Monty Hall Problem Calculator
Calculate the probabilities of winning by switching or staying in the classic Monty Hall game show problem. Demonstrates how revealed information changes conditional probabilities, a famous counterintuitive result in probability theory.
This free online monty hall problem calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Total doors in the game (classic version uses 3).
Number of losing doors the host opens after your initial pick.
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Monty Hall Problem Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Monty Hall Problem Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Monty Hall Problem Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Monty Hall Problem Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
- •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
- •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
- •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.
About This Calculator
The Monty Hall Problem Calculator is a free, browser-based calculation tool for engineers, students, and technical professionals. Calculate the probabilities of winning by switching or staying in the classic Monty Hall game show problem. Demonstrates how revealed information changes conditional probabilities, a famous counterintuitive result in probability theory. It implements standard formulas and supports both metric (SI) and imperial unit systems with automatic unit conversion. All calculations are performed instantly in your browser with no data sent to a server. Use this calculator as a quick reference and sanity-check tool during design, analysis, and learning. Always verify results against primary engineering references and applicable standards for any safety-critical application.
About Monty Hall Problem Calculator
The Monty Hall problem calculator demonstrates one of the most famous counterintuitive results in probability. Based on the game show 'Let's Make a Deal,' the scenario presents you with three doors: behind one is a car and behind the other two are goats. After you pick a door, the host (who knows what is behind each door) opens another door revealing a goat, then asks if you want to switch. Intuitively, people feel it should not matter, but switching actually doubles your winning probability from 1/3 to 2/3. This calculator extends the problem to any number of doors and revealed doors, showing that the advantage of switching increases with more doors. The problem highlights how new information changes conditional probabilities.
The Math Behind It
Formula Reference
Monty Hall Probabilities
P(stay) = 1/n; P(switch) = (n-1) / (n * (n-1-r))
Variables: n = total doors; r = doors revealed by host; assumes host always reveals a losing door
Worked Examples
Example 1: Classic 3-door Monty Hall
Three doors, one car, two goats. Host reveals one goat. Should you switch?
Switching doubles your winning probability from 33.3% to 66.7%. Always switch in the classic game.
Example 2: 100-door Monty Hall
100 doors, 1 car. You pick one door, the host opens 98 losing doors. Switch?
Switching gives a 99% chance of winning. With more doors, the advantage of switching becomes dramatically clearer.
Common Mistakes & Tips
- !Believing the probability is 50/50 after a door is opened -- this ignores that the host's choice is informed, not random.
- !Applying the Monty Hall logic to situations where the 'host' reveals information randomly rather than deliberately.
- !Forgetting that the solution depends on the host always knowing where the car is and always opening a losing door.
- !Not recognizing that if you picked the car initially (1/3 chance), switching always loses; but if you picked a goat initially (2/3 chance), switching always wins.
Related Concepts
Frequently Asked Questions
Should you always switch in the Monty Hall problem?
Yes, always switching maximizes your winning probability. With 3 doors, switching wins 2/3 of the time versus 1/3 by staying. The only scenario where switching and staying are equal is if the host opens a door randomly (not knowing where the car is) and happens to reveal a goat.
Why did mathematicians disagree about the Monty Hall problem?
When Marilyn vos Savant published the correct solution in her Parade magazine column in 1990, about 10,000 readers wrote in disagreeing, including nearly 1,000 with PhDs. The confusion arises because people treat the host's door opening as random rather than informed, and because our intuition about probability updating is poor. Computer simulations eventually convinced most skeptics.
Does the Monty Hall problem apply to real game shows?
The classic Monty Hall problem assumes the host always offers a switch and always reveals a losing door. In real game shows, the host might not always offer a switch, or might only offer it when you have picked the car (to trick you into switching away). The mathematical solution applies only when the host always follows the standard rules.