The Monty Hall Problem
You are a contestant on a game show. The host shows you 3 doors. He tells you that the prize behind one door is $1,000,000 and behind each of the other doors is a goat. He instructs you to choose a door; you will win whatever is behind it. You choose a door. “But wait,” he says. “Let me show you what’s behind one of the doors you didn’t choose.” He opens one of those doors to reveal a goat. “Would you like to switch your choice, or stick with what you have?” What should you do, stick or switch?
I put forth this problem on December 6, a month after the end of our Middle School Math Circle on Paradoxes in Probability, to make up our Hurricane Sandy class. Immediately, questions, conjectures, and assumptions started flying around the room:
- Can we assume that Monty Hall knows where the money is?
- Does he always reveal a goat?
- Should we assume that the money is more desirable than the goat?
- Are there situations in which the goat would be more desirable?
- (conjecture) Monty may be trying to trick the contestants.
- Using intuition might answer the question.
- There may be patterns in which door contains the money.
At this point, A mentioned that he had seen this problem addressed on the TV show Mythbusters. He said that the hosts ran experimental trials and determined that one particular strategy paid off more in the long run. I just happened to have with me some paper cups, paper goats, and paper million-dollar bills, so our group decided to run trials and collect data. The kids paired off, put the prizes under the cups, and played “Let’s Make a Deal.” C suggested that each pair adopt a different strategy (stick or switch) and record wins by strategy. After 10 minutes of this, the data seemed to clearly suggest that always switching resulted in more wins. “That’s all and good,” I whined, “but this is a Math Circle. Are the Mythbusters mathematicians or scientists?” The kids agreed that the Mythbusters are scientists and that in replicating their work, we weren’t exactly doing math. So we moved to the other side of the room to undertake a more mathematical approach.
We attempted to apply a bit of basic probability theory to the problem. It seems that each strategy should yield a 50% chance of success, right? So why do empirical results suggest an unequal chance? This problem, known as The Monty Hall Problem, is considered a paradox in probability. Our group discussed and questioned until the conversation found its way to randomness. The kids all realized that the choice of door is not really random since the host knows what’s behind each door and opens one according to his knowledge. This must be the reason for the paradox, but how does that work out mathematically? No one was sure. Discussion continued. Finally, A, who is always listening and thinking but does not speak up often, raised her hand high and said “I get it!” Everyone looked at her in surprise and excitement. She explained her reasoning. She had synthesized everyone else’s ideas into a solution, but not everyone understood. Once she explained it again, everyone understood. I’m not going to explain her reasoning here because the students left class excited to pose this question at home. If you haven’t already, give your kids a chance to pose the question to you. Since it’s been a while since this Math Circle took place, you may need additional explanatory help. You can find a concise clear explanation in Jim Tanton’s book Solve This, and a clear but lengthier explanation in his youtube video.
This Math Circle has spent six weeks discussing paradoxes and misrepresentations in probability and statistics. I think these students would find a recent study, seemingly on music and age, interesting. Researchers Simmons, Nelson, and Simonsohn demonstrated how easy it is to use statistics to support false hypotheses. The Penn Gazette explains in layperson’s terms how the researchers demonstrated that listening to The Beatles can make a person younger. Also, the older students in this group will probably enjoy our spring Math Circle for teens on Proofs. In it, we’ll continue this year’s Math Circle theme of how one can know for sure that something is true or false. It’s been a great six weeks! Wishing you all a Happy New Year.