## Boxing Up

- Lesson

Students explore the relationship between theoretical and experimental probabilities using an interactive tool known as a "box model." A "box model" is a statistical device that can be used to simulate standard probability experiments such as flipping a coin or rolling a die.

*Using the "Box Model"*

Before giving students the opportunity to use the Box Model for the actual lesson, they should be given time to explore the features of the Box Model.

Students can do the following:

- To Enter Data: Click on the number pad to enter numbered tickets into the box.
- To Randomly Draw Tickets: Click on the Start button to randomly draw tickets from this box (with replacement) and view, in real time, the experimental probability of drawing a given ticket.
- To Pause the Drawing: When you press the Pause button the "box model" pauses drawing. You can then click on any bar in the bar chart to display the current relative frequency.
- In pause mode, you are also able to scroll through the sample of the numbers drawn thus far.

This applet is also available on the Illuminations website: Random Drawing Tool - Individual Trials. Direct students to this applet when they are ready to begin the trials on their computers.

*Activity: Flipping a Coin*

Students should click on the 0 and the 1 to move them into the "box model." Next, they can click the "Show Theoretical Probability" checkbox to see the theoretical probability values displayed on the bar chart.

**In this model, 0 represents heads, and 1 represents tails.**

After reading the bar chart, students should answer the following questions:

- What is the theoretical probability for heads?
- What is the theoretical probability for tails?

Next, students should click on the Start button. (This begins a random draw, with replacement). Click on the Start button to pause the drawing after 10 draws.

After 10 draws, students should answer the following questions:

- what is the experimental probability of heads?
- What is the experimental probability of tails?
- Why might the experimental probability be different from the theoretical probability?

Next, students should begin the drawing again by pressing Start. Pause after 20 draws. Students should answer the following questions:

- Is the experimental probability closer to the theoretical probability than after 10 draws?
- Explain why more draws affects the closeness of the two values.
- Predict the number of draws that would bring the values "very" close to each other.
- Test your conjecture by beginning the drawing again and pausing after you reach your predicted number of draws. Repeat if necessary until you have gotten the two values "very" close to each other.
- What hypothesis can you make at this point about the number of draws it would take to ensure that the experimental and theoretical probabilities are equal?

As a class, discuss the students' individual responses to these questions.

The questions from this lesson are also available on the Flipping a Coin activity sheet.

*Alternative Activity*

As an alternative to the above box model, or in addition to, you could create your own real-life box model for the students to use. Take a box (a shoebox, or some similar box), and then write numbers (such as 0's and 1's) on index cards to put in the box. Students then pull out cards one at a time, and this is the "random number generator."

**References**

Copyright Notice: Applet generously provided by: L. O. Cannon, James Dorward, E. Robert Heal, Richard Wellman (Utah State University, www.matti.usu.edu). The USU MATTI project is supported by the National Science Foundation (Award #9819107). Copyright 1999.

**Credits**

*Writer:* Richard Wellman

*Contributors:*Enrique Galindo and Dasha Kinelovsky

*Web Design and Development:*

- Direction: Enrique Galindo
- Coding: Erdinc Cakiroglu and William Dueber
- Graphics: Canghui Song

**Extensions**

- The box model below allows you to replicate the drawing of many numbers and then investigate the distribution of the sum of draws or average of draws. Try it. Describe and explain any patterns you see.
This applet is also available on the Illuminations website: Random Drawing Tool - Sampling Distribution.

- Ask students the following question: "How could you use the box model to represent the distribution when rolling two standard dice?" Students can roll dice to find the experimental probability. Then, use the box model to yield similar results.

### Learning Objectives

Students will:

- Explore the relationship between theoretical and experimental probabilities

### NCTM Standards and Expectations

- Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.

### Common Core State Standards – Mathematics

Grade 7, Stats & Probability

- CCSS.Math.Content.7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.