Monty Hall Simulator – Test Stay vs. Switch

The Monty Hall Simulator is an interactive probability tool based on the famous Monty Hall problem. It lets you simulate the classic three-door game and compare two strategies: staying with your first choice or switching to the other unopened door.

The puzzle is simple. Imagine three doors: behind one is a prize, while the other two contain something you do not want. You choose one door. The host, who knows where the prize is, opens one of the other two doors and always reveals a losing option. You are then given a choice: stay with your original door or switch.

This creates a result that often feels surprising. In the standard version of the puzzle, your original choice has a 1/3 probability of being correct, while switching gives you a 2/3 probability of winning. A simulator makes this easier to understand because you can run many trials and observe how the results change as the number of simulations increases.

Use our Monty Hall Simulator to experiment with the puzzle yourself. You can test the stay and switch strategies, run repeated trials, and see probability in action.

What Is the Monty Hall Problem?

The Monty Hall problem is a probability puzzle named after Monty Hall, the host of the television game show Let’s Make a Deal. The mathematical puzzle became widely known because its answer seems counterintuitive at first.

The standard problem has three doors:

Only one door contains the prize.

You select one door, but you do not know whether you selected the winning door. The host then opens one of the remaining doors and shows that it does not contain the prize.

The important detail is that the host knows where the prize is and deliberately opens a losing door. You are then offered the opportunity to stay with your original choice or switch to the remaining unopened door.

That final choice is where the probability changes.

How Does the Monty Hall Simulator Work?

The simulator recreates the basic rules of the Monty Hall problem using repeated trials.

A typical simulation follows these steps:

1. The prize is randomly placed behind one of three doors.
2. You randomly select a door.
3. The host opens another door that does not contain the prize.
4. You choose whether to stay or switch.
5. The simulator records whether your strategy wins.
6. The process can be repeated many times.
7. The results show the winning percentage for each strategy.

Because each trial is independent, individual results can vary. However, when you run a large number of simulations, the overall results generally move toward the theoretical probabilities.

Why Does Switching Have a 2/3 Chance?

The easiest way to understand the Monty Hall problem is to focus on your first choice.

When you initially select one of three doors:

• Your selected door has a 1/3 chance of containing the prize.
• The two doors you did not select collectively have a 2/3 chance of containing the prize.

The host then opens one of those two unselected doors and reveals a losing door.

The host’s action does not make your original door more likely to contain the prize. Your original door still has a 1/3 chance of being correct.

The remaining unopened door effectively represents the original 2/3 probability assigned to the two doors you did not choose.

Therefore:

So, under the standard Monty Hall rules, switching wins about twice as often as staying when the game is repeated many times.

A Simple Example

Suppose you choose Door 1.

Before anything else happens:

• Door 1 = 1/3 chance of winning
• Door 2 or Door 3 = 2/3 combined chance of winning

Now suppose the host opens Door 3 and shows that it is empty.

There are now two unopened doors:

• Your original Door 1
• Door 2

It may look like each door should now have a 50% chance. But the host did not open a random door. The host intentionally opened a door that was known to contain no prize.

If your original Door 1 was correct, staying wins.

If your original Door 1 was wrong, the prize was initially behind Door 2 or Door 3. Because the host must reveal a losing door, switching leads you to the prize.

This is why switching wins in two of the three possible initial situations.

How to Use the Monty Hall Simulator

The simulator is designed to make the probability experiment quick and easy.

Try running only a few rounds first. Then increase the number of simulations substantially. You may notice that short runs can produce unusual results, while larger runs tend to move closer to the expected probabilities.

What Can You Learn From the Simulator?

The main purpose of a Monty Hall simulator is not simply to tell you the answer. It allows you to observe probability through repeated experiments.

Understand Probability

The simulator provides a practical example of probability instead of relying only on a mathematical explanation.

Compare Stay vs. Switch

You can compare the outcomes of the two strategies and see how frequently each one wins.

See the Law of Large Numbers

A small number of trials can produce results that are far from the theoretical probability. As the number of trials grows, the observed percentages generally become more stable.

Explore Counterintuitive Results

The Monty Hall problem is useful because the intuitive answer can differ from the mathematical answer. Running simulations can help explain why.

Learn Through Experimentation

Instead of simply reading that switching has a 2/3 probability of winning, you can test the idea repeatedly.

Manual Probability vs. Simulation

There are two common ways to study the Monty Hall problem.

A simulator is especially useful because it can perform hundreds, thousands, or more trials much faster than a person could play them manually.

Example of a Monty Hall Simulation

Imagine you run 30 trials.

Because this is a relatively small sample, your results might not be close to the theoretical values. For example, switching might win 19 times while staying wins 11 times.

Now imagine running 10,000 trials.

The exact results will still vary, but the percentages should generally become much closer to:

• Stay: approximately 33.3%
• Switch: approximately 66.7%

The simulator therefore demonstrates an important statistical idea: larger samples tend to provide more stable estimates of an underlying probability.

These numbers are theoretical expectations rather than a guarantee for every simulation. Random variation is always possible.

Why Don’t the Results Always Equal 33.3% and 66.7%?

