Math stories
The Monty Hall Problem: Why Switching Doors Wins
7 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
- Switching wins with probability 2/3; staying wins 1/3.
- The key is that the host knows where the car is and always opens a goat door.
- A simulation of a million games agrees.
- Changing the rules changes the answer.
You are on a game show with three doors. Behind one is a car and behind the other two are goats. You pick door 1. The host, who knows what is behind every door, opens door 3 and shows a goat. He asks: do you want to switch to door 2? Most people say it makes no difference. In fact, switching doubles your chance of winning.
The intuitive (wrong) argument
Two doors are left, so it must be 50-50. This treats the two remaining doors as equal. They are not, because the host’s choice depended on where the car is.
The reasoning
When you first picked door 1, you had a 1/3 chance of being right and a 2/3 chance that the car was behind one of the other two. Opening a goat door does not change your first pick’s 1/3 chance. But the 2/3 chance that the car was elsewhere now sits entirely on the one remaining door. So switching wins 2/3 of the time.
Every case
| Car is behind | You pick door 1 | Host opens | If you switch |
|---|---|---|---|
| Door 1 | ✓ | 2 or 3 | Lose |
| Door 2 | 3 (forced) | Win | |
| Door 3 | 2 (forced) | Win |
Switching wins in two of the three equally likely cases. When the car is behind the door you did not pick, the host has no choice about which goat door to open, and that forced choice carries the information.
A simulation
We simulated 1,000,000 games. Always staying won about 33.3% of the time (0.3332), and always switching won about 66.7% (0.6668), matching the theory of 1/3 and 2/3. You can run a small version yourself with a coin-and-cup game.
Try 100 doors
Imagine 100 doors. You pick one (1/100 chance). The host opens 98 goat doors, leaving yours and one other. Would you switch? The other door now holds the 99/100 chance. Most people see instantly that switching is right.
The rules matter
The answer assumes the host always opens a goat door and always offers a switch. If the host opened a door at random and it happened to show a goat, the probability would be 1/2. Probability problems often hinge on such details. Test related ideas with the probability calculator.
A little history
The puzzle is named after Monty Hall, the host of a TV game show, and became famous after a magazine column discussed it in 1990, drawing thousands of letters, many from people who insisted the answer was wrong. The lesson: intuition about probability is often unreliable, and simple checks like simulation help.
Frequently asked questions
Why is it not 50-50?
Because the host’s choice of door depends on where the car is, which gives extra information.
Does it matter which door I pick first?
No. The result is the same for any first choice.
What if the host does not know where the car is?
If he opens a door at random and it shows a goat, staying and switching each win 1/2.
Where can I learn more about probability?
Read our probability basics guide and try the probability quiz.
Try the tools
Probability CalculatorFind the probability of “not A”, “A and B”, “A or B” and conditional probability.Permutation and Combination CalculatorFind nPr, nCr, factorials and counts with repetition.
Percentage CalculatorFind a percent of a number, percent change, and increases or decreases.Learn the method
Probability Basics8 min readConverting Fractions, Decimals and Percents8 min readMean, Median, Mode and Range6 min readKeep reading
Lottery Odds Explained: What “1 in 14 Million” Really Means7 min readStatistics vs Probability: What Is the Difference?7 min readWritten and reviewed by the SolveCalcPro editorial team. Found an error? Tell us. See our editorial policy.