# Monty Hall Problem with 4 doors

• MHB
• MichaelLiu
In summary, the best strategy in this scenario is to switch after the host reveals a door with a goat.
MichaelLiu
You are given 4 doors. Behind 1 door is a car, and behind the others is a goat.

You will choose your first door, and the host will reveal another door which has a goat behind it.
Then, you can choose to either stay with your choice or switch. Then, the host will reveal a different door, containing a goat.
Again, you will choose to stay or switch. You may switch back to your original door.

What is the best strategy, staying twice, staying then switching, switching then staying, or switching twice?

Thanks a lot!

I would approach this scenario by first understanding the probability of each door containing the car. Initially, there is a 1/4 chance of choosing the correct door and a 3/4 chance of choosing a door with a goat behind it.

After the host reveals a door with a goat, there are now two doors left - the one you initially chose and the one that was not revealed. The probability of the car being behind either of these doors is now 1/2. This means that whether you choose to stay or switch at this point, the probability of winning the car is the same.

However, if you choose to switch after the host reveals a door with a goat, you are essentially combining the probabilities of both doors that were not initially chosen. This means that the probability of winning the car by switching is now 2/3, compared to the 1/4 probability of staying with your original choice.

Therefore, the best strategy in this scenario would be to switch after the host reveals a door with a goat. This strategy increases your chances of winning the car from 1/4 to 2/3. Switching twice or staying then switching would not change the overall probability of winning the car, as the probabilities are already equal at 1/2 after the first switch.

In conclusion, the best strategy in this scenario would be to switch after the host reveals a door with a goat, increasing your chances of winning the car to 2/3.

## 1. What is the Monty Hall Problem with 4 doors?

The Monty Hall Problem with 4 doors is a probability puzzle named after the host of the game show "Let's Make a Deal". It involves a hypothetical game where a contestant is presented with 4 doors, one of which contains a prize. After the contestant chooses a door, the host reveals one of the remaining doors that does not contain the prize. The contestant is then given the option to switch their choice to the remaining unopened door. The question is whether it is beneficial for the contestant to switch their choice or stick with their original selection.

## 2. What is the origin of the Monty Hall Problem?

The Monty Hall Problem was first introduced by Steve Selvin in a letter to the American Statistician in 1975. It gained widespread attention when it was featured in a column by Marilyn vos Savant in Parade magazine in 1990. The problem has since been the subject of numerous mathematical and statistical analyses, as well as debates and controversies.

## 3. What is the correct answer to the Monty Hall Problem?

The correct answer to the Monty Hall Problem is that it is more beneficial for the contestant to switch their choice to the remaining unopened door. This may seem counterintuitive, as it goes against our initial instinct to stick with our original choice. However, by switching, the contestant increases their chances of winning from 25% to 75%.

## 4. What is the explanation for the correct answer to the Monty Hall Problem?

The explanation for the correct answer lies in the fact that the host's reveal of one of the remaining doors provides new information. By eliminating one of the losing doors, the host is essentially telling the contestant that their initial choice was not the winning door. This means that the remaining unopened door has a higher probability of being the winning door, making it more beneficial for the contestant to switch.

## 5. How does the Monty Hall Problem relate to real-life situations?

The Monty Hall Problem is a prime example of how our intuition can sometimes lead us astray when it comes to probability and statistics. In real-life situations, it is important to analyze the available information and make an informed decision, rather than relying on our initial instincts. The Monty Hall Problem also highlights the importance of understanding and properly interpreting probability, as it can have a significant impact on our decision-making.

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