Tree Diagrams and Probability

Abstract

This lesson is designed to develop students ability to create tree diagrams and figure probabilities of events based on those diagrams.

Objectives

Upon completion of this lesson, students will:

• be able to create tree diagrams
• be able to figure probabilities based on tree diagrams
• practice adding and multiplying fractions
• be able to explain complementary probabilities

Standards

The activities and discussions in this lesson address the following NCTM standards:

Data Analysis and Probability:

• compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models
• understand and use appropriate terminology to describe complementary and mutually exclusive events

Number Sense:

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

• work flexibly with fractions, decimals, and percents to solve problems

Student Prerequisites

• Arithmetic: Students must be able to:
  • add and multiply fractions.
• Technological: Students must be able to:
  • perform basic mouse manipulations such as point, click and drag.
  • use a browser, such as Netscape, for experimenting with the activities.

Teacher Preparation

Students will need:

• Access to a browser
• pencil and paper

  Key Terms

  This lesson introduces students to the following terms through the included discussions:
  • complementary probability
  • compound event

  Lesson Outline

  1. Focus and Review

     Remind students what has been learned in previous lessons such as concepts of probability. Possibly use the example of rolling a die and the chances of that die rolling a specific number is 1/6. Ask students what the probability is of rolling a 1 or a 2. Mention the difference between the experimental probability and the theoretical probability. You can even have the students roll dice themselves, collect data for the class and figure the experimental probability for the class.

  2. Objectives

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

     • Today, class, we will be talking more about probability and how to determine the probability of multiple events, known as compound events. We will learn how to create tree diagrams to determine the probabilities related to compound events.

     • We are going to use the computers but please do not turn your computers on until I ask you to. I want to show you a little about the program first.

  3. Teacher Input

     Lead the students in a short discussion on trees as data structures.

     Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use our computer applets.

     • Open your browser to Racing Game with One Die in order to demonstrate this activity to the students.

     • Demonstrate some of the functionality yourself such as changing the length of the race, changing the game to an unfair race, and demonstrating how the multiple run panel works.

  4. Guided Practice

     Discuss and draw a tree diagram for a one-step and a two-step fair race. If you have a white board it is helpful to use a red and a blue marker to represent the two different colored cars.

     Make the race an unfair race by making the blue car move on rolls of 1 and 2 and the red car move on rolls of 3, 4, 5, and 6.

     Draw a tree diagram for a one-step unfair race. Mention that the sum of the end probabilities always equal one, which makes them complementary probabilities. Discuss why this must be so.

     Ask the students to go to their computers and create an unfair race as described above. In the multiple-run panel change the number of runs to 50,000. Have students run this configuration 5 or 6 times. Ask them to develop a hypothesis as to what the theoretical probability of an unfair two-step is based on the experimental data using the applet.

  5. Independent Practice

     Have the students create a tree diagram for an unfair two-step race to determine the theoretical probability.

     Have them show, based on their diagrams, the sum of the final probabilities equal one demonstrating they are complementary probabilities.

  6. Closure

     You may wish to bring the class back together for a discussion and verification of their findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

  Alternate Outlines

  This lesson can be rearranged in several ways if there is only one available computer. Place the students in groups of two or three. Run the multiple races yourself and have them develop the hypothesis with their partners.

  Extensions

  This lesson may be followed by:
  • Conditional Probability and Probability of Simultaneous Events
  • From Probability to Combinatorics and Number Theory