Monty Hall Problem Simulator

Probability and Randomness

Run simulations of the classic game show problem to see if you should stick with your choice or switch.

Examples

See how the probabilities play out with different numbers of simulations.

100 Simulations

Low Volume

A small number of simulations to get a quick feel for the problem.

Simulations: 100

1,000 Simulations

Standard

A standard number of simulations to see the probabilities start to stabilize.

Simulations: 1000

10,000 Simulations

High Volume

A high number of simulations that will closely reflect the theoretical probabilities.

Simulations: 10000

100,000 Simulations

Very High Volume

A very large number of simulations for a highly accurate statistical result.

Simulations: 100000

Other Titles
Understanding the Monty Hall Problem: A Comprehensive Guide
Delve into the logic, mathematics, and common misconceptions surrounding one of the most famous puzzles in probability.

What is the Monty Hall Problem?

  • The Game Show Setup
  • The Counter-Intuitive Choice
  • Why It's a Puzzle
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem statement is as follows: Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to switch your choice?
The Dilemma
At its core, the problem presents a dilemma that pits intuition against the laws of probability. Most people assume that after one door with a goat is revealed, the two remaining doors have an equal 1/2 chance of hiding the car. This intuitive, yet incorrect, assumption is what makes the problem so famous and such a great educational tool.

Step-by-Step Guide to Using the Monty Hall Problem Simulator

  • Running a Simulation
  • Interpreting the Results
  • Experimenting with Examples
How It Works
Our simulator allows you to test the Monty Hall problem thousands of times in an instant. Here's how to use it:
1. Enter the desired number of simulations into the input field. We recommend starting with at least 1,000 for a clear result.
2. Click the 'Run Simulation' button.
3. Observe the results for two strategies: 'Always Stay' with your initial choice and 'Always Switch' to the other unopened door.
Analyzing the Output
The results section will show you the number of wins, losses, and the overall win rate for both strategies. You will consistently find that the 'Always Switch' strategy yields a win rate of approximately 66.7%, while the 'Always Stay' strategy results in a win rate of about 33.3%, confirming the mathematical solution.

Real-World Applications of the Monty Hall Problem

  • Decision Making Under Uncertainty
  • Cognitive Biases
  • Scientific Research
While it may seem like a simple puzzle, the principles of the Monty Hall problem have applications in various fields.
Economics and Finance
In financial markets, investors often have to make decisions based on partial information. The Monty Hall problem teaches a valuable lesson about updating beliefs and strategies when new, relevant information becomes available, rather than sticking to an initial decision inflexibly.
Medicine
In medical diagnostics, doctors update the probability of a patient having a particular disease as they receive results from various tests. This process is analogous to the game show host providing new information by opening a door, changing the probabilities of the initial diagnosis.

Common Misconceptions and the Correct Logic

  • The '50/50' Fallacy
  • Why the Host's Knowledge is Key
  • The Power of Switching
The '50/50 Fallacy'
The most common misconception is that after the host opens a door with a goat, the remaining two doors each have a 50% chance. This is incorrect because the host's action is not random. The host always opens a door with a goat and never opens your chosen door. This act provides crucial information.
The Correct Logic
Your initial choice has a 1/3 chance of being correct. This means there is a 2/3 chance the car is behind one of the other two doors. When the host opens one of those other doors to reveal a goat, the entire 2/3 probability is concentrated on the single remaining unopened door. Your original choice's probability of being correct remains 1/3. Therefore, switching is the superior strategy.

The Mathematical Proof Behind Switching

  • Case-by-Case Analysis
  • Using Conditional Probability
  • Bayes' Theorem Application
A Simple Proof by Cases
Let's analyze the outcomes based on your initial pick:
Case 1: You initially pick the door with the car (1/3 probability). The host opens one of the two goat doors. If you switch, you lose. If you stay, you win.
Case 2: You initially pick a door with a goat (2/3 probability). The host must open the other door with a goat. If you switch, you are guaranteed to get the car. If you stay, you lose.
Summing this up, the 'stay' strategy wins only in Case 1 (1/3 of the time), while the 'switch' strategy wins in Case 2 (2/3 of the time).
Conditional Probability (Bayes' Theorem)
Let C1, C2, C3 be the events that the car is behind door 1, 2, or 3. Let H2 be the event that the host opens door 2. Assume you chose door 1. We want to find P(C3|H2), the probability the car is in door 3 given the host opened door 2. Bayes' theorem shows that this probability is 2/3, confirming that switching is the optimal choice.