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 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

Discrete Mathematics

  • Data Analysis and Probability

    • Competency Goal 2: 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.

Textbooks Aligned

7th

  • Module 6 - Flights of Fancy

    • Section 2: Probability
      • Reason for Alignment: The lesson Tree Diagrams shows students how to draw a tree diagram for multi-stage events and how to use tree diagrams to compute probability. Note: the textbook calls these problems Multistage Experiments while the terminology in the Interactivate lesson is Compound Events.

8th

  • Module 2 - At the Mall

    • Section 3: Exploring Probability
      • Reason for Alignment: Since the textbook also looks at tree diagrams as a tool to show possible outcomes. There is a detailed discussion of trees as data structures in the lesson.

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 for experimenting with the activities.

Teacher Preparation

  • Access to a browser
  • Pencil and paper

Key Terms

compound event

Two or more events that happen simultaneously

conditional probability

Conditional probability is the probability of an event occurring given that another event also occurs. It is expressed as P(A/B). It reads "Probability of Event A on condition of Event B." P(A/B) = P(A and B)/P(B), where P(B) is the probability of event B and P(A and B) is the joint probability of A and B

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 Outline

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.