Monty Hall Problem Calculator

Explore the counterintuitive probability paradox that stumped mathematicians and discover why switching doors doubles your chances of winning 🚪

Choose a door to start the game! The car is hidden behind one of the three doors.

Door 1

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Door 2

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Door 3

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Your Game Statistics

Games Played

Wins by Staying

Wins by Switching

Stay Win Rate

Switch Win Rate

Number of Simulations:

Strategy to Test:

Total Number of Doors:

Doors Opened by Host:

Welcome to the most famous probability paradox in mathematics! The Monty Hall Problem demonstrates how our intuition about probability can lead us astray, even confounding professional mathematicians when it was first popularized. You’ll master the counterintuitive logic behind this puzzle and understand why switching doors is always the mathematically superior strategy.

🎯 Quick Tip

If you find this probability puzzle fascinating, you might also want to try our Conditional Probability Calculator and Bayes’ Theorem Calculator for exploring related concepts in probability theory and decision-making.

Table of Contents

How to Explore the Monty Hall Problem

Follow our simple tutorial to understand this probability paradox through interactive gameplay and mathematical analysis:

Interactive Game Experience

Experience the paradox firsthand with our interactive simulation:

Choose your door: Click on any of the three doors to make your initial selection
Watch the reveal: The host will open one of the remaining doors, revealing a goat
Make your decision: Choose to stay with your original choice or switch to the other closed door
See the outcome: Discover whether you won the car and track your success rates
Analyze patterns: Play multiple rounds to see the 2/3 vs 1/3 probability ratio emerge

Probability Simulation Testing

Run large-scale simulations to verify the mathematical predictions:

Set simulation count: Choose how many games to simulate (1,000 to 1,000,000)
Select strategy: Test “always stay,” “always switch,” or compare both strategies
Run the simulation: Let the computer play thousands of games instantly
Review results: See how the win rates converge to the theoretical probabilities

Mathematical Analysis Tool

Explore generalized versions of the problem with different parameters:

Adjust door count: Increase the total number of doors (classic uses 3)
Modify doors opened: Change how many doors the host reveals
Calculate probabilities: See how the strategy effectiveness changes
Understand scaling: Learn why more doors make switching even more advantageous

Strategy Analysis and Mathematical Proof

🚪 Stay Strategy

33.33%

Logic: Your initial choice has a 1/3 probability of being correct. This probability never changes, regardless of what the host reveals.

Why it fails: You’re betting that your first random guess was lucky.

🔄 Switch Strategy

66.67%

Logic: The two doors you didn’t pick have a combined 2/3 probability. When one is eliminated, the other inherits the full 2/3 probability.

Why it works: You’re betting against your first random guess, which is usually wrong.

The Mathematical Proof

Let’s use the mathematical framework to understand why switching works:

Initial Probability Distribution

P(Car behind Door 1) = 1/3

P(Car behind Door 2) = 1/3

P(Car behind Door 3) = 1/3

Take a look at your data: When you choose Door 1, there’s a 1/3 chance it contains the car and a 2/3 chance the car is behind Door 2 or 3. When the host opens Door 3 (revealing a goat), the 2/3 probability doesn’t disappear—it transfers entirely to Door 2.

Conditional Probability Analysis

The key insight involves understanding conditional probability:

After Host Reveals a Goat

P(Car behind your door | Host action) = 1/3

P(Car behind switch door | Host action) = 2/3

The host’s knowledge and constraint (must open a goat door) creates this asymmetry. The host cannot randomly choose—they must avoid the car, which provides information that changes the probability distribution.

