A Brief Explanation of The Monty Hall Problem: To Switch Or Not To Switch

A Brief Explanation of The Monty Hall Problem: To Switch Or Not To Switch

Monty Hall Problem is a pretty popular as well as controversial problem sum. It’s a problem that’s purely based on the concept of probability.

This problem is pretty interesting from the point of view of mathematics. It’s not that difficult to comprehend as well. We would like to make something clear before we delve deeper into the subject.

You may or may not agree to the concepts that we are providing here after you have gone through the entire matter. That’s completely your prerogative and we respect that. Just keep in mind that we are only highlighting the concepts that are explained by the problem. We are not getting into the f act whether they are correct or not. We are leaving that decision to you. After all it’s just a game of chance. Hence, let’s begin this game with a statement that says, “May the odds always be in your favor”.

Introduction

This problem is said to be loosely based on an American TV game show known as “Let’s Make a Deal”. The problem is also named after the original host of the same show who was known as Monty Hall. The problem is based on a game chance and is made with the sole purpose of making the odds in the favor of the contestant.

Problem

Consider the fact that you are on a game-show. The game host has given a you a choice of 3 doors. You are informed that behind one of them there’s 5 million dollars and behind the other two there’s nothing. Pick the right door and you go home with 5 million dollars. What are your thoughts? You are thinking that it’s purely based on luck, isn’t it? You might win it or you might not. You are probably praying to your God to grant you that luck through which you’ll be richer by 5 million dollars within a matter of second. Now, that’s the emotional aspect of thinking. Logically, you should be thinking about the ways to make the odds in your favor. You can actually do that if the situation gets favorable from the point of view of the Monty Hall problem. We’ll come to the situation in the next paragraph.

Problem

The Situation

The game show host has asked you to pick any door of your choice. You have actually picked one say, for example the 1st door. Remember, there’s much at stake here. 5 million dollars, either you win it or you win nothing. Everything to lose! The host says such cheesy lines to initiate a minor doubt in your decision. You are tensed on one hand, excited on the other. You are waiting with bated breath. The host opens the 3rd door and shows you that nothing’s behind that door. You breathe a sigh of relief. After all, you have not picked the 3rd door; your choice is the 1st one. So you think that there’s still a chance for you to go home with that humongous sum of money. Next, the show host craftily asks you one question before he goes on to make the final discovery. The host wants to know whether you would like to change your choice or not. S/He is giving you another chance to think it over. Let’s stop at this point because this is where the discussion regarding the “situation” ends. This is where the “Monty Hall Problem” kicks in which is going to direct you to the path of logic. It’s time to traverse that path of logic.

Solution

At first we have to calculate the initial probability of your winnings.

Initial Calculation

Probability of the grand prize behind the 1st door= 1/3.

Probability that your grand prize is NOT behind the 1st door= 2/3= Probability of the grand prize behind the 2nd OR 3rd door.

Let’s move on to the final calculation now. The fact that you are aware of the situation that the grand prize of 5 million dollars is not behind the 3rd door is taken into consideration.

Final Calculation

Probability of the grand prize behind the 1st door= 1/3.

Probability that your grand prize is NOT behind the 1st door= 2/3= Probability of the grand prize behind the 2nd door.

Perhaps you are searching for that change now. If you can actually find that, you’ll get our point. But if you don’t, you are thinking that this discussion is meaningless. Don’t worry! We’ll help you out. Look closely at the 2nd calculation i.e. the final calculation part. The probability regarding the 1st door remains the same. What about the 2nd one? The probability regarding the 2nd door has INCREASED. The reason behind this is simply the elimination of the 3rd door since you know the fact that the 3rd conceals door no prize. Mathematically we can say this like-

2nd door probability + 3rd door probability= 2/3.

[Please note (if you come across terms such as “or”, “and” in mathematics):

or= +,

and = x.]

2nd door probability+ 0 =2/3

2nd door probability= 2/3.

Result and Conclusion

We’ll be showing you the result after the final calculation in a sequence for easy recognition:

1st door probability= 1/3.

2nd door probability= 2/3.

3rd door probability= 0.

Therefore, you see that you probably have more chance to grab your prize if you switch your choice to the 2nd door purely from the point of view of mathematics.

Please remember that this is still not guaranteed. Your odds have increased but there’s still a chance that the prize behind the 1st door. Since we are talking mathematics here, we would advise you to switch your choice to the 2nd door. We can say something for sure that if you play 10 such games you should get most of them right using this tactic.

Dependence

The Monty Hall Problem is dependent on a number of conditions.

  • The host of the show must open a door that’s not picked by the contestant for initial revelation.

  • The door that’s opened initially must reveal the thing that’s undesirable from the contestant’s point of view.

  • The game host must provide the contestant with a 2nd chance to alter his/her choice or not.

Monty Hall problem is an interesting approach to probability problems but the thing that makes it a stand out is the implementation of the same in real life problems. The lure of prize seems pretty enticing. If you successfully pull off perfect predictions through the application of this tactic, you would be the sensation among your friends. That would earn you endless bragging rights. You are actually not gambling. You are actually applying mathematics.

Before we conclude this article, we would like to say that such type of problems are pretty beneficial for students from the educational point of view. These problems are not abstract and are pretty eye-catching. Certain after-school activities and after-school online math lessons can also prove to be beneficial for children. With that we’ll sign off for now. Hope you had a good read.

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