Replacement and Probability

Abstract

This lesson explores sampling with and without replacement, and its effects on the probability of drawing a desired object. It is designed to follow the Conditional Probability and Probability of Simultaneous Events lesson to further clarify the role of replacement in calculating probabilites.

Objectives

Upon completion of this lesson, students will:

  • have taken a closer look at probability
  • have learned the difference between sampling with and without replacement

Standards

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

Data Analysis and Probability

Understand and apply basic concepts of probability

  • understand and use appropriate terminology to describe complementary and mutually exclusive events
  • use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations
  • compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models

Links to other standards.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • use addition, subtraction, multiplication and division to solve probability formulas
    • understand how tables can be used in multiplication
  • 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

All activities in the lesson are better experienced by using the software, with individual students or small groups of students having enough time to explore the games and find answers to the related questions. If the activities have to be set up physically, the following materials are necessary (one set of materials for each group of students that will be doing the activity):

  • Access to a browser
  • Pencil and Paper
  • Copies of the supplemental materials:
  • For the Marble Bag activity, each student/team needs:
    1. 10 to 20 marbles of varying colors
    2. A bag or some other type of container
    3. Marble Bag Worksheet
  • For the Two Colors game:
    1. three identical containers (e.g., small boxes or opaque cups)
    2. six objects of two different colors (three of each color), such as marbles or poker chips. The objects have to fit in the containers and have to be indistinguishable from each other by touch.
    3. The Two Colors Table to tally the results
    4. Two Colors Worksheet

Key Terms

This lesson introduces students to the following terms through the included discussions:

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 are going to learn about probability.
    • We are going to use the computers to learn about probability, 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
  4. Guided Practice

    • Begin by having the students experiment with a bag of marbles containing two different colored marbles to form a hypothesis about how replacement affects the probabilities on a second draw.
    • Next have the students experiment with the Marble Bag activity, asking them to validate the activity by comparing their computer results and their actual results.
    • Lead a discussion on Replacement to confirm that students understand the difference between sampling with and without replacement.

  5. Independent Practice

    • Then have them turn on the "multiple trials" feature on the Marble Bag to develope a sense of the theoretical probabilities.
    • Next have the students formulate a hypothesis about the results with more than 2 colors of marbles. Ask them to come up with a general formula or process.
    • Compare the results of the Marble Bag experiments to similar experiments with the Two Colors game.
    • Have the students write in their own words how replacement changes the probability of drawing objects.

  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 Outlines

This lesson can be rearranged in several ways.

  • Have students come up with their own versions of the Marble Bag game, and present their game and probability results to the class.

Suggested Follow-Up

After these discussions and activities, the students will have worked with conditional probability, sampling with and without replacement, and have seen the formula for the probability of simultaneous events. The next lesson, From Probability to Combinatorics and Number Theory, devotes itself to data structures and their applications to probability theory. Tables and trees are introduced, and some of their properties are discussed.