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 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.

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.

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.

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: Student 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

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, x^{3} 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

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.

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.

Teacher Input

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

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.

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.

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.

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.