Interactivate: 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 Addressed:

Textbooks Aligned:

Student Prerequisites

Arithmetic: Student 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 for experimenting with the activities

Teacher Preparation

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. Bag of lotto pieces with numbers. Lottery machine.

Key Terms

experimental probability	The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played
probability	The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability
random number generators	A device used to produce a selection of numbers in a fair manner, in no particular order and with no favor being given to any numbers. Examples include dice, spinners, coins, and computer programs designed to randomly pick numbers
theoretical probability	The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability

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 as The Racing Game with One Die 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 Outline

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 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.

Interactivate

Ideas that Lead to Probability