The Monty Hall Problem - Should You Switch?

🧮 The Probability Tree

Initial Pick (1/3 each door): ├── Car behind Door 1 (1/3) │ └── You picked 1 → STAY wins ├── Car behind Door 2 (1/3) │ └── You picked 1 → SWITCH wins └── Car behind Door 3 (1/3) └── You picked 1 → SWITCH wins STAY wins: 1/3 SWITCH wins: 2/3

🤔 Why Our Intuition Fails

After Monty opens a door, we see two closed doors and think "50/50!"

But Monty's action is NOT random. He MUST open a door with a goat. This gives us information that changes the probabilities.

The key insight: your initial 1/3 chance doesn't change when Monty opens a door—so the other door must have 2/3!

🚪 The Million Doors Version

Imagine 1,000,000 doors. You pick one. Monty opens 999,998 doors, all revealing goats.

Would you switch to the one remaining door?

Obviously yes! Your initial pick had 1/1,000,000 chance. The remaining door has 999,999/1,000,000 chance!

📜 The 1990 Controversy

Marilyn vos Savant published the answer in Parade magazine. She received 10,000 letters—including ~1,000 from PhDs—saying she was wrong!

Even Paul Erdős, one of history's greatest mathematicians, didn't believe it until he saw computer simulations.

The New York Times ran a front-page story on the controversy in 1991.

🎬 The Real Monty Hall

Monty Hall hosted "Let's Make a Deal" from 1963-2003. He knew about the problem!

His advice: "If the host is REQUIRED to open a door and offer a switch, then you should switch."

"But if he has the CHOICE whether to allow a switch... beware. Caveat emptor!"

⚠️ Critical Assumptions

The 2/3 answer requires these conditions:

Monty ALWAYS opens a door with a goat
Monty ALWAYS offers the switch
Monty knows where the car is
If you picked the car, Monty picks randomly between the two goat doors

🚪 The Monty Hall Problem

THE SETUP

Step 1: Pick a Door!

📊 Your Results

🔄 SWITCH Strategy

✋ STAY Strategy