Quick Answer: In the famous game show problem, you are asked to pick one of three doors. One has a car, two have goats. You pick Door 1. The host (who knows what is behind each door) opens Door 3 to reveal a goat. He asks: do you want to switch to Door 2? The answer is YES. Switching doubles your chances of winning from 1/3 to 2/3.
The Setup
This puzzle, based on the game show "Let's Make a Deal" hosted by Monty Hall, became famous in 1990 when Marilyn vos Savant declared in her column that contestants should always switch. Thousands of readers — including PhD mathematicians — wrote in to tell her she was wrong. They argued that with two doors left, it is a 50/50 chance. The mathematicians were wrong, and Marilyn was right.
Why It's Not 50/50
The key to the puzzle is understanding that the host's action is not random. The host MUST reveal a goat. This means the host uses his hidden knowledge of the car's location to filter out a wrong answer for you.
- Scenario A (You pick the Car from the start): There is a 1/3 chance of this. If you switch, you get a goat. You lose.
- Scenario B (You pick Goat A from the start): There is a 1/3 chance of this. The host MUST open Goat B. The remaining door is the Car. If you switch, you win.
- Scenario C (You pick Goat B from the start): There is a 1/3 chance of this. The host MUST open Goat A. The remaining door is the Car. If you switch, you win.
The Simple Explanation
When you make your first choice, there is a 1/3 chance you are right, and 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 2/3 probability doesn't change — it all collapses onto the single remaining unchosen door. Therefore, sticking with your door retains the original 33.3% chance, but switching yields a 66.7% chance.
The 100 Door Mental Model
If it still hurts your brain, imagine 100 doors. You pick Door 1 (1% chance). Monty Hall then opens 98 other doors, all with goats. You are left with Door 1 and Door 72. Do you stick with Door 1, or do you switch to the door that Monty deliberately avoided opening out of the 99 others? You switch! The same logic applies to 3 doors.