Cap takes on the Monty Hall Problem

I’m teaching probability this semester, and tomorrow I’ll be talking about that most headache-inducing of problems in probability, the Monty Hall problem.  I hadn’t been planning to go there (because the spring term classes are accelerated, a whole semester’s material in seven weeks, so I keep digressions to a minimum), but now I’ve been asked about it by several of my students. I’m writing this today to get my own thoughts in better order, and hopefully to illuminate the issue for you, dear reader.

If you’ve never heard of the Monty Hall problem (also called the Monty Hall paradox), you’re in for a doozy today. It’s infamous. You can read about it on wikipedia, or here, or here, or here, or in video form here (I could go on almost forever), or even on the website of Let’s Make a Deal (the game show which inspired the present form of the puzzle).  You can even play the game yourself here.

I suppose you could just follow a few of those links and be done with it; and if you don’t like my style then you probably should do just that (but then why are you here?). For my explanation, read on.

The right answer is actually not that complicated, but due to a known bug in human intuition, most people get the wrong answer. Moreover those who get the wrong answer have a tendency to be extremely vociferous in defending their answer. I have actually seen fistfights break out over the answer to this problem. So, discuss this problem with your friends and family only if you’re brave.

(I blog not to unite, but to divide?)

The Monty Hall “paradox” is the story of the following game.  All of my games today involve two characters: Alice (the contestant) and Hatter (the host).

Game 1 Script (Original)

There are three doors.  One leads to a car, and the other two lead to goats.  (The locations of the car and the goats are assigned at random in advance, and Hatter is aware of what is where.)

Alice chooses one door (which she hopes leads to the car) but does not open it.

Hatter, knowing where the car is, opens one of the doors that Alice did not choose, showing her a goat.  (That is, if she picked a goat, he’ll show her the other goat; if she picked the car, he’ll arbitrarily choose a goat to show her.)

Now there are two doors, and Alice knows for sure that one leads to a car and one leads to a goat.  Hatter then offers Alice the option to either switch her choice to the other door or stick with the door she had chosen at the start.

Assuming that Alice wants a car and doesn’t want a goat, is it in her interest to take the prize behind her original door, or is it in her interest to take the prize behind the other door, or does it not matter?

The correct answer is that Alice is twice as likely to get a car if she switches as if she stays; most people think that it shouldn’t matter, that the chances of getting the car are 50-50 either way, so it doesn’t matter.  If that was you, you’re in good company.  Read on for an explanation that will, I hope, be both gentle and convincing.

Dear readers, I invite you to temporarily put aside the original game and consider two others.

Game 2 Script

There are three doors.  One leads to a car, and the other two lead to goats.  (The locations of the car and the goats are assigned at random in advance, and Hatter is aware of what is where.)

Alice chooses one door but does not open it.

Hatter then offers Alice the option to either open her door and take the car if it’s there, or else to open both of the other doors and take the car if it’s there.

Assuming that Alice wants a car, should she keep her original door, or switch to the other doors?

In this case, the strategy is clear. The Hatter is giving Alice either the door of her choice or two doors of her choice. With no information to distinguish the doors, two doors are better than one!

Game 3 Script

There are three doors.  One leads to a car, and the other two lead to goats.  (The locations of the car and the goats are assigned at random in advance, and Hatter is aware of what is where.)

Alice chooses one door but does not open it.

Hatter then asks Alice to guess whether her door leads to a car or not.  If she guesses right, she wins a car.  If she guesses wrong, she goes home empty-handed.

Assuming that Alice wants a car, what should she say (yes or no)?

The best thing to do is say “no”. Alice’s original guess is correct with probability 1/3 and wrong with probability 2/3, so she’d be wise to claim that her guess was wrong.

Games 2 and 3 are, I think, uncontroversial and, one might think, uninteresting. Hardly a suitable warm-up, you might think, for the frumious Monty Hall problem.

But now I invite you to just notice that all three of these games are really the same game. (Take a moment, mull that over.) The first version is designed, I think, to make switching and not switching seem as symmetric as possible (which confuses our intuition), while the other two are less obfuscatory. But there is really only one thing going on, in all of these games. Alice makes a guess (which is correct with probability 1/3), and then Hatter invites her to bet on the correctness of her original guess. Naturally, if she bets against her first guess (that is, if she switches doors), her likelihood of success is 2/3, twice as good as not switching.

