Let’s Make a Deal is a long-running TV game show that debuted in 1963 and is still on
the air today.
The early versions of the show featured contestants who made bargains with the show’s co-creator
and longtime host, Monty Hall.
If the contestants were lucky and skilled at making deals, they could win cash and other
prizes, but if they weren’t, they received undesirable “zonk” prizes such as goats
instead.
Who would have known that decades after the show premiered, a probability puzzle loosely
based on it would stir up so much controversy in academic and intellectual circles.
We will examine this intriguing puzzle today, in this episode of The Infographics Show,
“The Monty Hall Problem.”
The probability puzzle, also known as the Monty Hall Problem, is similar to the “three
curtain” game of Let’s Make a Deal, except that it features three doors in place of the
curtains.
One door has a car hidden behind it, while a goat is hidden behind each of the other
two doors.
You pick the door that you think will win you the car.
After you select a door, Monty Hall opens one of the other two doors and reveals a goat
behind it.
At this point, he gives you a choice: you can stay with the door that you selected,
or you can select the other door instead.
Should you stay with your original choice, or should you switch?
On the surface, this seems like a simple question, but the three main solutions to this problem
have been a source of debate for many years.
1.
The “Common Sense” Solution
If you know that a goat is behind one of the three doors, then there are only two doors
left to choose from.
A goat is behind one of the doors and the car is behind the second remaining door, so
your odds of picking the door with the car is 50-50.
This is the solution that is the easiest for most people to understand, but now many experts
think it is wrong.
2.
Marilyn vos Savant’s Solution
Marilyn vos Savant is known as the woman with the world’s highest recorded IQ.
She has worked for Parade magazine since 1986.
In her column called “Ask Marilyn,” she responds to a wide variety of intellectually
challenging questions from readers.
On her website, she devotes an entire page to the “Game Show Problem” that a reader
sent to her in 1990:
Suppose you’re on a game show, and you’re given the choice of three doors.
Behind one door is a car, behind the others, goats.
You pick a door, say #1, and the host, who knows what’s behind the doors, opens another
door, say #3, which has a goat.
He says to you, "Do you want to pick door #2?"
Is it to your advantage to switch your choice of doors?
This was her solution to the reader’s version of the Monty Hall Problem:
Yes; you should switch.
The first door has a 1/3 chance of winning, but the second door has a 2/3 chance.
Here’s a good way to visualize what happened.
Suppose there are a million doors, and you pick door #1.
Then the host, who knows what’s behind the doors and will always avoid the one with the
prize, opens them all except door #777,777.
You’d switch to that door pretty fast, wouldn’t you?
Some additional explanation will further clarify her solution.
Because there are three doors, each one of them has a 1/3 probability of having the car
behind it.
The opening of a door that reveals a goat does not cause the probability to be equally
divided between the two remaining doors.
As vos Savant puts it, “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.”
Instead, the probability of the opened “goat” door having the car behind it is transferred
only to the door that you do not select, which gives the unselected door a 2/3 probability
that it has the car behind it.
This solution revealed the Monty Hall Paradox: the obvious “common sense” solution to
the Monty Hall Problem is not the correct one.
This paradox upset a lot of people because, as one reporter put it, “it’s an example
of math completely contradicting my gut instinct.”
Marilyn vos Savant received widespread criticism from shocked Ph.D.’s across the country.
In their letters, they made disparaging comments to her like “You blew it!”and “Your
answer is clearly at odds with the truth.”
But vos Savant did not back down.
She defended her answer in additional columns.
She presented her critics with the following table showing what she called the “benefits
of switching.”
The first three games show that you win 2/3 of the time if you switch, while the second
three games show that you win 1/3 of the time if you stay.
However, these results are based on the assumption that the “host always opens a loser.”