[Photo by MontyPython.]
You can get a good argument going in almost any group of people with the infamous Monty Hall problem:
Imagine you are on a TV game show, and the host lets you choose between three closed doors. One of the doors hides a fancy sports car, and if you pick that door, you win the car.
You pick door #1.
The host opens door #3 to reveal a goat. Then he gives you a chance to switch your door for the unopened door #2.
Should you switch?
What if you say you’re going to switch, and then the host offers to give you $5,000 instead of whatever is behind door #2?
Try the game for yourself at the Stay or Switch website.
Door #1
Most people will say that the game show host, by opening that door, has reset the problem. At the beginning, you had a 1/3 chance of picking the right door, but now you know there was a goat behind door #3, and that new information changes your estimate of probability. The car has to be behind one of the two remaining doors, and they are both equally likely to have the grand prize. It doesn’t matter if you switch, because you have a 50% chance of the car either way.
The best explanation that I’ve seen refuting this argument is the 50 doors (or 50 bowls) demonstration.
Door #2
People who understand probability will argue that these choices are not independent events. That is, which door the host opens depends on which door you pick first, and your choice at the end depends on which door the host opens. So the only way to figure the true chance of winning is to write out all the possibilities and count the times it is better to switch.
Since there was a 2/3 probability that your first choice was wrong but only a 1/3 chance that it was right, you will win twice as often by switching as by keeping your first door.
Door #3
Still other people claim that the host will never give you the chance to switch to the winning door. He only offers a switch if he knows you picked the right door to start with. So you should never switch.
The true probability of winning the car by switching or not depends on the host: Does he know where the car is? Will he always offer a switch, or only sometimes?
To see all the possibilities, check out this Wikipedia chart.
What About the Money?
The usual Monty Hall problem doesn’t mention money, but when New York Times columnist John Tierney interviewed Monty Hall himself, they ended up playing a game…
Tierney picked Door 1. Mr. Hall opened Door 3, revealing a goat. Tierney was about to switch to Door 2 when Mr. Hall pulled out a roll of bills.
“You’re sure you want Door No. 2?” he asked. “Before I show you what’s behind that door, I will give you $3,000 in cash not to switch to it.”
“I’ll switch to it.”
“Three thousand dollars,” Mr. Hall repeated, shifting into his famous cadence. “Cash. Cash money. It could be a car, but it could be a goat. Four thousand.”
“I’ll try the door.”
“Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousand dollars.”
“Let’s open the door.”
How do you think it turned out? Go read the story for yourself.
No other statistical puzzle comes so close to fooling all the people all the time.
— Massimo Piattelli-Palmarini
on the Monty Hall problem, quoted in The Power of Logical Thinking
In no other branch of mathematics is it so easy for experts to blunder as in probability theory.
— Martin Gardner
quoted in Behind Monty Hall’s Doors: Puzzle, Debate and Answer?
If the host is required to open a door all the time and offer you a switch, then you should take the switch. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.
My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home.
— Monty Hall
quoted in Behind Monty Hall’s Doors: Puzzle, Debate and Answer?
To Be Continued…
Read all the posts from the July/August 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.
Denise Gaskins' Let's Play Math 
Hi, I’m glad you liked my picture and thanks that you linked to it. 🙂 I already heard about this statistical problem (it’s called “Ziegenproblem” [“goat problem”]) here in Germany and it’s like you said: It’s really easy to have an argument about it. I had with one my girlfriend, who always says “It doesn’t matter what statistic says about anything, it has nothing to do with reality! If you pick the goat it doesn’t help you that the probability was 2/3!” etc. 🙂
Author Andrew Vazsonyi in his book “Which Door Has the Cadillac” tells of the reaction of the famous Paul Erdos to the Monty Hall conundrum. I retell the story towards the end of the page http://www.cut-the-knot.org/ctk/ErrDisc.shtml. The topic of the page is another probabilistic puzzle: do boys have more sisters than the girls?
I try to explain it like this (because I really had trouble accepting it)
It’s not only you who has a problem, it’s the host as well.
If you pick the car door first (1/3), you’ll make the host choose a door te open.
If you pick a goat door first (2/3) the host won’t have to choose.
Therefor, two-thirds of the time the host doesn’t have to make a choice. And when he doesn’t; switching = succes.