# The Monty Hall Problem

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## Objective

SWBAT understand the subtle complexity of probability.

#### Big Idea

We can understand complex problems using simple models.

## Start Up

20 minutes

As students enter the room, I play the game show music for the game show Let's Make a Deal and I display the first slide from Monty Hall. Once the students settle in, I start the "game show". I show the three doors and tell them that behind one door is a car and behind the other two doors are goats. When I say the word "goat" I cue up this video:  http://youtu.be/kh6c0SOfkH4. It has nothing to do with the problem, but it makes them laugh every time.

Then, I have a "contestant" play the first round. He/she picks a door. I show them a goat behind one of the other doors and ask if they want to change their mind. They get to take three suggestions from the audience and then they make up their mind. The classic misconception comes out right away, essentially students say "it doesn't matter because one door has the car and the other has the goat. So its a 50-50 chance either way."

We play two rounds of the game and discuss their ideas around the problem. My questions for them are essentially built around "how can we figure out if it matters or not?" I help them realize that at least one way to figure this out it to run some trials and analyze the empirical or experimental probability.

## Play the game

25 minutes

The goal is for students to play the game and gather data about the results of their actions. We give each partnership 3 cards and enough pictures of goats and cars to set up four rounds of the game. We tell each partner to put a car on one card and the goats on the other two cards. Then, they shuffle and play. One partner is guessing and the other hosting. As a control, we ask them to run successive trials in which they always switch and other trials where they never switch. Then they enter their data on the whiteboard for the class to see.

As we collect data, I tend to stop groups or the class and point out interesting outcomes, like "wow, that table always won when they switched."

The table is always set up simple. I usually make two columns, one saying "wins from switching" and the other saying "wins from not switching."

## Summary

15 minutes

We start the summary by discussing the table that we generated as a class. We discuss their observations and I point them towards the fact that each groups individual results seem to be all over, where one group one more from switching and the next from not. However, we look at the group as a whole, it seems that switching had significantly more wins. To verify our findings, we would write out the empirical probabilities of winning from switching and not switching.

"Why might a larger group reveal a trend in probability?" This gets them discussing the nature surrounding the law of large numbers.

Then we move towards our second method of making sense of probability: modeling. In this case, we use the tree diagram (in the powerpoint from the start up). This model clearly shows that there is a 2/3 chance of winning when switching.

The lesson ends with a brief discussion on their reaction to this observation. We return to their original idea that it doesn't matter if you switch or not since the car is behind one door and the goat is behind the other. However, the original condition, your first pick, has a significant impact on the probability. This is an introduction into the world of conditional probability.