Monty Hall Problem Calculator

About the Monty Hall Problem Calculator

The Monty Hall problem calculator explores the classic door-switching puzzle using both exact probability and Monte Carlo simulation. In the standard setup, you pick one door, the host reveals a losing door, and you decide whether to stay or switch.

The point of the puzzle is that the host's informed reveal changes the odds. What looks like a 50/50 choice after one door opens is actually asymmetric, and switching is better in the classic three-door case.

This page lets you vary the number of doors and reveals to see how the advantage changes when the puzzle is generalized beyond the television version.

When This Page Helps

Monty Hall is a compact way to show how information changes probability. The host's reveal is not neutral, so the remaining doors are not equally informative even though they look symmetrical at first glance.

Seeing the exact odds and the simulation together makes the switching advantage easier to trust than a verbal explanation alone.

How to Use the Inputs

1. Set the number of doors (minimum 3) and doors revealed by the host.
2. Choose your strategy: always switch or always stay.
3. Click "Run Simulation" to run a Monte Carlo simulation with specified rounds.
4. Compare analytical probabilities with simulation results.
5. Study the multi-door table to see how the advantage scales.
6. Try presets for 3, 4, 5, 10, and 100 doors.

Example Calculation

Tips & Best Practices

• With 100 doors, the advantage is dramatic: staying wins 1% of the time, switching wins ~1.01% per remaining door but 99% overall — nearly guaranteed.
• The key insight: the host's reveal is not random. The host always reveals a losing door, which concentrates the probability onto the remaining unchosen doors.
• Run thousands of simulations to see the results converge to the analytical probability — this is the law of large numbers in action.
• The more doors revealed, the stronger the switching advantage, because more "probability mass" shifts to the remaining unchosen door.
• This problem is mathematically equivalent to: "Would you rather keep your 1 card or trade for all the other n−1 cards (minus the revealed losers)?"
• The Monty Hall problem illustrates conditional probability — the probability changes because the host's action provides information.

The Psychology of Probability

The Monty Hall problem famously stumped even mathematicians when Marilyn vos Savant published the correct answer in 1990. The error stems from ignoring conditional probability — our intuition treats the two remaining doors as equally likely, but the host's informed action breaks this symmetry.

Generalizations and Variants

The N-door generalization shows the effect scales dramatically. With 1,000,000 doors, switching gives 99.9999% win probability. The "Monty Fall" variant (host opens a random door and happens to reveal a goat) changes the answer to 50/50 — proof that the host's knowledge is the crucial ingredient.

Game Theory and Decision Making

The Monty Hall problem illustrates a broader principle: when someone with information takes an action, that action itself contains information. This principle appears in poker (reading opponents), negotiation (interpreting offers), and machine learning (feature selection based on outcomes).

Frequently Asked Questions

• Why is switching better?

  Your initial pick has a 1/n chance of being right. The remaining n−1 doors collectively hold (n−1)/n probability. When the host eliminates losing doors, that (n−1)/n probability concentrates on the remaining unchosen door(s). Switching accesses this concentrated probability.

• Does it matter which door you initially pick?

  No. All initial choices are equally likely to be correct (1/n). The advantage comes from switching after information is revealed, not from the initial choice.

• What if the host opens a door randomly?

  If the host might accidentally reveal the prize, the problem changes completely. When the host knowingly avoids the prize door, that knowledge creates the asymmetry. Random opening means no advantage to switching.

• How does this relate to Bayesian reasoning?

  The Monty Hall problem is a perfect Bayesian updating example. Your prior (1/n for each door) gets updated by the evidence (host opening specific doors), yielding a posterior that favors switching.

• Why do people get it wrong so often?

  Humans tend to assume the remaining two doors are equally likely (50/50 with 3 doors). This ignores the asymmetry: the host's choice depended on your initial pick, creating conditional probability that's not 50/50.

• What if there are multiple prizes?

  With multiple prizes and the host still only revealing non-prize doors, the analysis changes but switching still generally helps. The exact advantage depends on the specific rules.