## Deal or No Deal to Teach Probability

Oftentimes, the class periods right before breaks are so crazy that it is hard to get much done. I have gotten in the habit of using these periods to introduce mathematical topics that might not otherwise be covered. This year, before thanksgiving, I set up a game of deal or no deal, so I could introduce my students to probability and expected value.

Deal or No Deal is a press-your-luck style game show where contestants try to win money. A contestant is shown a bunch of cases that contain between \$1 and \$1 million dollars. At the beginning of the game the contestant chooses a case but does not see its contents. Then they open other cases, each time revealing more cases which cannot be the one they chose. After opening each case they are offered a deal and they can take that deal or continue opening cases. Here's Howie's explanation:

I always found this show fascinating and thought it would be a fun game to play with students. I wanted the students to be playing for something tangible, so instead of imaginary money, I offered them skittles. I prepared 10 cases and blindly labeled them with letters. They represent 1, 5, 10, 25, 50, 100, 200, 400, 800, and 1000 skittles.[2]Logistically, counting more than 100 skittles is a waste of time. So instead, I found the mass of an average skittle and then used a scale to measure out the winning numbers. As you'll see below, I didn't have to do this often. In case you are curious, each skittle weighs about 0.0388 ounces. The image below shows sets of "cases" for each of my four classes.

This game was fun. Really fun! The kids had a blast and it was really exciting. Here's how they did:

Highest offer: 351 skittles
Result: 10 skittles
Percent of highest offer: 2.84 %

Highest Offer: 183 skittles
Result: 25 skittles
Percent of highest offer: 13%

Highest Offer: 155 skittles
Result: 5 skittles
Percent of highest offer: 31%

Highest Offer: 560 skittles
Result: 400 skittles
Percent of highest offer: 71.4%

Not a single group took a deal. Instead they kept opening cases until only their case was left. At each decision point, the class had to vote, so students used varying rationals to convince their peers. Some students used the gambler's fallacy while others used probability, expected value, and game theory. A few students even argued that risking a large amount of skittles would be worth it because they could gloat if they won a lot. These discussions presented lots of great opportunities to discuss probability topics.

I knew I needed a computer program to offer the deals or my students would jump on my case -- saying that's not a fair offer. So I wrote a program in python (you can try it out here). In the end, my program was probably a little too cruel when it came to offering deals. I had the program calculate the expected value based on the cases left and offer a percentage of the EV. At first I only offered 50% of the EV, but this climbed to 90% of the EV by the final deal. The next time I try this -- and there will definitely be a next time -- I am going to have the percentage of the EV climb more quickly.

Let me know if you have ever tried Deal or No Deal with your kids. I am also happy to help with the coding if you want to try this in your class.

Others' Ideas

Sarah wrote about using Deal or No Deal with her students. I really like that she had her students thinking about not just expected value, but the probability of earning more than a specific threshold: 100,000. My students were thinking about thresholds a lot. Most groups made some version of the argument, "Let's keep opening until we all get at least 10 skittles."

Thanks to Dan Meyer, Mr. Lucchese wrote about the mathematics of Deal or No Deal. He argues that the "banker" is trying to minimize the amount paid out by extending each contestant's time playing the game since more contestants means more money paid out.