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:
- 10 to 20 marbles of varying colors
- A bag or some other type of container
- Marble Bag Worksheet
- For the Two Colors
game:
- three identical containers (e.g., small boxes or opaque cups)
- 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.
- The Two Colors Table
to tally the results
- Two Colors
Worksheet
Key Terms
This lesson introduces students to the following terms through the included discussions:
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 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.
- Teacher Input
- 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.
- 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.
- 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.
|