March/April 2000 Copyright © Information Today, Inc.
	Let’s Make a Deal Math: The Study of Probability
by Linda C. Joseph, Columbus (Ohio) Public Schools, Library of Congress

[Editor's note: URLs mentioned in this article appear in the chart that follows.]

"Heads, you go first; tails, I do.” “I need to roll a seven to win.” “If I can spin and land on a green space the jackpot is mine.” How many times have you played a game and hoped to win? What is the probability of winning? Games can be fair or unfair, depending on the rules. Why are some choices better than other choices? Introduce your students to some fascinating problems and learn about probable outcomes.
 

Probability Introduction
Begin your study of probability by having students flip a coin 50 times. Have them keep a record so they can compare their results using a tally sheet. M&M activities are another engaging way to introduce fractions, percents, and probability.

Ken White’s Coin Flipping Page
Choose a penny or a dime to flip. Input the number of times to flip the coin and click to start flipping. A page will display the coins as heads and tails with the totals at the bottom. In addition, results are given for the entire number of coins flipped.

Coin Flipping
In this simulation you flip a quarter and a statistical record is kept for heads and the probability of getting a head—P(Head). Questions are posed regarding the expected probability and their comparison to the found probability. Answers are linked on a separate page.
 
  

m&m Candy Comparison Making a Pie Graph with ClarisWorks 1. Highlight the data, colors, total, and percent in the M&M spreadsheet. 2. Pull down File and select Copy. 3. Scroll to the right to a new spreadsheet section. 4. Click on G1 and paste the information. 5. Highlight colors, total, and percent. 6. Pull down Options and select Make Chart. 7. Select Pie from the Gallery, then click on Series. 8. Click on the box next to Label data. Make sure % of slice is selected. 9. Click on Labels and type a title for the graph. Click OK. Customizing Colors and Text 1. Click on the first color located in the legend box. 2. Pull down Window and select Show Tools. 3. Click on the color palette and select the color. 4. Pull down Format and select Text Color. Repeat steps 3 and 4 for the remainder of the legend.

M&Ms: In the Bag Graphing Activity
Predicting the number of each color in a bag of M&Ms is a popular activity to do with students. In this slight twist on the activity, have students input their results into a table on the Web, reload the page, and see the graphed results of the colors. You can also have them calculate the percentages online, then compare their results with those of the Mars Company.

Mighty M&M Math
Another great place to visit is the Mighty M&M Math Project, a self-sustaining Web site that eliminates the need for registrations or deadlines. Here, students can upload their M&M color data and compare their results with other classes around the world. Great instructions for completing the activity are provided, along with a link to the M&Ms network.

Probability Investigations
After establishing the concept of probability, challenge students to tackle more complex problems that involve more imagination and reason. What may or may not seem obvious at first glance could lead to intriguing investigations and solutions.

The Birthday Problem
How likely is it that two people in your class have the same birthday? How likely is it in other classes? A concise explanation is provided with a formula for calculating the probability of a match with N birthdays. If you want something a bit easier, run 10 trials and observe what happens. The material at this site is geared for middle and high school students.

The Cereal Box problem
“Hey kids,” the announcer barks, “collect all eight of these great prizes in specially marked boxes of cereal.” How many boxes would you have to purchase to get all of the advertised prizes? Use this lesson as a springboard to explore experimental and theoretical values. Run the cereal box simulation program that allows you to select the number of prizes and log the trials. Trial statistics include the number of trials, average, maximum, minimum, and range. After several trials, collect the data and begin to draw conclusions.

The Cliff Hanger
Homer Simpson stands at the precipice of a cliff. One wrong step will spell disaster. What advice might you be able to provide Homer about his chances of escaping?
 

Monty Hall Problem
“Would you make a deal to trade up to $500 in cash for one of these three doors, knowing behind one of them is $3,254 in cash or valuable merchandise? Several people may have to make that decision during the next few minutes as we bring you the Marketplace of America—Let’s Make a Deal! And now, here’s America’s top trader, TV’s Big Dealer, Monty Hall!”

The well-known Monty Hall Problem is based on the Let’s Make a Deal television show of the 1960s and 1970s. Show host Monty Hall would ask a contestant to pick one of three doors. Behind one of these doors was a large prize. Behind the other two doors were lesser prizes or a booby prize like a group of goats. Once the contestant picked a door, Monty would open one of the remaining two doors that did not have a good prize. Then, he would offer the contestant a chance to switch doors.

Take a box and cut three slots for doors. Put cardboard dividers in the back. Label the doors, Door #1, Door #2, and Door #3. Find two plastic toy goats. Use these for the booby prizes. Find candy money for the good prize. Pick one of your students to play the part of the host. Then choose contestants from the class. Before playing, have them predict whether or not it is better to switch doors after one is revealed. Tally the results on the Monty Hall Tally Table. Next, jump on the Web and have your students try one of the Monty Hall simulation games. What happened? Compare the results in your classroom.

Monty Hall, Three Doors
Monty Hall, Three Doors is a fast-paced simulation game that records the number of times you keep or switch your door and the experimental probability to win. A treasure-trove of information regarding many aspects of probability is one click away. Included are a variety of simulations that will stretch the minds of young mathematicians.

Monty Hall
In this rendition of the Monty Hall Problem, you select one of three boxes, then keep or trade your box. A summary of wins and losses along with percentages is displayed after each game. One additional choice is available for those who must know the answer before making a decision. By clicking on the cheat button the winning box will be revealed.

The Monty Hall Problem
High school students will be challenged to think about solutions to various scenarios associated with the Monty Hall Problem. Several formulas are suggested for calculating the best strategies to use when keeping or switching doors. Experiment with these strategies while playing the simulation game. Keep a journal of the results and compare with others in the class.

Now Playing: Let’s Make a Deal
Read the story behind the Monty Hall controversy that began with a reader’s question to Marilyn Vos Savant’s Sunday Parade column. View photographs of the famous game show host and columnist. Then, play yet another version of the game, this time, with photographs of goats, and a bright red car as the prizes behind the doors.
 

Lessons

AITLC Teacher Lesson Plan: Flipping Coins
This lesson is designed for grades 3-6 and focuses on fractions using the Ken White Coin Flipping site.

Bumped Again! Why Does Delta Overbook?
The setup for this problem states that 90 percent of the customers that have purchased a ticket check in for a flight with 113 seats. If they sell 120 tickets, what is the probability that at most 113 customers will check in? In this lesson a TI-83 calculator is used for random-number generation and finding theoretical solutions.

Lessons on Probability from Math Goodies
Many facets of probability are presented and explained utilizing illustrations and a few interactive examples. At the end of each lesson is a quiz covering the material for each topic. Students will need lots of guidance from teachers in order to understand and take advantage of this site.

Be sure to visit the MultiMedia Schools Home Page (http://www.infotoday.com/MMSchools) with active links to all of the Web sites mentioned in this article. Then fly over to CyberBee (http://www.cyberbee.com) for  the Let’s Make a Deal WebQuest, more curriculum ideas, treasure hunts, research tools, and activities to use with your students and staff.

Linda Joseph is the author of Net Curriculum: An Educator’s Guide to Using the Internet, published by CyberAge Books. The recipient of numerous awards, in addition to her work in the Columbus Public Schools and the Library of Congress, Linda is a part-time instructor for Ohio State University. Communications to the author may be addressed to her at Columbus Public Schools, 737 East Hudson Street, Columbus, OH 43211; 614/365-5277; ljoseph@iwaynet.net.
 
 

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