Doors Resource

Door Puzzles

THE LYING GUARDIAN TWO DOOR PROBLEM

The Lying Guardian Two Door Problem, sometimes known as the Door to Heaven problem, is as follows. You are presented with two doors, each with a guardian standing beside it. One door leads to heaven, the other to hell. You want to go to heaven, and are allowed to ask only one question of one of the guardians before you open a door. The catch is that one guardian always tells the truth and the other guardian always lies.

Question: What question should you ask?

Answer: Go up to either guardian, and say 'Which door would the other guardian say leads to heaven?'. Then take the other door.

Explanation: If the guardian you ask is the lying guardian, he will lie about what the other (truth-telling) guardian will say, and therefore direct you to the wrong door. If the guardian you ask is the truth-telling guardian, he will correctly say that the other guardian would mis-direct you.

THE MONTY HALL THREE DOOR PROBLEM

A door problem which is altogether more puzzling and surprising is the Monty Hall Three Door Problem. It is a probability problem loosely based on the TV show Let's Make a Deal, and named after that show's presenter, Monty Hall. Originally posed in a letter by Steve Selvin to the American Statistician in 1975, the Monty Hall Three Door Problem was widely publicised by Marilyn vos Savant in her 'Ask Marilyn' column in Parade magazine in 1990. Marilyn vos Savant was listed as having the world's highest IQ (228) from 1986 until 1989 at which date the category was discontinued by Guinness on the grounds that IQ tests are not reliable enough to designate a single world record holder.

In the Monty Hall Three Door Problem the contestant stands in front of three doors set up on the stage. The host, who knows what lies behind each door, explains that behind one door there is a car, and behind each of the other two doors there is a goat. The contestant, who wants to win the car, is allowed to open one door and will win what lies behind it. The contestant is asked to decide which door to open, and point to it. Let us suppose the contestant points to Door 1 (the left hand door). The host then, to help the contestant, opens one of the other doors, say Door 3 (the right hand door), revealing a goat. The host then asks the contestant whether he would like to change his mind, and choose Door 2 instead of Door 1.

Question: Would it increase the contestant's chances of winning the car if he switches his choice from Door 1 to Door 2?

Answer: Yes. It would double his chances from 1/3 to 2/3.

Explanation: This answer, which has been verified by various mathematical proofs and by practical experiment, amazes the majority of people. Most think that because there are only two closed doors left, each must have a 50/50 chance of concealing the car. But this is not the case. One explanation goes as follows. Consider the three doors, at the outset of the game, as comprising two groups: Door 1, and a second group comprising Doors 2 and 3. At that stage the first group (Door 1) has a 1/3 chance of concealing the car, and the second group (Doors 2 and 3) has a 2/3 chance of concealing the car. After the host opens Door 3 (revealing a goat) the second group (Doors 2 and 3) still has a 2/3 chance of concealing the car. Because we know the car is not behind Door 3, the whole of this 2/3 chance now falls to Door 2. For a much fuller explanation and discussion fo the Monty Hall Three Door Problem, please see the article on this page of Wikipedia.

A man wanted to enter an exclusive club but did not know the password that was required. He waited by the door and listened. A club member knocked on the door and the doorman said, "twelve." The member replied, "six " and was let in. A second member came to the door and the doorman said, "six." The member replied, "three" and was let in. The man thought he had heard enough and walked up to the door. The doorman said ,"ten" and the man replied, "five." But he was not let in.

Question: What should he have said?

Answer: Three. The doorman only lets in those who tell him the number of letters in the word he said.

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