Game 4 Script

There are three doors.  One leads to a car, and the other two lead to goats.  (The locations of the car and the goats are assigned at random in advance, and Hatter is unaware of what is where.)

Alice chooses one door (which she hopes leads to the car) but does not open it.

Hatter opens a different door (not the one Alice picked, but otherwise arbitrary).  If the opened door happens to lead to the car, then the whole thing starts over (the prizes are secretly reshuffled and Alice picks a fresh door), repeating until the Hatter opens a goat door.  If and when that happens, we proceed.

Now there are two doors, and Alice knows for sure that one leads to a car and one leads to a goat.  Hatter then offers Alice the option to either switch her choice to the other door or stick with the door she had chosen at the start.

Assuming that Alice wants a car and doesn’t want a goat, is it in her interest to take the prize behind her original door, or is it in her interest to take the prize behind the other door, or does it not matter?

In this case, it doesn’t matter!  The odds are 50-50 either way! It turns out, if you can talk to a person long enough to get through to their inner thought processes, most people were really thinking about this game all along. (If it’s not clear why this is different than game 1, or how the probability of Alice’s door being the car can increase from 1/3 to 1/2, then read on to the very end.)

The Moral of the Story

If the confusion here is caused by a bug in human intuition about probability, is there a lesson to be learned?  What can we take away from this, and make our intuition better in the future?  This blog is, after all, not so much about mathematical facts for their own sake as it is about what they mean and why they are interesting. If this post has a moral, it’s this.

New information can do more than rule out possibilities that are now impossible; new information can also change the relative likeliness of possible outcomes.

Perhaps the problem is that for so many of us, our first exposure to probability involves dice or cards or marbles in a jar, some scenario in which all the outcomes are equally likely. So perhaps it’s natural, when in doubt, to list everything that’s possible and treat the alternatives as equally likely, because it’s familiar.

Consider the following illustrative example. Suppose that I have seven dice. One die is rigged so that it always comes up 1, the second die always comes up 2, and so on up to the sixth die which always comes up 6. The seventh die is an ordinary fair die. All seven die are identical to a casual glance, and they are sitting in a dish on my desk. I have no idea which is which, and I grab one at random. I toss that die, and it happens to come up 6. How likely is it that I’m holding the fair die?

Some things are obvious. Right before I throw the die, there are seven equally likely alternatives (for what die I’m holding). Right after I throw the die, there are only two possible alternatives. So someone very naive might guess that the likelihood I’m holding the fair die is 50%. But I have some information about this die. I know that it came up 6, and that tells me that it is much more likely to be the always-6 die than the fair die. (Invoke your common sense here; suppose I asked you to guess which die you thought you were holding, and I offered to bet real money on it, at even odds; I hope you’d realize that I’m a sucker, and bet that you had the always-6 die.) So not only does rolling a 6 disqualify five of the seven possibilities, it also advantages one remaining possibility over the other.

(It turns out that probability I’m holding the fair die is 1/7, and the probability that I’m holding the always-6 die is 6/7; this reflects the fact that the always-6 die is six times as likely to turn up 6 than a fair die is. But I don’t really care about numbers today, just the qualitative statement that the two possibilities are no longer equally likely — one is now much likelier than the other.)

In Game 1, the fact that Hatter can show Alice a goat doesn’t give her any information about the correctness of her choice. Hatter would have shown her a goat with the same probability (i.e., with absolute certainty) no matter which door she picked. So watching Hatter open a goat door, which he had already promised to do, should not in any way change Alice’s attitude about the correctness of her door. Initially, she was 1/3 to pick the right door, and nothing’s changed: she has the right door with probability 1/3.

But in Game 4, the fact that Hatter’s random door-opening reveals a goat does contain information. A priori, it might have been the car. The fact that it was a goat actually does shed light on Alice’s original choice. Hatter is more likely to pick a goat if Alice picked the car than he is if she picked a goat, so Alice should now feel more confident about her guess than she previously did. This accounts for the rise in probability from 1/3 to 1/2 in this case.

P.S.

Feel free to comment, strenuously object, or post other ways of looking at it in the comments.

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