# The Monty Hall Puzzle

## The Problem

A game show presents a contestant with a choice of three doors. Behind one door is a new car. Behind the others are goats. The host knows where the car is and has scripted the scenario in advance:

• The contestant chooses a door.
• The host opens one of the other doors and shows the contestant a goat. (Remember that behind at least one of the unchosen doors must be a goat and that the host knows where the car is.) Two doors remain unopened.
• The host offers the contestant a choice: Keep the door you originally chose or switch to the other unopened door.

Question: Should the contestant switch? Does it matter?

Note: This is only a demonstration. We are not giving away any cars (or goats).

## The Analysis

The contestant should switch; this doubles the chances of winning.

The chance that the car is behind the originally chosen door is one in three (about 33.33%). This chance never changes. When the host shows a goat behind one of the other doors, it merely increases the chance that the car is behind the remaining door to two in three (about 66.67%).

The key is that the host is not randomly opening one of the doors. The host knows what's behind each door and is purposely opening a door that will reveal a goat.

Imagine the order of events were slightly different. After the contestant initially chooses a door, what if the host offered the option of trading that one door for both of the others. Clearly this doubles the chance of winning the car to two in three--it also gives the contestant a 100% chance of getting at least one goat. If the host then proved that a goat was behind at least one of the two doors, this would not alter the chance of winning the car.

Still don't get it? You are not alone. Try playing the game.

## Playing the Game

You play the role of the contestant. You choose from three doors, numbered 1 through 3. One of the remaining doors is then opened to show a goat. You are then given the option of staying with your original choice or switching to the other unopened door. After you make this final choice, your door is opened to reveal a car or a goat. Statistics in the bottom part of the window are updated with the latest results. You can choose to see just results for the current session or to see historical results (since about June 1, 2003).

To automatically play the game multiple times, enter a number in the field to the right of the doors and then click the Auto Play button. The program automatically plays the number of games you specified and updates the displayed results. During auto play, the contestant chooses a door randomly and then chooses to switch about half the time.

If you enter an especially high number, you might experience some delay as the games are processed.