The Monty Hall Problem Explained: Should You Stay or Switch?
Imagine you’re standing in front of three closed doors.
Behind one door is a brand-new car.
Behind the other two doors are goats.
You choose Door 1.
The host, who knows what is behind every door, opens Door 3 and shows you a goat.
Now he gives you a choice:
Do you stay with Door 1, or switch to Door 2?
At first, the answer seems obvious: there are two doors left, so surely each one has a 50% chance of hiding the car.
But that is where this famous math problem becomes surprisingly difficult.
The mathematically correct answer is to switch.
And the reason comes down to probability.
What Was the Probability When You First Chose?
When you selected Door 1 at the beginning, you had a 1-in-3 chance of choosing the car.
That means there was a 2-in-3 chance that the car was behind one of the other two doors.
Your original choice does not become more likely just because the host opens another door.
You started with:
- Door 1: 1/3 chance
- Door 2 or Door 3 together: 2/3 chance
The host then uses his knowledge to open a door that he knows contains a goat.
That is the important part.
He isn’t randomly opening one of the two doors.
He deliberately opens a door that cannot contain the car.
What Happens If You Never Switch?
Suppose you play the game 100 times and always keep your original choice.
Your first choice is correct only about one-third of the time.
So you would expect to win approximately:
33 out of 100 games.
You would lose approximately:
67 out of 100 games.
Your original choice has only a 1-in-3 probability of being correct.
What Happens If You Always Switch?
Now imagine you use a different strategy.
Every time you choose a door, you immediately switch after the host reveals a goat.
If your original choice was correct, switching makes you lose.
That happens about 1/3 of the time.
But if your original choice was wrong, the host is forced to reveal the other goat, leaving the car behind the only remaining unopened door.
That happens about 2/3 of the time.
So when you always switch, you win approximately:
67 out of 100 games.
That’s twice the winning probability of always staying.
Why Does It Feel Like 50–50?
This is what makes the Monty Hall Problem so confusing.
After one goat is revealed, you can see only two unopened doors.
It feels like:
“There are two doors, so each must have a 50% chance.”
But probability isn’t determined only by how many possibilities remain.
It also depends on how those possibilities were created.
The host knew where the car was.
He didn’t randomly open a door.
He intentionally removed a losing option while preserving information about your original choice.
That is why the original 1/3 probability remains attached to your first door, while the 2/3 probability effectively transfers to the remaining unopened door.
A Simple Way to Understand It
Imagine the game had 100 doors instead of three.
You choose one door.
Your chance of picking the car is just:
1 out of 100.
That means there is a 99 out of 100 chance that the car is somewhere among the 99 other doors.
Now imagine the host knows where the car is and opens 98 doors, showing you goats behind every one.
Only two doors remain closed:
your original door and one other door.
Would you really believe your original door suddenly became a 50% chance?
Probably not.
Your original door still has its original 1% probability.
The other remaining door carries the other 99% probability.
The three-door version works according to the same principle.
The Answer
So, if you ever encounter the classic Monty Hall Problem and the host follows the usual rules, switching doors gives you the better strategy.
Staying gives you approximately a:
1/3 chance of winning.
Switching gives you approximately a:
2/3 chance of winning.
The lesson isn’t really about doors or goats.
It’s about how easily our intuition can misunderstand probability.
Sometimes a decision that feels like a 50–50 choice is not 50–50 at all.
And that is exactly why the Monty Hall Problem has remained one of the most famous probability puzzles for decades.
Meta description:
The Monty Hall Problem seems like a 50–50 choice, but probability says otherwise. Learn why switching doors gives you a 2/3 chance of winning.
