You are a contestant on a game show. The host, Monty Hall, presents you with three doors. Behind one of the doors is a car (the prize you want), and behind the other two doors are goats. You choose one door, and Monty, who knows what is behind each door, opens one of the other two doors, revealing a goat. Monty then gives you a choice: you can either stick with your original choice or switch to the other remaining unopened door.

Objective:
Determine whether you should stick with your original choice or switch doors to maximize your chances of winning the car.

Solution

The Monty Hall Problem is a classic probability puzzle that often defies intuitive reasoning. The key to solving it lies in understanding the probabilities associated with each choice.

Step-by-Step Solution for the Monty Hall Problem:

Step 1: Initial Choice Probability

When you first choose a door, there is a 1/3 chance that you picked the door with the car behind it and a 2/3 chance that you picked a door with a goat.

Step 2: Monty Reveals a Goat

Monty then opens one of the two remaining doors, revealing a goat. This action provides additional information that alters the probabilities:

• If your initial choice was correct (1/3 chance), switching would cause you to lose.
• If your initial choice was incorrect (2/3 chance), switching would cause you to win.

Step 3: Analyze the Outcomes

Let’s break down the possible scenarios:

• Scenario 1: You initially picked the car (1/3 chance).
  • If you stick with your choice, you win the car.
  • If you switch, you lose (because you switch to a door with a goat).
• Scenario 2: You initially picked a goat (2/3 chance).
  • If you stick with your choice, you lose (because you stick with the goat).
  • If you switch, you win (because you switch to the door with the car).

Since Monty always reveals a goat behind one of the unchosen doors, switching doors effectively gives you a 2/3 chance of winning the car, while sticking with your original choice gives you only a 1/3 chance.

Step 4: Conclusion

Based on the probabilities, it is always better to switch doors after Monty reveals a goat. This strategy gives you a higher chance of winning the car.

Final Answer

To maximize your chances of winning the car, you should always switch doors after Monty reveals a goat. The probability of winning by switching is 2/3, compared to a 1/3 chance if you stick with your original choice.