The Monty Hall Problem

The Monty Hall Problem is a brain teaser in the form of a PROBABILITY puzzle.

This problem was originally posted in a letter by STEVE SELVIN to the American Statistician in 1975.

The problem got its name from a host of the American Television Game Show named "LET'S MAKE A DEAL".

THE QUESTION:

" Suppose you are on a game show, and you are given a chance to choose one door amongst the three doors : Behind one door there is a car and behind other two they have kept goats. you have to pick a door to win the price behind it. Lets say you chose door no.1, now the host opens one of the doors other the one you have chosen. As, he knows which one has goat, he opens a door having one of the goats in it, say door no.3. Now you have a choice of switching your choice or staying with your choice. So, what will you do? "

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Take your time to think over the problem........
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THE ANSWER: 

We definitely switch the doors to gain more probability of winning a car.

• Before choosing a door let's see what is the probability of choosing a car: It would be one third. Why? simple math. As there are three doors and you can choose only one, therefore, the probability becomes one over three which is one third.But you choosing a goat is two third or 66% as there are two goats. Hence, you have higher possibility to choose a goat over the car.
• So, now let's assume you choose not to switch and you go on with the door you chose at your first opportunity to choose. This keeps the probability of you winning a car over goat same which only 33%. It is of no help to you. Obviously,  you may win but the possibility of that happening is way too low.
• Now, let us take the case where you choose to switch your doors after the host shows you a goat. In this case, you increase your chances of winning the car. How??  Here's how:-
  
  -In this case, you have two cases wherein you have first chosen the car and switch the door, or,
  
  -You have chosen a goat and then switched the door(THE MAIN PART OF THE PROBLEM).

1. Assuming, its car behind the door you initially chose. But as you had decided to switch you lose. but again remember this comes in that lower 33% of the chance. So it does become a rare case.
2. In, this case you have chosen a door which has a goat behind it. As, you are going to switch the doors and the host has only one door to open(as you have chosen a door that has a goat in it, he is left with only one door to open).So, now if you switch you win a car. And you get a 66% chance of winning it.
   As a probability of you choosing a goat is 66% in original case but by switching you decrease it to 33%. You have a 33% chance of winning a goat by switching and choosing the car first time whereas, you have 66% chance of choosing to win a car by choosing a goat first and then switching.
   
   THIS IMAGE BELOW SHOULD HELP YOU UNDERSTAND THE WHOLE SCENARIO.    

~Jay Mehta

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Jay Mehta.

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