# MATH04: MONTY HALL SIMULATOR

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Monty Hall simulator is a program that allows you to simulate the famous Monty Hall problem in a virtual environment. The simulator is based on the probability puzzle where a game show contestant is asked to choose one of three doors, behind which one door has a car, and the other two doors have goats. After the contestant makes their choice, the game show host, Monty Hall, opens one of the remaining two doors to reveal a goat, and then offers the contestant the opportunity to switch their choice to the other unopened door.

The Monty Hall simulator allows you to play out this scenario by randomly assigning a car and goats behind three virtual doors. You get to choose one of the doors, just like in the real problem, and then the simulator reveals one of the other two doors to have a goat behind it, just like Monty would do. At this point, the simulator gives you the option to switch your choice to the other unopened door, or stick with your original choice.

The purpose of the Monty Hall simulator is to help you understand the counterintuitive solution to the Monty Hall problem, which is that you should always switch your choice to the other unopened door. The simulator does this by showing you the probability of winning the car if you stick with your original choice, versus the probability of winning the car if you switch your choice. The results will show that, over a large number of simulations, switching your choice results in a higher probability of winning the car.

Overall, the Monty Hall simulator is a useful tool for demonstrating the principles of conditional probability and showing how seemingly simple problems can have counterintuitive solutions.

WHAT IS MONTY HALL PROBLEM?

The Monty Hall problem is a famous probability puzzle that is named after the host of the television game show “Let’s Make a Deal,” Monty Hall. The problem is often used to illustrate the concept of conditional probability.

Here’s the problem: You are a contestant on a game show, and there are three doors in front of you. Behind one of the doors is a car, and behind the other two doors are goats. The game show host, Monty Hall, asks you to pick one of the doors. After you have made your choice, Monty opens one of the other two doors to reveal a goat. Now Monty offers you a chance to switch your choice to the remaining door, or stick with your original choice. The question is: Should you switch your choice or stick with your original choice?

The answer is that you should always switch your choice. Here’s why: When you first make your choice, you have a 1/3 chance of picking the door with the car behind it. That means there is a 2/3 chance that the car is behind one of the other two doors. When Monty reveals one of the other two doors to have a goat behind it, he is effectively giving you information about where the car is not. Therefore, the 2/3 chance of the car being behind one of the other two doors now becomes concentrated behind the remaining unchosen door.

If you stick with your original choice, you still have a 1/3 chance of winning the car. But if you switch your choice, you now have a 2/3 chance of winning the car, because the car is now more likely to be behind the door you did not originally choose.

This problem is counterintuitive because it seems like switching your choice should not make any difference to your chances of winning. But the key to understanding the solution is to recognize that Monty’s decision to reveal a door with a goat behind it is not random – it is based on his knowledge of what is behind the doors. This changes the probability of where the car is, and therefore changes the optimal strategy for the contestant.