it has on the probability that one player will win over the other.
- You might also challenge the students to find the combination of race length and numbers needed to cause one player to have a specific probability of winning.
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.

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.