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 probabilities.

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 Addressed:

Grade 10

Statistics and Probability

The student demonstrates a conceptual understanding of probability and counting techniques.

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.

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 9-12

Data Analysis and Probability

Understand and apply basic concepts of probability

Student Prerequisites

Arithmetic: Student 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 for experimenting with the activities

Teacher Preparation

Access to a browser

Pencil and paper

Copies of the supplemental materials:

For the
Marble Bag activity, each student/team needs:

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.

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 above materials
are necessary (one set of materials for each group of students that will be doing the
activity).

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

probability

The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability

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

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.

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 Outline

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.