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.

THE PASSWORD

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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