Probability: Playing with Fire

Abstract

Students use probability to set a simulated forest on fire.

Objectives

Upon completion of this lesson, students will:

  • understand possible outcomes of a probability experiment

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

  • 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

    • Use probability to evaluate outcomes of decisions

Grades 3-5

  • Data Analysis and Probability

    • Develop and evaluate inferences and predictions that are based on data
    • Understand and apply basic concepts of probability

5th Grade

  • Data Analysis and Probability

    • The student will demonstrate through the mathematical processes an understanding of investigation design, the effect of data-collection methods on a data set, the interpretation and application of the measures of central tendency, and the application of b
    • The student will demonstrate through the mathematical processes an understanding of investigation design, the effect of data-collection methods on a data set, the interpretation and application of the measures of central tendency, and the application of basic concepts of probability.

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.

3rd Grade

  • Probability and Statistics

    • 3.23 The student will investigate and describe the concept of probability as chance and list possible results of a given situation.

Student Prerequisites

  • 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

  • access to a browser
  • pencil and graph paper
  • dice

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

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

    Review the difference between experimental and theoretical probability.

  2. Objectives

    Students will be able to model possible outcomes of the controlled burning of a forest both by hand and by using computer applets.

  3. Teacher Input

    • Inform students that foresters use probability when setting what are known as controlled burns. In order to do a controlled burn, foresters need to be able to model the forest burning. They do this by assigning a certain burn probability to each tree. The burn probability depends on a variety of climatic conditions.

  4. Guided Practice

    • Have the students draw four 5x5 squares on a sheet of graph paper.
    • Tell the students to draw a triangle inside of each block in the 5x5 squares
    • Explain to the students that the triangles represent trees.
    • Also make sure the students understand that only trees directly on top of, beneath, to the left and to the right of an already burning tree can catch on fire.
    • Tell the students that the trees in the first 5x5 square have a 1/6 probability of catching on fire after the center tree has been set on fire.
    • Tell the students that the trees in the second 5x5 square have a 1/3 probability of catching on fire after the center tree has been set on fire.
    • Tell the students that the trees in the third 5x5 square have a 1/2 probability of catching on fire after the center tree has been set on fire.
    • Tell the students that the trees in the fourth 5x5 square have a 2/3 probability of catching on fire after the center tree has been set on fire.
    • Have the students use dice to decide wether or not each tree touching the burning tree catches on fire. For example, if a tree has a 1/2 probability to catch on fire then it will burn on rolls of 1, 2, and 3 and not burn on rolls of 4, 5, and 6.
    • Work through one model on the board as a class
    • After you have completed the burn, show the students how to calculate the percent burn.

  5. Independent Practice

    • Have the students work through all four of their forests.
    • Monitor the students.
    • As they finish have them open up the Fire Applet and do several burns with the burn probabilities they modeled using dice, pencil, and paper.
    • After all the students have had a chance to work through their models on paper, write all the percent burns on the board in columns under their corresponding burn probability.
    • Ask the students why they think there is such a large variation in percent burns for certain probabilities and such a small variation for others.
    • Have the students use the Fire Applet to try to figure out which probability is most unpredictable.
    • Ask the students if any of them can think of a way to model a burn if when the wind is blowing in a certain direction. (Set higher probabilities in the direction the wind is blowing while setting lower probabilities in the opposite direction of the wind.)
    • Have the students open the Directable Fire Applet.
    • Tell the students to try experimenting with the applet to see how they might manipulate the burn probabilities to model the effects that wind might have on a burning forest.
    • Monitor the students' progress and have a couple of the students share their model with the class

  6. Closure

    • Draw connections between the probability that each tree will catch on fire and the percent of trees burned
    • Be sure to point out that a 1/2 probability a tree will catch on fire does not imply that 50% of the trees will burn.
    • Point out that it was much quicker and easier to model forest burns on a computer rather than by hand.