Monty Hall Simulation

What is the Monty Hall Problem?

The Monty Hall Problem is a famous probability puzzle named after the host of the television game show "Let's Make a Deal." The problem goes as follows:

1. You are presented with three doors. Behind one door is a car (the prize), and behind the other two doors are goats.

2. You pick a door, say door #1, hoping to win the car.

3. The host (who knows what's behind each door) opens another door, say door #3, which has a goat.

4. The host then gives you a choice: Do you want to stick with your original choice (door #1), or switch to the remaining unopened door (door #2)?

The counterintuitive result is that you have a 2/3 chance of winning if you switch doors, but only a 1/3 chance if you stick with your original choice. This simulation demonstrates this probability in action.

The reason for this is that when you first choose a door, you have a 1/3 chance of picking the car. The host then eliminates one incorrect door, but they can't eliminate the door with the car. This means that if you initially picked a goat (2/3 probability), switching will always win you the car. If you initially picked the car (1/3 probability), switching will lose.

Why Isn't It 50/50 For A Single Game?

Many people argue that once the host opens a door, you're choosing between two doors, so it must be a 50/50 chance. This is incorrect because it ignores the crucial information provided by the host's actions. Here's why:

The Key Point: The Host's Knowledge
The host MUST show you a goat, and MUST offer you a chance to switch. This is not random - they are forced to follow these rules. This means their action carries important information that changes the probability.

Let's Break It Down:

1. Your Initial Pick (Door 1)
   • If you picked the car (1/3 chance) → Host can open either goat door → Switching loses
   • If you picked a goat (2/3 chance) → Host MUST open the other goat door → Switching ALWAYS wins

Think About It This Way:
When you first pick a door, you have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. The host's action doesn't change these original probabilities - instead, it concentrates the 2/3 probability of the car being behind "not your door" into the single remaining door they didn't open.

A Different Perspective:
Imagine if instead of three doors, there were 100 doors, one car, and 99 goats. You pick one door, then the host opens 98 doors, all showing goats, leaving just your door and one other door. Would you switch? In this case, it's more obvious that your initial pick had a 1/100 chance of being right, and therefore the remaining door must have a 99/100 chance of containing the car.

This is why even in a single game, switching gives you a 2/3 chance of winning, while staying gives you a 1/3 chance. The host's actions, guided by their knowledge and rules, provide real information that makes this different from a simple 50/50 choice.