Monty's Dilemma
Let's Make A Deal
There is a TV game show in which the contestant is
asked to choose one of three doors. Behind one of the doors is a whopping big
prize, and behind the other two doors there is junk.
After the contestant chooses one of the doors, the
game show host shows him what is behind one of the other two doors, always
showing a "junker." Then the contestant is presented with the
following dilemma:
Would you like to keep the door you chose, or
switch to the other ( still veiled) door?
Monty's Strategies
Let us pose a mathematical (probabilistic) problem from this dilemma
Which of these three strategies is most likely to lead the contestant
to the winning door?
1) Just stay put, and keep the original door you chose, after the
junk door is disclosed.
2) Choose again by randomly selecting
a door from the remaining two closed doors.
3) Choose again by switching from the
door you chose to the other closed door.
How could we simulate the playing of
Monty's game many times over?
The answer is (3) - You should always
switch!
The first door has a 1/3 chance and the second door has a 2/3 chance.
This problem first appeared in the Ask Marilyn column in Parade magazine.
It has also been featured in the Mathematics Teacher and other math journals. Many
professional mathematicians wrote in to say that she was wrong. Here
is Marilyn vos Savant's reply to a reader who said that the odds would go up to 1/2:
Dear Readers,
Good Heavens! With so much learned opposition, I'll bet this one is going to keep
math classes all over the country busy on Monday.
My original answer is correct. But first, let me explain why your answer is wrong.
The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door.
To illustrate this, let's say we play a shell game. You look away, and I put a pea under one of three shells.
Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed?
Then I simply lift up a shell from the remaining other two. As I can (and will) do this regardless of what
you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger.
The benefits of switching are readily proven by playing through the six games that exhaust
all the possibilities. For the first three games, you choose #1 and "switch" each time; for the second three
games, you choose #1 and "stay" each time; and the host always opens a loser. Here are the results:
|
|
Door 1
|
Door 2
|
Door 3
|
|
| Game 1
|
AUTO
|
GOAT
|
GOAT
|
Switch and you lose
|
| Game 2
|
GOAT
|
AUTO
|
GOAT
|
Switch and you win
|
| Game 3
|
GOAT
|
GOAT
|
AUTO
|
Switch and you win
|
|
|
Door 1
|
Door 2
|
Door 3
|
|
| Game 4
|
AUTO
|
GOAT
|
GOAT
|
Stay and you win
|
| Game 5
|
GOAT
|
AUTO
|
GOAT
|
Stay and you lose
|
| Game 6
|
GOAT
|
GOAT
|
AUTO
|
Stay and you lose
|