Random simulations do not guarantee exact percentages.

For example, if you flip a fair coin 10 times, you do not have to get exactly five heads and five tails. You could get seven heads and three tails.

The same principle applies to the Monty Hall problem.

A small simulation might produce:

• Stay: 40%
• Switch: 60%

Another simulation might produce:

• Stay: 30%
• Switch: 70%

Neither result changes the underlying probability. As the number of trials increases, the observed percentages generally become more stable around their expected values.

Common Uses for a Monty Hall Simulator

A Monty Hall simulator can be useful in several situations.

Students

Students can use it to understand probability, conditional probability, experimental probability, and random trials.

Teachers

Teachers can use the simulation as an interactive classroom example. Students can make a prediction before running the experiment and then compare their prediction with the results.

Probability Practice

Anyone learning probability can use repeated simulations to connect mathematical theory with experimental evidence.

Programming Projects

The Monty Hall problem is also a popular programming exercise because it can be simulated with simple random-selection logic.

Decision-Making Experiments

The puzzle provides a simple demonstration of how additional information can change the interpretation of probabilities.

Tips for Getting Better Results

For meaningful simulation results, keep these points in mind:

• Run more trials when you want a more stable estimate.
• Compare both strategies rather than testing only one.
• Understand the host’s rules. The standard puzzle assumes the host knows where the prize is.
• Do not treat one short simulation as proof.
• Repeat experiments to see how random variation affects the results.
• Compare observed results with theoretical probability.

The host’s behavior is particularly important. If the host does not know where the prize is, or does not always reveal a losing door, the probability analysis can change.

Standard Monty Hall Rules

The classic probability result assumes several specific conditions.

If you change these rules, you may no longer have the standard Monty Hall problem.

For example, if the host randomly opens one of the other doors and could accidentally reveal the prize, the probabilities need to be analyzed differently.

Monty Hall Problem Formula

The basic probability can be expressed simply.

When you initially select one door from three:

Probability your first choice wins:

1/3

Therefore, the probability that the prize is behind one of the two doors you did not select is:

2/3

After the host reveals one losing door, switching transfers your choice to the remaining unopened door. Under the standard rules:

P(stay wins) = 1/3

P(switch wins) = 2/3

This is why the switch strategy has twice the theoretical winning probability of the stay strategy.

Monty Hall Simulator vs. Manual Calculation

Mathematical analysis explains why switching works. Simulation lets you see the pattern through repeated experiments. Using both approaches can provide a clearer understanding of the problem.

Frequently Asked Questions

What is a Monty Hall simulator?

A Monty Hall simulator is an online tool that recreates the three-door Monty Hall probability problem. It can repeatedly test the stay and switch strategies and show the resulting outcomes.

Is switching really better in the Monty Hall problem?

Under the standard rules, switching has a 2/3 probability of winning, while staying with the original choice has a 1/3 probability.

Why is switching 2/3 instead of 1/2?

Your original door starts with a 1/3 probability of containing the prize. The other two doors collectively have a 2/3 probability. When the host deliberately reveals a losing door, the remaining unopened door carries that 2/3 probability under the standard rules.

Does the Monty Hall simulator use real randomness?

A simulator can use random selection to place the prize and choose the initial door. The exact implementation depends on the tool, but repeated random trials are used to demonstrate the theoretical probability.

How many simulations should I run?

There is no single required number. A few trials are useful for seeing individual outcomes, while hundreds or thousands of trials can provide a more stable estimate of the theoretical probabilities.

Can I use the simulator for a probability lesson?

Yes. It can be a useful educational demonstration because students can first predict the result and then run repeated experiments to compare experimental and theoretical probability.

What happens if the host does not know where the prize is?

That is a different probability scenario. The classic 2/3 switching result depends on the host knowing the prize location and deliberately opening a losing door.

Does switching guarantee a win?

No. Switching does not guarantee that you will win every round. It gives you a higher probability of winning over repeated trials under the standard rules.

Is the Monty Hall problem a trick question?

It is often described as counterintuitive because the correct probability can conflict with an initial 50/50 intuition. The result follows from the rules of conditional probability rather than a trick.

Can I run the Monty Hall experiment online?

Yes. You can use our free Monty Hall Simulator to run the experiment and compare staying with switching.

External Resources

For additional information about probability and statistical reasoning, the National Institute of Standards and Technology (NIST) provides educational and technical resources related to probability and statistics.

You can also explore probability concepts through the Encyclopaedia Britannica, which provides general reference information on mathematics and probability.

Conclusion

The Monty Hall Simulator turns a famous probability puzzle into an interactive experiment. Instead of simply memorizing that switching has a 2/3 theoretical probability of winning, you can run repeated trials and observe how the results develop.

The key idea is that your first choice has only a 1/3 chance of being correct. The two doors you did not choose collectively have a 2/3 chance. When the host deliberately reveals a losing door, switching to the remaining unopened door preserves that larger probability under the standard rules.

For students, teachers, probability learners, and anyone curious about the puzzle, an online simulator provides a simple way to explore the relationship between theoretical probability and experimental results.

Try the Monty Hall Simulator and run your own trials to see the probability pattern for yourself.