The Monty Hall Problem

Introduction: The Monty Hall Problem

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Back in 1963, a new television show premiered in the United States called Let’s Make a Deal. In this show the host, Monty Hall, would choose people from the audience to play games where they could win fabulous prizes from large amounts of money, furniture, and new cars. They also had a chance to win a ZONK (a bad prize that means the person lost). Many of these games seemed to be based on random chance. You may have to choose between something in a box and something behind a curtain, which gives a 50/50 (50%) chance of getting the good prize. However, there was one game that seemed to follow these rules but instead boggled mathematicians for decades.

The game begins with Monty Hall showing 3 doors to which he knows what is behind each door. Behind 2 doors is a Goat (the ZONK) and behind one door is a brand-new car. The person picks their door (Door #1), but before it is opened Monty Hall opens another one of the doors (Door #2) to show a goat. Monty Hall then asks the person if they would like to switch their door or keep what they started with. What would you do?

Many would think that they should keep their original door as to go with their “gut instinct” as it’s a 50/50 chance now that one door is opened. One door has a car and one door has a goat, but there have always been 2 goats in play. Math shows us then that it is actually better to switch as you actually have a 2/3 chance of getting the car.

Think of it like this:

There are 100 doors with 99 having goats behind them and only 1 has a car. You have a 1/100 chance of getting the car on the first try. Monty Hall then opens every other door except for door number 100 and your door to show the goats. He either chose that door randomly since you were lucky and got the car with a 1/100 chance OR he chose it because it has the car behind it, which is at a 99/100 chance. Which odds would you choose?

Key Terms

Probability – the extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

Supplies

- 1 piece of Paper

- Pen/Pencil/Other Writing Utensil

- Scissors

- Ruler

- Coloring Utensils

Step 1: Building Directions (Part 1)

: Take the piece of paper and mark the 11in (8 cm) side every 3in (8 cm) with the ruler. Use the back of the ruler as a straight edge to make a line all the way across the paper. With the scissors cut along the lines to create three 3 in (8 cm) wide papers and 1 smaller strip.

Step 2: Building Direction (part 2)

On the smaller piece of paper, mark every 2in (5cm) on the 8 ½ in (22 cm) side. Use the back of the ruler as a straight edge to make lines across. Use the scissors to cut along the lines to make even boxes. You will only need to use 3.

Step 3: Decorate

Decorate the large pieces to look like doors and label them “1”, “2”, and “3”. On 2 of the small pieces, label them as a “ZONK” and the other as the prize of your choice. Feel free to be as simple or complex in your decorations as you desire.

Step 4: How to Play the Game

Choose one person to be the host (Monty Hall). This person will hide the prizes behind the doors without the other person knowing. The host must be able to remember what prizes are behind what doors.

Step 5: Player's Choice

Have the player choose one door as “Their Door”. Have the Host open one of the doors with a ZONK behind it that is not the player’s chosen door. The host will then offer to for the player to switch doors if they choose. (The trick is based on the 2/3 chance, they should but it will only work for the long term).

Step 6: The Big Reveal!

The Host can now show the other doors and their prizes to reveal if the player won or lost. Repeat the experiment over a different number of times (10, 20, 30, etc.) and see how many times the person wins after switching and after not switching. Do they win more by switching or keeping their original door?

Step 7: ​Further Exploration and Sources

Further Exploration and Sources

The Monty Hall Problem on Numberphile – Video Playlist

https://www.youtube.com/playlist?list=PLt5AfwLFPxWLzNG4Ttv9-qZj86xt-J9_W

The Monty Hall Problem - VSauce (14 mins)

https://www.youtube.com/watch?v=TVq2ivVpZgQ&t=168s

The Monty Hall Problem – Explianed – AsapSCIENCE (2 ¾ mins)

https://www.youtube.com/watch?v=9vRUxbzJZ9Y

The Monty Hall Problem – Science City Video*

https://youtu.be/G8Wir585zuI

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