Monty Hall Simulator: Play It, Then Run 10,000 Rounds

This Monty Hall problem simulator lets you play the puzzle yourself and then run 10,000 rounds in a second. Switching doors wins the car 2 times in 3, and staying wins only 1 time in 3, because the host's choice tells you where the car probably is. Try 3 doors, or 100, and watch the totals settle on the exact chances.

What is the Monty Hall problem?

A car is behind one of 3 doors and goats are behind the other 2. You pick a door. The host, who knows where the car is, opens one of the other doors and shows a goat. He then asks whether you want to stay with your door or switch to the one that is left. Most people say it makes no difference. It does: switching wins 2 times in 3.

The puzzle is named after Monty Hall, the host of the game show Let's Make a Deal, and became famous in 1990 when Marilyn vos Savant answered it in Parade magazine and readers, including mathematicians, wrote in to say she was wrong. She was right.

Why is the chance not 50/50?

Because the host does not open a door at random. He opens one he knows hides a goat. Your first pick was right 1 time in 3, and nothing the host does changes that. The other 2 times in 3 the car was behind one of the doors you did not pick, and the host, unable to open the car's door, leaves exactly that door closed.

Two of the three cases are wins for switching, and only one is a win for staying. That is the whole argument. Counting the cases is the reason the answer is not 1/2: the two closed doors are not equally likely, because one of them was your blind guess and the other was left closed by someone who knew.

Does the Monty Hall problem actually work?

Yes, if the rules are exactly the classic ones. The host must know where the car is, must always open a door that hides a goat, and must always offer the switch. Change any of those and the answer changes. If the host opens a door at random, and it happens to show a goat, switching and staying are equal at 1 in 2. The simulator above uses the classic rules, so its totals match the maths.

Monty Hall with 100 doors

The easiest way to feel the answer is to raise the door count. With 100 doors you pick one, and the host opens 98 of the others, all goats, leaving one closed. Your door had a 1 in 100 chance. The door he left closed carries the other 99 in 100. Almost nobody would stay. Pick 100 in the simulator, run 10,000 rounds, and switching wins about 99% of them.

How the simulation works

Each simulated round places the car at random, makes a random first pick, and lets the host open every door except your pick and one other, never the car. The same round is then scored twice, once as if you had stayed and once as if you had switched, so the two totals come from identical rounds. A run of 10,000 rounds usually lands within a percentage point or two of the exact chance. With a few rounds it can wander a lot, which is why one lucky or unlucky game proves nothing either way.

Use Play it yourself to feel it, and expect your own tally to look messy for the first dozen games. Then run many rounds to see the same maths at scale.

Is this the same as a game in Gift Club?

No. This is a probability puzzle, not one of our games. Gift Club's Magic Door has five doors and no host who opens any of them, so there is no stay-or-switch decision and nothing on this page applies to it. The link between the two is only that both are worth understanding as probability rather than luck; for that, see gambling odds explained.

Frequently asked questions