From Probability to Combinatorics and Number Theory

Abstract

The activities and discussions in this lesson are devoted to data structures and their applications to probability theory. Tables and trees are introduced, and some of their properties are discussed.

Objectives

Upon completion of this lesson, students will:

  • seen how division is used to help solve probability problems
  • used tables as data structures used to count outcomes and to compute probabilities
  • seen how trees are a type of data structure

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.

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

Advanced Functions and Modeling

  • Data Analysis and Probability

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

6th Grade

  • Data Analysis and Probability

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

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.

Secondary

  • Algebra II

    • AII.01 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices.

Textbooks Aligned

7th

  • Module 8 - Heart of the City

    • Section 2: Permutations and Combinations
      • Reason for Alignment: The From Probability to Combinatorics lesson shows how the use of tables and trees can be utilized to compute and understand probability. These types of data structures are a good tie to the textbook lesson.

8th

  • Module 4 - Inventions

    • Section 5: Counting Techniques
      • Reason for Alignment: This lesson goes into probability theory, along with tables and trees. It ties with the combinations portion of this section. The word "combinatorics" is used in the lesson, but it is explained in the discussions.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • use division to count outcomes in probability problems
    • use multiplication in working with data structures
  • 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

Key Terms

combinatorics

The science that studies the numbers of different combinations, which are groupings of numbers. Combinatorics is often part of the study of probability and statistics

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

superscript

In mathematics, superscripts are numbers or letters written above and to the right of other numbers or letters or symbols indicating how many times the latter is to be used as a factor. When typing, one can represent a superscript by using the ^ symbol to indicate raising the number. For example, x3 is the same as x^3, which equals x * x * x

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 of what they 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.

  2. Objectives

    Let the students know what they will be doing and learning today. Say something like this:

    • Today, class, we will learn how to use division to solve probability problems.
    • We are going to use the computers to help us, 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

    • Explain the Racing Game with Two Dice , which will introduce the concept of data structures and computing particular probabilities.

  4. Guided Practice

    • Have the students begin with the Racing Game with Two Dice Several players "race to the finish" using the software or on paper. For every round, each player makes either one or two steps depending on the outcome of the roll of two dice. Each group of students can come up with its own way of randomly choosing which players make one or two steps.
    • Lead a discussion based on Tables and Combinatorics, discussing tables as data structures.
    • Have a discussion about divisibility as it can be used in probability. The discussion is based on the Dice Table activity.
    • Lead the discussion: Tree as a data structure. This discussion introduces and develops the idea of trees as data structures. It is based on all the other parts of the lesson. Plan it as a "live" discussion where students have an opportunity to ask their own questions, because the topic tends to be interesting to many people and it can lead to various investigations in math and computer science.

  5. Independent Practice

    • Have the students work alone or in small groups and play the Dice Table activity, where students research tables as data structures and use tables to count outcomes and compute probabilities.

  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.

  • Use the text in the Tables and Combinatorics discussion to prepare for a "live" discussion that can take place while students are using the Dice Table activity.
  • Have students read the divisibility discussion independently, or use the text to prepare for a "live" discussion.

Suggested Follow-Up

After these discussions and activities, the students will have seen how data structures such as tables and trees can be used when solving probability problems. You may want to introduce student to the idea of expected value next.