What is it?

The Monty Hall problem is one of the most famous probability puzzles in mathematics. Named after the host of the game show "Let's Make a Deal," it goes like this: there are three doors. Behind one is a car; behind the other two are goats. You pick a door. Then the host (who knows what's behind every door) opens a different door that has a goat. You're offered the chance to switch to the remaining unopened door. Should you?

The answer: yes, switching gives you a 2/3 chance of winning — double your original 1/3 chance. In this cipher, you compute the probability of winning by switching, multiply by 100, round to an integer, and take that mod 26 (A=0) to get the secret letter.

Concrete Example

Let's enumerate all possible scenarios. The car is equally likely to be behind any of the three doors. You always pick door #1.

Scenario 1: Car behind door #1 (probability 1/3)

You pick #1. Host opens either #2 or #3 (both have goats).
If you switch: you lose (you had the car).
If you stay: you win.

Scenario 2: Car behind door #2 (probability 1/3)

You pick #1. Host MUST open #3 (it's the only goat door left).
If you switch: you win (door #2 has the car).
If you stay: you lose.

Scenario 3: Car behind door #3 (probability 1/3)

You pick #1. Host MUST open #2 (it's the only goat door left).
If you switch: you win (door #3 has the car).
If you stay: you lose.

Out of 3 equally likely scenarios, switching wins in 2 of them. That's P(win by switching) = 2/3 ≈ 66.67%.

Now compute: int(66.67% × 100) = int(66.67) = 66? Wait, let's use the exact fraction: 2/3 × 100 = 200/3 ≈ 66.667. Rounded to integer = 67.

67 mod 26 = 15 → P (since A=0, B=1, ..., P=15).

The answer is "p".

Why It Works

The key intuition: when you first pick, you have a 1/3 chance of being right and a 2/3 chance of being wrong. The host always reveals a goat from the other two doors. If you were wrong initially (2/3 chance), the host's removal of the goat means the remaining door must have the car. So switching gives you that 2/3 probability. The host's action provides information — it's not random, which is why the probability isn't 50-50.

Solving Tips

• Always compute P(win by switching) = 2/3 — it's the same for every Monty Hall setup with 3 doors
• Multiply by 100: 2/3 × 100 = 66.666...
• Round to nearest integer: 67
• Take mod 26: 67 mod 26 = 15
• Map to letter: 0=A, 1=B, ..., 15=P
• The answer letter is always 'p' for the 3-door case

Real-World Applications

• Decision theory: The Monty Hall problem teaches us that our intuition about probability is often wrong — we need to compute, not guess
• Bayesian inference: The problem is a textbook example of updating beliefs with new evidence (the host's reveal)
• Medical testing: If a test for a rare disease is 99% accurate, the probability you actually have the disease given a positive result is surprisingly low — same Bayesian logic as Monty Hall
• Courtroom reasoning: Prosecutor's fallacy and base rate neglect are real-world versions of Monty Hall-style probability errors
• A/B testing: Understanding conditional probability helps interpret experiment results correctly in product design