The Monty Hall problem: beyond closed doors




Monty Hall(1921~)
The host of 'Let's Make a Deal'

The Monty Hall problem: beyond closed doors

Monty Hall problem is a probability puzzl. Puzzle Monty Hall problem was orginated the american television game show ‘Let’s Make a Deal’. It became famous in 1990, a reader’s letter quoted in Marilyn vos Savant’s “Ask Marilyn” column in Parade magazine.

 

 

   

Problem is as follows.

You’re given the choice of three doors. Behind one door is a car, behind the others, goats. First pick a door and them the host who knows where the car is opens another door which the goat behind. And he sats to you “You can change your choice. Do you want to switch?”. Here, is it to your advantage to switch your choice?
 

 Many people easy to misunderstand the solution is probability is same whether open it or not. Because the host open door and we pick one of the remainders. So we can think probability is 1/2 when we pick the door which the car behind. But that is not the anwser. You have to change choice. The probability of earning the car when switch choice is 2/3.

 Here is the solution.
 The probability of you choose the door the goat behind is 2/3 and then host show you another door the goat behind. So if you switch your choice you can get a car. But if you choose the door the car behind that probability is 1/3 and switch, you can’t get a car. And to conclusion you need to change your choice.
 

Marilyn vos Savant's column
'Ask Marilyn'

 

 

 When Vos Savant dealed with this problem, many readers of her column refused to believe switching is beneficial despite her explanation. Interestingly, approximately 10,000 readers wrote to magazine, most of them claiming she was wrong and 1,000 people among them were PhDs.

 This is conditional probability puzzle. Many the other puzzle about conditional probability is interesting that we can easily confused. So if you understand some conditional probability puzzle, you can have another  viewpoint when you meet the problem like this.