Monty Hall simulator
Play the Monty Hall game yourself, or simulate thousands of games in one click, and see why switching doors wins about two thirds of the time. The page explains the maths in plain English.
Play the game
Loading the game. Switch on JavaScript to play, or read the explanation below.
Or simulate many games
How to use it
- Click a door to choose it. A car is behind one door and goats are behind the other two.
- The host opens one of the other doors and shows you a goat.
- Decide: stay with your first door or switch to the other closed one.
- Play several times to see your own tally, then run the simulation for a big sample.
The problem in one paragraph
You are on a game show with three doors. Behind one is a car, behind the others are goats. You pick a door, say door 1. The host, who knows what is behind every door, opens another door, say door 3, to reveal a goat, and offers you the chance to switch to door 2. Should you? Most people think it makes no difference, because two doors remain. The answer is that you should switch: it doubles your chance of winning.
How it works
At the start your door has a one in three chance of hiding the car, and the other two doors together have two chances in three. The host then opens one of those two doors, always a goat. Opening it does not change the odds of your first pick, which stays at one in three, so the whole two-in-three chance now sits on the one remaining door. Switching wins exactly when your first pick was wrong, which is two times out of three.
A helpful way to see this is the 100-door version. You pick one door out of 100. The host, knowing where the car is, opens 98 goat doors and leaves one other door closed. Your door had a 1 in 100 chance; the other closed door is almost certainly the car. The three-door game works the same way on a smaller scale.
| Car is behind | You pick door 1: stay | You pick door 1: switch |
|---|---|---|
| Door 1 | Win | Lose |
| Door 2 | Lose | Win |
| Door 3 | Lose | Win |
Staying wins in one of the three equally likely cases and switching wins in two. The simulator above does not use this shortcut: each game randomly places the car, picks a first door, has the host open a goat door that is neither the car nor your pick, and then records the outcome of staying and of switching.
Worked examples
One game
The car is behind door 2. You pick door 1 and the host opens door 3. Staying gives a goat; switching to door 2 wins the car.
A lucky first pick
The car is behind door 1 and you pick it. The host opens door 2 or 3. Staying wins, switching loses. This happens one time in three.
A long run
In 10,000 simulated games, expect switching to win somewhere near 6,667 times and staying near 3,333, with random variation of a hundred or so either way.
Did you know?
The puzzle is named after Monty Hall, host of Let's Make a Deal, and became famous in 1990 when the columnist Marilyn vos Savant published the correct answer in Parade magazine and received a flood of letters insisting she was wrong.
If the host does not know
The 2/3 answer depends on the host knowing where the car is and always opening a goat door. If the host opens a random door and it happens to be a goat, then staying and switching are equally likely to win, at one half each. The details of the rules really matter in probability puzzles.
Tips and common mistakes
- Do not judge by a handful of games: a lucky streak can make staying look as good as switching.
- Remember the host's choice is not random: it depends on where the car is.
- Try the 100-door version in your head if the three-door logic still feels wrong.
- Use the simulation for large samples and compare with the 2/3 and 1/3 prediction.
Who this is for
- Anyone who has argued about this puzzle at a party.
- Students learning conditional probability.
- Fans of popular maths who want to see the result with their own eyes.
Related tools and pages
- Birthday paradox calculator: another result that defies intuition.
- Hello World: Hannah Fry on algorithms and decision-making.
- No One Knows What They're Doing: her 2027 book on doubt and decisions.
- What to expect: the Come Fry With Me show.
- All tools: planners, finders and maths toys.
Questions about this tool
Should I switch doors in the Monty Hall problem?
Yes. Switching wins the car two times out of three, while staying wins only one time in three. That is true when the host always opens a goat door you did not pick.
Why does switching work?
Your first pick is right only one time in three. If it is wrong (two chances in three), the host must open the only other goat door, so the remaining door hides the car. Switching therefore wins whenever your first pick was wrong.
Is it really 2/3 and not 50/50?
Yes. The two doors left are not equally likely because the host's choice depends on where the car is. The simulator shows the 2/3 versus 1/3 split over thousands of games.
What if the host does not know where the car is?
If the host picks a door at random and it happens to show a goat, the odds change to 50/50. The classic puzzle assumes the host knows and always reveals a goat.
Where does the name come from?
It is named after Monty Hall, the host of the TV game show Let's Make a Deal. The puzzle became famous in 1990 when Marilyn vos Savant answered it in her Parade magazine column.
How does the simulator work?
Each game hides a car at random, picks a random first door, has the host open a goat door, then records whether staying or switching wins. It repeats that for as many games as you ask.
Sources and method notes
- Monty Hall problem (Wikipedia): history, solutions and variants.
Last reviewed 2026-10-06. Spotted a mistake? Tell us. This tool is free and runs entirely in your browser: nothing you enter is sent anywhere.