# Monty Hall Problem

## Monty Hall problem :

The Monty Hall problem is a probability puzzle named after the host of the game show “Let’s Make a Deal.” The problem involves a contestant who is given the choice of three doors: behind one door is a prize, while behind the other two are goats. After the contestant picks a door, Monty Hall opens one of the remaining doors to reveal a goat. The contestant is then given the option to stick with their original choice, or switch to the other remaining door. The question is: should the contestant switch doors to increase their chances of winning the prize?
To illustrate the problem, let’s imagine that our contestant, Alice, is given the choice of three doors. Behind one door is a brand new car, while behind the other two are goats. Alice picks door number 1. Monty Hall then opens door number 3, revealing a goat. Alice is now given the option to stick with door number 1, or switch to door number 2. Should she switch?
The answer is yes, Alice should switch. Here’s why:
When Alice first picked door number 1, she had a 1 in 3 chance of choosing the car (since there is only one car and three doors). After Monty Hall opened door number 3, the chance of the car being behind door number 1 remained the same (since the car’s location has not changed), but the chance of the car being behind door number 2 increased. This is because when Monty Hall opened door number 3, he eliminated one of the two remaining options for the car (door number 3), leaving only door number 2 as a possibility. Therefore, by switching to door number 2, Alice’s chances of winning the car increase from 1 in 3 to 1 in 2.
Let’s consider another example to further illustrate the Monty Hall problem. Imagine that our contestant, Bob, is given the choice of three doors. Behind one door is a trip to Hawaii, while behind the other two are goats. Bob picks door number 2. Monty Hall then opens door number 1, revealing a goat. Bob is now given the option to stick with door number 2, or switch to door number 3. Should he switch?
The answer is yes, Bob should switch. Here’s why:
When Bob first picked door number 2, he had a 1 in 3 chance of choosing the trip to Hawaii (since there is only one trip and three doors). After Monty Hall opened door number 1, the chance of the trip being behind door number 2 remained the same (since the trip’s location has not changed), but the chance of the trip being behind door number 3 increased. This is because when Monty Hall opened door number 1, he eliminated one of the two remaining options for the trip (door number 1), leaving only door number 3 as a possibility. Therefore, by switching to door number 3, Bob’s chances of winning the trip increase from 1 in 3 to 1 in 2.
In conclusion, the Monty Hall problem shows that switching doors can increase a contestant’s chances of winning a prize. This is because, after one of the losing doors is revealed, the remaining doors have an equal chance of containing the prize. By switching to the remaining unopened door, the contestant is essentially “trading” a losing door for a potentially winning one.