Monty Hall Problem Simulator
Interactive simulator for the famous Monty Hall probability puzzle. Play the game yourself or run simulations to understand why switching doors gives you a better chance of winning.
Calculate Your Monty Hall Problem Simulator
Monty Hall Simulator
There are three doors. Behind one door is a car, behind the others are goats. Choose a door:
Run Simulations
Run multiple simulations to see the probability difference between staying and switching:
Always Switching
Win Rate: 0.00%
Wins: 0
Losses: 0
Always Staying
Win Rate: 0.00%
Wins: 0
Losses: 0
What is the Monty Hall Problem?
The Monty Hall Problem is a famous probability puzzle named after the host of the television game show "Let's Make a Deal," Monty Hall. It presents a counterintuitive scenario that has stumped mathematicians, statisticians, and even probability experts.
The Classic Scenario
The classic version of the problem goes like this:
- You're on a game show, and you're given the choice of three doors.
- Behind one door is a car (the prize); behind the others are goats (the joke booby prizes).
- You pick a door, say Door 1, but the host, who knows what's behind the doors, doesn't open it yet.
- Instead, the host, who must always open a door with a goat, opens another door, say Door 3, which has a goat.
- The host then asks you: "Do you want to stick with your original choice of Door 1, or switch to Door 2?"
The question is: Is it to your advantage to switch your choice to the other unopened door?
The Surprising Answer
Although it seems intuitive that switching or staying should make no difference (with a 50/50 chance of winning), the mathematically correct answer is that you should always switch doors. By switching, you increase your probability of winning from 1/3 to 2/3.
Why Switching is Better: The Explanation
There are several ways to understand why switching gives you a better chance of winning:
Explanation 1: Initial Probabilities
When you make your initial choice, you have a 1/3 probability of selecting the door with the car and a 2/3 probability of selecting a door with a goat.
If you initially select a goat (which happens 2/3 of the time), the host will be forced to open the other door with a goat, leaving the car behind the remaining door. So, switching in this case would lead to winning.
If you initially select the car (which happens 1/3 of the time), the host will open either of the two goat doors, and switching would lead to selecting a goat.
Therefore, switching wins when your initial selection is a goat (2/3 of the time) and loses when your initial selection is the car (1/3 of the time).
Explanation 2: The Host's Knowledge
The key insight is that the host's action of revealing a goat is not random—they know where the car is and must reveal a goat. This introduces new information that can be leveraged by switching.
By staying, you're betting that your original 1/3 chance selection was correct. By switching, you're betting that your original selection was wrong, which happens 2/3 of the time.
Verification Through Simulation
The best way to verify this counterintuitive result is through simulation. By running the scenario thousands of times, we can empirically demonstrate that switching indeed wins approximately 2/3 of the time, while staying wins only about 1/3 of the time.
Our calculator provides both an interactive simulation where you can play the game yourself and track your results, as well as a Monte Carlo simulation that can run thousands of trials automatically to verify the probabilities.
Historical Context and Controversy
When this problem was first presented in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990, her correct answer that switching doors is the optimal strategy sparked a massive controversy. Thousands of readers, including many with Ph.D.s in mathematics and statistics, wrote letters insisting she was wrong. This demonstrates how counterintuitive the problem is, even for those with training in probability theory.
The Monty Hall Problem has since become a classic example used in probability courses to illustrate conditional probability, Bayes' theorem, and the importance of carefully analyzing the conditions of a problem.
Using the Interactive Simulator
Our interactive simulator allows you to:
- Play the Monty Hall game multiple times
- Choose whether to switch or stay after the host reveals a goat
- Track your statistics for both strategies
- Run automated simulations of thousands of games to verify the probabilities
By engaging with the simulator, you can develop an intuitive understanding of this famous probability paradox.
Frequently Asked Questions
Many people intuitively think that once the host reveals a goat, you face a simple choice between two doors, giving you a 50/50 chance regardless of whether you switch. However, this intuition doesn't account for the host's constrained behavior - they must always reveal a goat, not a random door. This creates an asymmetry that favors switching. Your initial choice has a 1/3 probability of being correct, and this probability doesn't change when the host reveals a goat (because they would always reveal a goat regardless). The remaining 2/3 probability must be concentrated on the other unopened door, making switching the better strategy.
Yes, in the standard formulation of the Monty Hall Problem, the host always knows the location of the prize and intentionally reveals a door with a goat. This is a crucial aspect of the problem. If the host were to randomly select a door (potentially revealing the prize and ending the game), or if they could only open your door first, the probability calculations would be different. The host's omniscience and the rules constraining their behavior (always revealing a goat, never your chosen door) are what create the advantage for switching.
Yes, but the advantage of switching would actually increase. In a generalized version with n doors, where the host still opens all but one of the unchosen doors (all containing goats), the probability of winning by staying remains 1/n, while the probability of winning by switching becomes (n-1)/n. For example, with 10 doors, your initial choice has a 1/10 chance of being correct. After the host reveals 8 goats, staying with your original choice still gives you a 1/10 chance of winning, while switching gives you a 9/10 chance. The more doors in the game, the greater the advantage of switching.
The Monty Hall Problem is a perfect example of Bayesian updating - revising probabilities based on new information. Initially, each door has a 1/3 probability of containing the prize. When the host reveals a goat behind one of the doors you didn't choose, this provides new information that should update your probabilities. Using Bayes' theorem formally: P(Car in Door 2 | Host opens Door 3) = P(Host opens Door 3 | Car in Door 2) × P(Car in Door 2) / P(Host opens Door 3) = 1 × (1/3) / (1/2) = 2/3. This shows that after the host's action, the probability of the car being behind the other unopened door increases to 2/3.
Yes, the Monty Hall Problem illustrates important concepts that apply to many decision-making scenarios involving updating beliefs with new information. It has applications in medical diagnosis (interpreting test results given prior probabilities), investment strategies (knowing when to change investments based on new market information), and scientific research (updating hypotheses based on experimental results). It also demonstrates how intuition can lead us astray in probability problems and the importance of careful mathematical analysis. The core insight—that new information can change optimal strategies—is widely applicable in fields like economics, statistics, and artificial intelligence.
Yes, the Monty Hall Problem has been demonstrated empirically countless times through simulations and experiments. MythBusters featured it in a 2011 episode and verified the 2/3 probability for switching through repeated trials. Numerous classroom experiments, computer simulations, and even public demonstrations have consistently shown results matching the theoretical prediction. That said, on the actual "Let's Make a Deal" show hosted by Monty Hall, the exact scenario rarely if ever occurred in the same way as the theoretical problem, as the rules and constraints were different. When properly set up with the right constraints, however, the empirical results always support the mathematical conclusion that switching is better.
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