### Simulating Probability Situations Using Box Models

The interactive tool in this i-Math investigation is a "box model" to explore the relationship between theoretical and experimental probabilities. 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.

To use the "box model":

• 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.

Sample Activity: Flipping a Coin

• Click on the 0 and the 1 to move them into the "box model."
• Click the "Show Theoretical Probability" checkbox to see the theoretical probability values displayed on the bar chart.
• Read the bar chart to answer the following questions: What is the theoretical probability for heads? For tails?
• 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, what is the experimental probability of heads? Of tails? Why is this different from the theoretical probability?
• Begin the drawing again by pressing Start. Pause after 20 draws.
• 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 insure that the experimental and theoretical probabilities are equal?

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.

References

Last updated: April 16, 2003

© 2000 National Council of Teachers of Mathematics