Monty Hall problem facts for kids

The Monty Hall problem is a famous brain teaser about probability. It's based on an old American TV game show called Let's Make a Deal, hosted by Monty Hall. The puzzle became very popular in 1990 when it appeared in a magazine column by Marilyn vos Savant.

Here's the puzzle:

Imagine you're on a game show. You see three closed doors. Behind one door is a brand-new car! Behind the other two are goats. You pick one door, let's say No. 1. The host, Monty Hall, knows where the car is. He then opens one of the other doors (not the one you picked), and it always shows a goat. Let's say he opens No. 3, and there's a goat. Then he asks you, "Do you want to switch your choice to No. 2?" Is it better to switch doors, or stick with your first choice?

Marilyn vos Savant said you should always switch doors. She explained that by switching, you have a much better chance of winning the car. Most people think the chances become 50/50 between the two remaining closed doors. But that's not quite right! Switching actually gives you a 2/3 (two out of three) chance of winning, while sticking with your first choice only gives you a 1/3 (one out of three) chance.

This puzzle is tricky because it goes against what many people first think. Even smart people, including many scientists, disagreed with Savant at first!

Understanding the Monty Hall Game Rules

To solve the Monty Hall problem, we need to understand the exact rules of the game. These rules are very important for the probabilities to work out:

• The host must always open a door that was not selected by the contestant.
• The host must always open a door to reveal a goat and never the car.
• The host must always offer the chance to switch between the door chosen originally and the closed door remaining.
• The car is placed randomly behind one of the three doors.
• If the player initially picks the car, and the host has two goat doors to choose from, the host picks one of those goat doors randomly.

If these rules are followed, switching your choice gives you a 2/3 chance of winning the car. Not switching means you only have a 1/3 chance.

Why Switching Doors Increases Your Chances

Let's look at why switching doors is a smart move. When you first pick a door, there's a 1/3 chance you picked the car. This means there's a 2/3 chance the car is behind one of the other two doors.

In search of a new car, the player chooses a door, say 1. The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player switch from door 1 to door 2.

Now, Monty opens one of the other doors and shows a goat. This is key! Monty knows where the car is. He will never open the door with the car. So, when he opens a goat door, he's actually giving you valuable information about the two doors you didn't pick.

Think of it this way:

• If you first picked the car (1/3 chance), and you switch, you will lose.
• If you first picked a goat (2/3 chance), and you switch, you will win! This is because Monty had to open the other goat door. So the remaining closed door must have the car.

The host's action doesn't change the initial 1/3 chance for your chosen door. Instead, it concentrates the 2/3 chance from the two other doors onto the one remaining closed door.

Visualizing the Outcomes

Let's imagine all the possibilities:

As you can see, if you always switch, you win the car in 2 out of 3 situations! If you always stay, you only win in 1 out of 3 situations.

Car has a 1/3 chance of being behind the player's pick and a 2/3 chance of being behind one of the other two doors.

The host opens a door, the odds for the two sets don't change but the odds become 0 for the open door and 2/3 for the closed door.

The Million-Door Example

To make it even clearer, imagine there are a million doors! You pick Door 1. The car is behind one of them, and 999,999 goats are behind the others. Monty, who knows where the car is, then opens 999,998 of the other doors, all revealing goats. Only two doors are left closed: your original Door 1, and one other door (let's say Door 777,777).

Would you stick with your first choice, Door 1? Or would you switch to Door 777,777? It's highly unlikely you picked the correct door out of a million on your first try. So, the car is almost certainly behind the other remaining closed door. You would switch very quickly! This is the same logic, just on a bigger scale.

Three initial configurations of the game. In two of them, the player wins by switching away from the choice made before a door was opened.

Testing the Monty Hall Problem with Simulations

A great way to prove that switching works is to try it out! You can simulate the game using three playing cards.

• Use one special card (like an Ace) to represent the car.
• Use two other cards (like 2s) to represent the goats.

Here's how to play:

1. Shuffle the three cards and place them face down (these are your "doors").
2. You pick one card.
3. The "host" (a friend, or you if you're playing alone) looks at the other two cards. The host then turns over one of those two cards that is not the Ace (a goat card).
4. Now, you decide: do you stick with your original card, or switch to the other face-down card?
5. See if you won the Ace (car) or a 2 (goat).

Repeat this game many times. Keep track of how many times you win when you switch, and how many times you win when you stay. You'll find that switching wins about two-thirds of the time! This is a fun way to see the math in action.

The History and Impact of the Monty Hall Problem

The Monty Hall problem first appeared in a letter to a statistics magazine in 1975, written by Steve Selvin. He named it after the host of the game show Let's Make a Deal.

The problem became famous in 1990 when Marilyn vos Savant answered it in her "Ask Marilyn" column in Parade magazine. Her answer, that switching was better, caused a huge stir! Thousands of readers, including many with advanced degrees, wrote in to say she was wrong. They believed the chances were 50/50.

Even famous mathematicians like Paul Erdős didn't believe the solution at first. He only accepted it after seeing a computer simulation that showed Savant's answer was correct. This shows how tricky and counter-intuitive probability can be!

The Monty Hall problem is a type of paradox where the correct answer seems impossible but is actually true. It's a great example of how our intuition can sometimes mislead us in probability.

How Changing the Rules Affects the Game

The solution to the Monty Hall problem depends entirely on the specific rules of the game. If the host doesn't follow the standard rules (like always opening a goat door, or always offering a switch), then the probabilities can change.

For example, if the host only offers you a switch when you've picked the winning door, then switching would always make you lose! But if the host only offers a switch when you've picked a goat, then switching would always make you win!

The original problem assumes the host must follow the rules we discussed. This is why understanding the rules is so important in probability puzzles!

See also

In Spanish: Problema de Monty Hall para niños

• MythBusters Episode 177 "Wheel of Mythfortune" – Pick a Door
• Principle of restricted choice – similar application of Bayesian updating in contract bridge

Similar puzzles in probability and decision theory

• Boy or Girl paradox
• Sleeping Beauty problem
• Two envelopes problem