Scaling the Problem

The advantage of switching becomes even more dramatic with more doors:

100 doors: Stay = 1% win rate, Switch = 99% win rate
1000 doors: Stay = 0.1% win rate, Switch = 99.9% win rate
General formula: Stay = 1/n, Switch = (n-1)/n

Real-World Applications and Lessons

Decision-Making Under Uncertainty

The Monty Hall Problem teaches valuable lessons about probability and decision-making:

Intuition vs. Mathematics: Our gut feelings about probability are often wrong
Information value: New information can dramatically change optimal strategies
Conditional probability: Understanding how probabilities change given new constraints
Systematic thinking: The importance of mathematical analysis over intuitive guesses

Business and Investment Applications

This probability framework applies to many real-world scenarios:

Investment diversification: Spreading risk across multiple options
A/B testing: Understanding when to switch strategies based on data
Medical diagnosis: How additional tests change probability assessments
Quality control: Updating probability estimates with new information

⚠️ Common Misconceptions

Many people incorrectly assume that after the host opens a door, the probability becomes 50/50 between the remaining doors. This ignores the crucial fact that the host’s choice is constrained by knowledge of the car’s location, creating the asymmetric probability distribution that makes switching advantageous.

Frequently Asked Questions

Why isn’t it just 50/50 after the host opens a door?

The key is that the host’s choice isn’t random—they know where the car is and must avoid opening that door. Your original door still has a 1/3 probability, while the remaining door gets the full 2/3 probability from the eliminated possibilities. The host’s constraint creates this asymmetry.

What if the host doesn’t know where the car is?

If the host opens doors randomly, the problem changes completely. There’s a chance they might accidentally reveal the car, ending the game. In this scenario, the probabilities would indeed become closer to 50/50, but the original Monty Hall Problem specifically requires that the host knows and never reveals the car.

Does the strategy change with more doors?

Switching becomes even more advantageous with more doors. With 100 doors, staying gives you a 1% chance while switching gives you a 99% chance. The general formula is: Stay probability = 1/n, Switch probability = (n-1)/n, where n is the total number of doors.

What if the host opens multiple doors?

The principle remains the same. If the host opens k doors (all containing goats), the staying probability is still 1/n, and the switching probability becomes (n-1)/n distributed among the remaining unopened doors you didn’t choose. If only one remains, it gets the full (n-1)/n probability.

Why do even mathematicians get this wrong initially?

The problem challenges our intuitive understanding of probability. Our brains tend to treat the situation as “two closed doors = 50/50 chance” while ignoring the crucial information provided by the host’s constrained choice. Even trained mathematicians can fall into this intuitive trap without careful analysis.

Is there any real-world evidence for this?

Yes! Computer simulations consistently show the 2/3 vs 1/3 split, and the problem has been tested on actual game shows. The mathematical proof is also ironclad. Every large-scale simulation or experiment confirms that switching wins approximately 67% of the time.

What if I play many games and stick to one strategy?

Over many games, you’ll see your win rate converge to the theoretical probabilities: approximately 33% for always staying and 67% for always switching. The more games you play, the closer your results will match these predicted ratios.

How does this relate to Bayes’ theorem?

The Monty Hall Problem is essentially an application of Bayes’ theorem. We update our probability estimate based on the new information (the host’s action). The host’s constraint to avoid the car door provides information that changes our probability assessment through Bayesian updating.

Can this help with other probability problems?

Absolutely! The key insights—understanding conditional probability, recognizing how constraints affect probability distributions, and distinguishing between random and informed choices—apply broadly to statistics, game theory, medical diagnosis, and decision-making under uncertainty.

What’s the best way to convince someone who doesn’t believe the solution?

Use the 100-door version: “Choose 1 door out of 100. I’ll open 98 doors showing goats. Do you want to stick with your 1 door, or switch to the 1 remaining door that represents 99 other possibilities?” This makes the probability difference much more intuitive.

🎯 Master Probability Paradoxes

Understanding the Monty Hall Problem opens your mind to the counterintuitive nature of probability and the importance of mathematical reasoning over intuition. These skills translate directly to better decision-making in business, science, and everyday life. Use the interactive tools above to fully grasp this fundamental concept, and you’ll be better equipped to tackle complex probability scenarios with confidence.