Questions : What is the Monty Hall problem and its solution?
Answer:
The best strategy is to switch, giving you a 2/3 chance of winning the car, compared to a 1/3 chance if you stay.
Additional Information
The Monty Hall problem is a classic probability puzzle based on the American TV game show Let's Make a Deal. It's famous because the correct answer is counterintuitive to many people.
The Problem
- The Setup: You're on a game show. There are three doors.
2 Behind one door is a car, and behind the other two are goats. - Your Choice: You pick a door (let's say Door #1).
3 - Monty's Reveal: The host, Monty Hall, knows what's behind each door.
4 He always opens a different door than the one you chose, revealing a goat. Let's say he opens Door #3. - The Dilemma: Monty now gives you the option to switch your choice to the remaining closed door (Door #2). Should you switch, or should you stick with your original choice?
The Solution
You should always switch. Switching doubles your chances of winning the car.
Why Switching Works (The Intuition Breaker)
Most people think the odds are 50/50 after Monty reveals a goat. However, that's not the case. Here's why:
- Initial Choice: When you initially chose a door, you had a 1/3 chance of picking the door with the car and a 2/3 chance of picking a door with a goat.
- Monty's Action: Monty's action of revealing a goat doesn't change the initial probabilities. The 2/3 probability that you initially picked a goat door is now concentrated on the other closed door. Monty has essentially given you information. He hasn't changed the odds; he's revealed where the higher probability lies.
- Switching: By switching, you're taking advantage of that initial 2/3 probability. You're betting that you were initially wrong (which you had a 2/3 chance of being), and the remaining door is the one that has the car.
Simplified Explanation
Imagine the doors are numbered 1, 2, and 3. You pick door 1. There are two possibilities:
- You picked the car initially (1/3 probability): If this is the case, switching loses you the car.
- You picked a goat initially (2/3 probability): If this is the case, Monty must reveal the other goat. Switching wins you the car.
Since the probability of picking a goat initially is twice as high as picking the car, switching doubles your probability of winning.
In short: Monty's knowledge and deliberate action of revealing a goat is what makes switching the optimal strategy. He's not randomly opening a door; he's giving you valuable information.
