Be smarter than a theoretical physicist

Here is a puzzle that will give you lots of self-esteem by solving it: several Professors of Physics got it wrong when I tried it with them, and one well-known theoretical physicist actually had to run a Monte Carlo simulation in order to become convinced of the solution (hardly laughed more in my whole life - will explain in due time). But it is not hard, if you think outside the box.

You are at a TV show. The anchor shows you three closed doors, A,B, and C. He says there is a Ferrari behind one door, and nothing behind the other two. He asks you to choose one among doors A, B or C: if you get the right door, you will own a Ferrari!

You choose door A. But now the anchor, who knows where the Ferrari is, tells you he is going to make it easier for you: he will open a door with no Ferrari behind among the two you did not choose. So he opens door B, showing that indeed there is no Ferrari there.

He now asks you if you want to change your mind, and choose door C, or if you want to stick with your original plan, and keep door A.

What should you do ? Does it matter ? What are your chances of driving home in style ? And, for physicists: What happens to the Ferrari(x) wave function upon observation that F(B)=0 ? Does QM apply after all ?

 

Comments

[Wim]

It's a nice puzzle. Had it on highschool. And I also wasn't convinced at first.

[claire]

We had a big argument over this last year :) It's a great one!

[little miss demosthenes]

ah, the monty hall problem. a classic, no doubt. i like the bit about the simulation - must have been amusing indeed ;)

cheers!
-demie

[Tommaso Dorigo]

Well, I am surprised to see that this little puzzle is so well known! Ten years ago nobody knew about it, probably since then the word has spread...

Demie, what is "monty hall" ?

T.

[Jochen Weller]

This is the sort of puzzle I was amazed to discuss in high school, which was back in 1988 :)
But it is still intruiging.

[Marco]

The page is in italian, but there's a nice script which simulates the process...

[Marco]

I forgot the link, sorry:
http://utenti.quipo.it/base5/probabil/montyhall.htm

[Helge]

Smarter is one thing. The theoretical physicist will still know a lot of stuff, I have no clue on.

[David]

Your comment on how this relates to quantum physics caught my attention because I have specifically thought about it before, in the context of some of the ideas floating around in the quantum information community.

"And, for physicists: What happens to the Ferrari(x) wave function upon observation that F(B)=0 ? Does QM apply after all ?"

Although quantum physics doesn't actually apply here, some quantum information researchers would argue that the situation described here is a little like quantum measurement in the following sense: They argue that quantum physics is actually a theory about information we have about a physical system, not about the physical system as such. So what happens here is that you get extra information when the host of the show opens the door to the empty room. And that information changes what you know about the system and you can update your choice. Some quantum information people also argue that when you make a measurement, you might not actually change the system, as such, just the way you think about it. (This is a quick and dirty version which is not absolutely correct in the details but tries to give a flavour of the argument.)

I find the best way to understand how this puzzle actually works is to understand exactly why the host is giving you extra information by opening the door. (I won't give that away - it's fun to figure out why.) If you understand that, you know exactly what you should do in response.

[little miss demosthenes]

it can be concluded from the above empirical observations [aka: time-to-stop-reading alert] clear that the host is giving you extra information by opening the door for the very reason that if he [or she, i didn't want to upset any transvestites] didn't do so, there would be no such thing as this flavor of the problem we are discussing, and therefore, your post on the given topic would not exist.

all of this is was done so the very things unraveling before your eyes are happening. you're right -- it's fun to figure out why. or is it because it's so obvious?

what should i do in response?

well, respond to you.

-demie

ps: the above post was done on excessive amounts of mint chocolate chip ice cream. therefore, any drunkenness that seeps out is not the authoress' fault. *NOT*. do not blame her.

[David]

Demie: You make an interesting point about the ways information can be conveyed. The method you describe is not essential to solving the problem (i.e. you don't need to rely on the fact that a solution must exist to find the solution). There is indeed something more specific about the information communicated by the host.

There are some really interesting logic problems that use communication of information in bizarre ways. One example is the monks with marks on their foreheads problem (which I should try to find described somehere but here is a rough version):

A group of monks are trying to decide whether they have marks on their own foreheads only by looking at other monks who may or may not have marks on their foreheads. (no mirrors or cheating, etc.) Nobody is allowed to speak but a bell rings each minute and, at those times, the monks are allowed to either declare that they know whether they have a mark, or to keep silent. It is possible for the monks to work out whether they have marks on their foreheads. But where is the information flow here? It's much trickier. (A hint is that silence can communicate information. Also, the fact that the bells ring is essential.)

