Interactivate: 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 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:

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:
  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
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.

Teacher Input

Review the Probability of Simultaneous Events and Conditional Probability discussions.

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 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.

Interactivate

Replacement and Probability