Wednesday, December 21, 2011

The Monty Hall Problem. . . . .

Ok, this blog requires two clarifications by way of introduction. One is that I grew up watching too much tv and one of the shows that I recall seeing many times was "Let's Make a Deal," with beloved Canadian MC Monty Hall, (who is still alive, by the way, and is 90 years old). That game show has recently be revived hosted my the charming and funny Wayne Brady. The other point of clarification is that, as my readers know, I have very little faith in any organized institutions in general, and of the so-called sciences in particular.

These two introductory points bring us to the so-called Monty Hall Problem (or Monty Hall Paradox as some people call it.) The Monty Hall Problem was first explained in popular terms by Marilyn von Savant in Parade magazine in 1990. She explains it thus


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 No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

It turns out, despite the apparently counter-intuitive nature such a problem, it is indeed to your advantage to which door if you are given the choice. There are a number of ways to explain this claim which at first glance seems so patently false. But I am no mathematician and so I choose an easier way to explain it that makes sense to those of us who cannot do it with numbers and equations. The explanation is simple - in the initial choice you have a one in three change of winning the car. After one door has been opened, if you choose not to switch doors, you still have a one in three chance of winning the car. However, if you switch doors you have a fifty-fifty chance of winning the car. The initial problem is so counter-intuitive that I did the experiment myself. I tried the experiment two hundred times, one hundred of which I didn't change my door and one hundred of which I did. When I switched doors I won 14 more times than when I didn't change doors.

Besides the fascinating issue of the problem itself, what I find interesting about the Monty Hall Problem is that, as pointed out by von Savant in her book The Power of Logical Thinking, "even Nobel physicists systematically give the wrong answer, and they insist on it, and they are ready to berate in print those who propose the right answer."

I will let people draw their own conclusions concerning the implications of these interesting issues. One thing that I think von Savant's problem has demonstrated is that there is more to the work of Thomas Kuhn than many people might think.

9 comments:

Anonymous said...

If there is a paradox I don't believe your description captures it.

You write "After one door has been opened, if you choose not to switch doors, you still have a one in three chance of winning the car."

In fact, once one door has been ruled out your odds have improved to one in two. Getting more information changes things.

kirbycairo said...

Anonymous, I don't know if my description captures the idea, it did for me and allowed me a simple way of seeing the issue more clearly. I invite you to see von Savant's full description of the Monty Hall Problem. And of course, technically it is not a paradox, it is only a kind of counter-intuitive demonstration.

But I think you are wrong about the odds changing. I think that is what von Savant is demonstrating - the odds are only changing conceptually but not de facto, while changing the door choice, changes the odds de facto and that is why you win more often by changing your choice and not when you don't change your choice.

kirbycairo said...

The thing is, if you don't change your door choice, you continue to win approximately 33 percent of the time. But if you do change your door choice those odds slightly improve. Argue with the numbers not with me.

The Mound of Sound said...

I've heard it explained this way. The odds are fixed when you first choose your door = 1:3. You have a one chance in three of winning.

When the empty door is revealed it doesn't change your odds one bit. they're still 1 in 3. The fact that one of the other doors has been opened only increases the chances that the other unopened, unchosen door is the winner.

The door that is supposedly gratuitously opened by Monty Hall is chosen based on prior knowledge that it's a loser. He's not risking opening the grand prize door nor is he willing to open the contestant's chosen door. He's only opening a known loser.

The contestant's choice remains a 1 in 3 longshot. The odds on Monty Hall's side alone have changed. Your odds improve significantly if you switch.

Anonymous said...

After learning about this 'problem' in Phil 110, I always tried to think about it from the odds of picking a goat.

Whatever door you initially pick has a 2/3 probability of being a goat.

When the host reveals another goat, the odds that you initially picked the goat remain at 2/3 (it doesn't fall to 1/2, which is an assumption people make).

And since you don't want to win the goat, the sensible choice is to switch.

Marley52 said...

The interesting thing to me about the Monty Hall problem is that if Monty knows where the prize is and intentionally reveals a goat then the switch option gives a 2/3 chance of winning, whereas if Monty, not knowing where the prize is, accidently reveals a goat then the switch option gives no advantage(i.e it's 50/50 if you switch or stick). From the contestant's viewpoint though the situation is identical: he's picked a door, Monty has revealed a goat behind one of the other 2 doors, and yet his odds of winning the car if he switches are either 2/3 or 1/2 dependant NOT on what Monty has done but on what Monty knows. Weird or what?

kirbycairo said...

Marley - Thanks for the comment but I am not sure I understand. Can you try that again?

doconnor said...

The conclusion you should reach is that you should trust the scientific method, which you correctly followed by doing the expirement yourself, even if you can't always trust sciencists.

kirbycairo said...

Ha ha, touché doconnor.