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:

Grade 10

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 6

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 7

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 8

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Grade 9

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

Seventh Grade

Statistics and Probability

Investigate chance processes and develop, use, and evaluate probability models.

Statistics and Probability

Conditional Probability and the Rules of Probability

Understand independence and conditional probability and use them to interpret data

Use the rules of probability to compute probabilities of compound events in a uniform probability model

Making Inferences and Justifying Conclusions

Understand and evaluate random processes underlying statistical experiments

Make inferences and justify conclusions from sample surveys, experiments, and observational studies

Using Probability to Make Decisions

Calculate expected values and use them to solve problems

Use probability to evaluate outcomes of decisions

Third Grade

Measurement and Data

Represent and interpret data.

Grades 6-8

Data Analysis and Probability

Understand and apply basic concepts of probability

Grades 9-12

Data Analysis and Probability

Understand and apply basic concepts of probability

Advanced Functions and Modeling

Data Analysis and Probability

Competency Goal 1: The learner will analyze data and apply probability concepts to solve problems.

Discrete Mathematics

Data Analysis and Probability

Competency Goal 2: The learner will analyze data and apply probability concepts to solve problems.

Grade 6

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 4: The learner will understand and determine probabilities.

4th grade

Data Analysis and Probability

Standard 4-6: The student will demonstrate through the mathematical processes an understanding of the impact of data-collection methods, the appropriate graph for categorical or numerical data, and the analysis of possible outcomes for a simple event.

7th Grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of the relationships between two populations or samples.

8th grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.

4th Grade

Probability and Statistics

4.19.a The student will predict the likelihood of outcomes of a simple event, using the terms certain,
likely, unlikely, impossible

4.19.a

7th Grade

Probability and Statistics

7.14 The student will investigate and describe the difference between the probability of an event found through simulation versus the theoretical probability of that same event.

7.15 The student will identify and describe the number of possible arrangements of several objects, using a tree diagram or the Fundamental (Basic) Counting Principle.

8th Grade

Probability and Statistics

8.11 The student will analyze problem situations, including games of chance, board games, or grading scales, and make predictions, using knowledge of probability.

8.11 The student will analyze problem situations, including games of chance, board games, or

Reason for Alignment: Ideas that Lead to Probability is an introductory lesson on probability which matches the initial probability lesson in the text. The Racing Game activity used in the lesson is simple, yet illustrates many concepts of probability. It is also a preview of a fair game, which is introduced later in Book 1.

Reason for Alignment: This lesson contains the basics for understanding probability on an intuitive level. It employs a number of different experimental situations to get students started with these concepts.

The following objects or their pictures may be used in activities or to illustrate the
discussions.

Dice with various numbers of sides.

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.

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 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.

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.