Introduction to the Concept of Probability

Abstract

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 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.

Grade 10

  • 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

Grade 6

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

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

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.

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.

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
    • 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.

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.

  • 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
    • 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.

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.

4th Grade

  • Probability and Statistics

    • 4.19.a
    • 4.19.b
    • 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.

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

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

Textbooks Aligned

Grade Six

  • How Likely Is It?

    • Investigation Four: Theoretical Probabilities
    • Investigation Five: Analyzing Games of Chance

6th

  • Module 2 - Math Detectives

    • Section 1: Probability
      • 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.

7th

  • Module 2 - Bright Ideas

    • Section 5: Probability
      • 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.

8th

  • Module 2 - At the Mall

    • Section 3: Exploring Probability
      • 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: 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 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 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

  1. 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?

  2. 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

  3. 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."
    • Lead a discussion on Statistics vs. Probability

  4. Guided Practice

    • 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:
      1. # number of games played
      2. # number of games each player won
      3. 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.
      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?
    • Next, initiate a discussion about Events and Set Operations

  5. Independent Practice

    • 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?
    • 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.

  6. 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
  • 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.