Thursday, June 12, 2025

The Monty Hall Problem

Monty Hall and Lets make a deal
The Monty Hall Problem is a probability puzzle based on a game show scenario. Here's a simplified version:

  1. You are a contestant on a game show. There are three doors. Behind one door is a car (the prize you want), and behind the other two doors are goats.
  2. You choose one door, but the host (Monty Hall) opens another door, revealing a goat.
  3. Now, you are given the option to stick with your original choice or switch to the remaining unopened door.

The surprising result is that statistically, you are more likely to win the car if you switch doors. This goes against the intuition of many people, as it seems like the probability should be 50-50 after one door is opened.

The optimal strategy is to always switch doors, as it gives you a 2/3 chance of winning the car, compared to a 1/3 chance if you stick with your initial choice. The counterintuitive nature of this problem has led to many discussions and debates among mathematicians and the general public.

This probability puzzle is named after the host of the television game show "Let's Make a Deal," Monty Hall. The problem gained fame when it was presented by Marilyn vos Savant in her column in Parade magazine in 1990. While the solution is straightforward, the counterintuitive nature of the problem has led to much debate and confusion.

Here's a more detailed explanation of the Monty Hall Problem:

  1. Initial Scenario:

    • You are a contestant on a game show. There are three doors. Behind one of the doors is a car (the prize you want), and behind the other two doors are goats.
  2. Your Choice:

    • You choose one of the three doors, but the door is not opened immediately.
  3. Host's Action:

    • The host, Monty Hall, who knows what is behind each door, opens one of the other two doors, revealing a goat. Importantly, Monty always opens a door with a goat behind it, and he never opens the door you initially chose.
  4. Switching or Sticking:

    • Now, you are given a choice: stick with your original choice or switch to the remaining unopened door.
  5. The Counterintuitive Result:

    • The counterintuitive solution is that you are better off switching doors. If you switch, your probability of winning the car is 2/3, while if you stick with your initial choice, your probability is only 1/3.

Explanation:

The key to understanding the solution lies in the fact that Monty Hall's action of revealing a goat provides additional information. When you initially choose a door, there is a 1/3 chance that the car is behind your chosen door and a 2/3 chance that the car is behind one of the other doors.

When Monty reveals a goat behind one of the other doors, the initial probability distribution doesn't change. If your first choice was a goat (1/3 chance), switching doors will win you the car. If your first choice was the car (1/3 chance), switching doors will result in a loss. However, if your first choice was a goat (2/3 chance), switching doors will always win you the car because Monty will reveal the other goat, leaving the car behind the remaining unopened door.

In summary, by switching doors, you capitalize on the 2/3 chance that your initial choice was a goat, leading to a higher probability of winning the car. This solution often surprises people because the intuition might suggest a 50-50 chance after one door is opened, but the probabilities favour switching doors.

Source: Some or all of the content was generated using an AI language model

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