I also know of a few mathematical problems that are much easier to solve if you are allowed to assume a solution exists. For example, it is relatively easy to prove that a circle is the shape that surrounds the most area for a given perimeter, but only if you know that there is such a shape. If you aren't allowed to assume that there is a maximal shape, then the problem becomes much more difficult. There are a handful of other geometrical problems that are also much easier if you are allowed to assume a solution exists.

Isn't it amazing where a discussion about puzzles (mathematical or logical) can go? If you're into this kind of thing, you have to read the books by Raymond Smullyan. He is the master of fiendishly difficult logic problems but lets you work up to the hardest starting from easier versions.

[little miss demosthenes]

actually, that was meant to be a joke. i didn't want to blatantly state that aloud because i thought you physicists would get it, but apparently not. i even added the drunk thing to, you know, hint at that. still no.

nice story though.

-demie

[sandra]

Tommaso, we discussed this puzzle during statistical classes at the University. I don't remember the year, but I finished university in 1989. By the way, I remember I found the solution really amazing, at that time. Good idea to post it.
Ciao, Sandra

[David]

Demie: I know you meant it as a joke, but you actually touched on something interesting, even if unintentionally... that mint choc chip icecream inspires all sorts of hidden thoughts!

[claire]

Is there a prize for most commented-on blog entry? If so, I think this one wins. Nice one Mr T!

[Aaron]

LOL!!!!!! I heard this problem a long time ago, but the first time I really discussed it was a few years ago, in a class that was nominally about infinite sets but actually covered all sorts of quirky math. I remember arguing loudly and forcefully that it shouldn't matter whether you switch or stay, but when I drew out the probability tree to prove my point, I promptly changed my mind! At least now I have the comfort of knowing that I didn't have to simulate it... :)

[Tommaso]

Quite funny. I can see you at the blackboard, arguing and then making a U turn...
The thing is, this problem does seem to be just as hard for very knowledgeable people as it is to laymen.
That's what is fascinating... Any math-educated person is able to compute the decision tree in their mind, but for some reason nobody even tries to do that.

Cheers,
T.

[Enrico Scalas]

This year, I used the "Monty Hall quiz" to teach the subtleties of conditional probabilities. By the way, there are rumours that also Erdos, a well-known mathematician, was only convinced by a Monte Carlo simulation.

The problem can be generalized in various ways. There is a large number of sites devoted to Monty Hall, to its generalizations, to why it is so difficult to find the right answer and how we succeed in getting the right answer. Here is a couple of them:

http://mathworld.wolfram.com/MontyHallProblem.html
http://math.ucsd.edu/~crypto/Monty/monty.html

The problem became very popular in the US after a column by Marylin Vos Savant in Parade:

http://www.willamette.edu/cla/math/articles/marilyn.htm

Warm wishes

Enrico

[Ramaswamy Hari]

About the Ferrari problem, I think you should always choose to change your mind. Here's my explanation. When you first chose between doors A,B & C your probability to get it right was 1/3rd. By not switching yoy have not changed your probability. But if you switch, you are choosing between 2 doors which makes the probability of getting it right to 1/2. So from a probability point of view, you have increased your chances. This may sound weird, but that is the strange thing about probability.
I am curious what your answer is. I would appreciate it if you could send me a reply with your answer.
-hari

[Tommaso Dorigo]

Hi Ramaswamy,

you are right. You have to change your mind to increase your chances: your skin feeling is correct.
However, when you change door you do not increase your chances from 1/3 to 1/2, but to 2/3 !!
The reason is that the probability that the Ferrari is still behind the door you first chose has not changed, as you said. So, since the Ferrari HAS to be somewhere, and only a third of it is behind the first door you chose, the remaining two thirds are on the other one. This may sound weird, unless you understand that by switching doors you effectively choose two doors at once, since you are taking the one you pick AND the one the anchor opened: picking two doors you'd always have at least one empty door...

Cheers,
Tommaso

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