Okay, despite my declaration that I'd devoted enough time/space to this, I'm still getting messages arguing The Monty Hall Problem from odd angles, some disagreeing with what I said was the correct answer, some offering convoluted explanations of why it's right. I'm going to give this one more try. It's really a lot simpler than some people think.
You have a choice of three doors. Behind one is a car. Behind two are goats. You get to pick one and you'll win the prize that is behind it. You have no idea where the car is but your goal is to win it and not a goat.
After you make your selection but before that door is opened, the host (Monty) opens one of the doors you didn't pick to reveal a goat. Monty knows what's behind each of the doors and he always opens one to show you a goat. (This, by the way, is not exactly the way Let's Make a Deal ever worked.)
So now two doors have not been opened — yours and the one you didn't pick. Monty gives you the option of staying with your original door or swapping for the unopened door. Should you swap?
Yes, you should swap. Contrary to what you might think at first glance, it does make a difference. If you swap, you double your chances of going home with the car.
To prove this, let us consider the odds if you don't swap…
- One-third of the time, you pick the door with the car
- Monty opens a door that reveals a goat
- You don't swap
- You win the car
- Two-thirds of the time, you pick a door with one of the goats
- Monty opens a door to reveal the other goat
- You don't swap
- You win a goat
So one-third of the time, you go home with the car. Two-thirds of the time, you go home with a goat. Do we all agree on this? Fine. Now, here's the situation if you do swap…
- One-third of the time, you pick the door with the car
- Monty opens a door that reveals a goat
- You swap the door you picked for the other one that is not yet opened
- You win the other goat
- Two-thirds of the time, you pick a door with one of the goats
- Monty opens a door to reveal the other goat
- You swap the door you picked for the other one that is not yet opened
- You win the car
Got it? Two-thirds of the time, the swapper will go home with the car. That's because there's a one-in-three chance you'll start with the right answer and a two-in-three chance you'll start with the wrong answer…and swapping will turn the right answer into the wrong answer and the wrong answer into the right answer.
This is my final post on the subject. I hope.