An activity and two discussions of this lesson introduce the concept of probability and the basic
set operations that are useful in solving probability problems that involve counting outcomes.
This material is the basis of the so-called
naive probability theory. In contrast with axiomatic probability theory that deals with abstract,
axiom-driven concepts, the naive theory is built upon intuitive and experimental knowledge.

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

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

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.

3rd Grade

Data Analysis and Probability

The student will demonstrate through the mathematical processes an understanding of organizing, interpreting, analyzing and making predictions about data, the benefits of multiple representations of a data set, and the basic concepts of probability.

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.

6th Grade

Data Analysis and Probability

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

Measurement

The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine unit rates; and the use of scale to determine distance.

The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine

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.

Measurement

The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and volume; and the use of conversions within and between the U.S. Customary System and the metric system.

The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and

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.b The student will determine the probability of a given simple event, using concrete materials.

4.19.a

4.19.b

5th Grade

Probability and Statistics

5.17b The student will predict the probability of outcomes of simple experiments, representing it with fractions or decimals from 0 to 1, and test the prediction

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

Computation and Estimation

8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data, such as finding a discount and discount prices and balancing a checkbook.

8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data,

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: The Introduction to the Concept of Probability lesson contains a great discussion of theoretical and experimental probability. This is really useful at this point in the textbook. The lesson could be used as a further discussion, or independent practice. This one seems to be more exploration than introductory, for sixth graders at least. This lesson contains an excellent discussion on data, statistics and probability, and their use and common errors, which would be good for the teacher's background as well as the student's information.

Reason for Alignment: The Introduction to Concept of Probability lesson formalizes probability with discussion examples for key terms and concepts. It incorporates the use ofthe Crazy Choices activity with the use of accompanying worksheets. The lesson could feasibly be used with a parent at home for extra help along with the text, as needed.

Reason for Alignment: This is another rather basic look at probability, but this lesson takes the students a little further into some investigations. It should be useful at this time in the text.

Student Prerequisites

Arithmetic: Student 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 for experimenting with the activities

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 Sets 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

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

output

The number or value that comes out from a process. For example, in a function machine, a number goes in, something is done to it, and the resulting number is the output

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

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:

In science have any of you ever done an experiment that you thought would turn out one way,
but it ended up doing something different?

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 probability

We are going to use the computers to learn about 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

In science you do things over and over to make sure you have the correct results. Well, in
math you repeat things over and over to make sure your experimental results are as close to
the theoretical result as possible. The actual results from your experiments are called
statistics. While the possibility you may or may not get a certain result has to do with
probability.

Lead a discussion, based on the Crazy Choices Game, of
Outcomes and Probability to introduce the ideas of "outcome" and "probability."

Have the Students play the
Crazy Choices Game to show how probabilities can be compared experimentally, and to help students understand the
definition of probability.

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:

# number of games played

# number of games each player won

experimental probability of winning

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)

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)

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.

In the Crazy Choices Game, each player won in so many outcomes out of so many total
outcomes. How can we define an outcome?

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?

What happens to experimental probabilities when we collect more and more data on the same
game?

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.

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?

What is the intersection of sets?

If each set describes an event, what events are described by the union and the
intersection?

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.

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.

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

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.