There’s an article about the notorious Marilyn vos Savant Monty Hall Problem incident at Priceonomics that I saw linked from the Diversity in Physics and Astronomy Facebook group. If you’re not familiar, I recommend both that article and (perhaps first) the Wikipedia page on the problem.
The short version is that Marilyn vos Savant (what a name!), billed in Parade magazine as the smartest person in the world, posed the Monty Hall problem to her readers, with the solution. The problem is (my phrasing):
A game show. Three large doors: one hides a car, the other two hide goats. You, the contestant, may choose any door, and receive whatever is behind it as a prize. No matter which you choose, the host, following the format of the game, lets you know the location of one of the goats by opening one of the doors you didn’t pick. Then you have the option of switching to the third door. Does switching improve your chances of winning the car?
In her column, vos Savant posed and solved the problem (the answer is yes, you should switch). A huge backlash ensued over her (correct) solution, mostly because it is counter-intuitive. Over the next few columns she explained the answer in detail and published some of the responses she got, many of which were nasty (I think it’s safe to assume the nastiest ones never saw print).
I remember getting this series of columns as a case study in my first probability class. I got very defensive over my contention that her solution was wrong wrong wrong, and I’m surprised how much I still get riled up about it. Now, this was the first time I had seen this material, so it’s not surprising I didn’t get the right answer. But what really galled me (OK, it still galls me so much I’m writing a blog entry about it) was that I was thrown by her wording in the original problem.
The problem only works if you know the host’s rules. WHY does he open door #3? Does the host open it every single time a contestant plays, or only if they pick a certain door? Does the host always open a door with a goat?
Here’s how vos Savant phrased it:
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.
It’s the word “say” that threw me. It means that what comes next is arbitrary, specific to the example she’s giving. But does it refer to just the number on the door, or also to what’s behind it? Is it “another door, (say #3, which has a goat)” or “another door (say #3), which has a goat.”
If it’s clear to you from her wording that the host always:
a) must show you a door, which:
b) you didn’t pick AND
c) has a goat
Then her original column is fine. But that wasn’t how I interpreted it.
Now, I probably would have gotten it wrong even if I had interpreted it correctly, but the fact that the wording was ambiguous gave me a face-saving way to protect my fragile young ego. Yes, I had the wrong answer, but that’s because I was solving the wrong problem!
Only when I got to the column where she explained the rules clearly (by suggesting a way the reader could prove it to themselves) did I understand how the problem was supposed to work, and then I felt duped. So I remember being incredibly annoyed by her intransigence in subsequent columns — why couldn’t she just acknowledge that her original column was ambiguous?
I’m not defending her detractors here; I agree that the criticism of her column was over the top. I’m sure many of the complainers understood the problem just fine and simply got it wrong, even (especially) the math and probability professors who should have known better.
Anyway, in the past when this incident came up I was always quick to point out the syntactic ambiguity and criticize her presentation of the problem (I guess I still am!). But the framing of the Priceonomics article makes it clear that there was a good deal of sexism involved in the backlash, which is something that I hadn’t noticed before. I was a less enlightened, young lad then — if the original columns had been by Richard Feynman or George Gamow perhaps I wouldn’t have felt so embarrassed about misreading them, and perhaps I would have learned more, instead of developing a 20 year-grudge?
At any rate, safe to say that my probability professor gave problems that really made me think.
Bonus: I vividly remember another problem from that class that has also stuck with me all these years, and made me better at understanding probability and its proper role in science. A black box flashes (only) x’s and o’s. The sequence you see is:
XXOXOXOOOXOXOOOXOXOOOX…
what is the probability that the next symbol will be an O? Highlight the white text below for the answer:
There is no “correct” answer because the problem is somewhat ill defined. If you notice that the series is that of the prime and composite numbers (X’s are prime) then you would say that the answer is 100% (24 is composite). If you like, you can estimate the (undefined) probability that the series just appears to be the sequence of prime and composite numbers and use that in your answer (maybe in a Bayesian way). Or you could assume that the series is random, and that each symbol is an independent draw If you do, you can estimate the chances that the next symbol is an O by adding up the total number of X’s and O’s, giving you 13/22.
But the point is that probability theory is a model that we apply when we think we can describe some event as “random”, and as being drawn from a distribution we can characterize. The correctness of your answer is subject to the correctness of your model.
Thanks Aki Roberge and Nicole Sullivan for spotting errors in post.
The phrase you quote, “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”, is not actually the way vos Savant phrased it, it’s directly from the question that Craig F. Whitaker originally asked her:
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?
So… let me tell a different story.
Suppose you’re at a buffet restaurant, and you and your friend approach the dessert table. You walk on one side of the table, your friend on the other. There are three plates on the table, all with domes (to prevent flies landing on the desserts) and windows (so the dessert can be seen). It happens that all the windows are pointing to your friend’s side of the table. On one plate is chocolate cake, on the others, apple pie. You pick a plate, say the one the one to your left, and your friend, who knows what’s on the plates, picks another plate, say the one that was in the middle, which has apple pie.
Is it clear from this dessert story that your friend *always* picks apple pie, if the same setup as described here, happened to occur the next day? He has knowledge, but does the story indicate he has any compulsions driving his action, that he *must* pick pie?
Marilyn’s own words (archived at https://web.archive.org/web/20130121183432/http://marilynvossavant.com/game-show-problem/) are “the original answer defines certain conditions, the most significant of which is that the host …”. Well, that spells out the problem very clearly, right there. What defines the conditions of a question is not the answer, what defines the conditions of the question is the question. A person may ask the question, “There are two odd numbers, what can you say about the sum?” – if he gets the reply, “What you can say about the sum is, it’s 10”, there’s going to be backlash. A defense of “The original answer defines certain conditions, that the odd numbers are 7 and 3” does lead to people agreeing that she has answered the question “There are two odd numbers, they are 7 and 3, what can you say about the sum?”… but the original reply is still not an answer to the original question, and where the difference between the original question and the conditions-added question can manifest in subtle phrasing, confusion between the questions can then arise.
As to your bonus problem; I’ll give you another story. I give you a coin which lands (only) on heads (H) and tails (T). The sequence you see after the first few tosses is: HHTHTHTTTHTHTTTHTHTTTH…. Care to take a guess what the probability is for the next toss?
Oh… did I forget to say that, attached to the coin, is an accelerometer, a microprocessor, and weighted motors? Is it unreasonable of me to think you might not have assumed that from the way I phrased the question? If you start assuming that the person posing the question is unreasonable, or using language in bad faith, then you can pretty much ignore the question. But to engage in good-faith rational discussion and sincerely answer a given question exactly as the given question is, you really have to start from a position of trust, that what is said is taken to be true, and also that what is said is taken to be the full extent of the truth as can be used in reaching a reasonable answer to a question posed.
Now, in an interactive scenario, it’s perfectly reasonable to look askance at the person asking a question and say “… Really? Is that the question you’re really interested in, or do you really mean to ask this other question?”. In a non-interactive scenario, like having text on a page which forms a question you’re to answer, that’s not an option. In a non-interactive scenario, for a bonus, after the given question is answered, an addendum might start “This is similar to another question, which I will now discuss: …”. But that discussion not a place to start a reply as if it were a genuine, full answer to the given question.
Correction: one half of one of which
The problem here is that the problem is not mathematical, it is philosophical and conceptual.
Because we know, in advance, that one door and one goat _will_ be shown, we also know in advance that the odds are 50: 50. They began that way, not as 1/3-2/3. Another way of looking at it is that, rather than three doors, there are two classes, one of which is pre-ordained to be tossed away. Hence the odds are not 1/3-2/3, but 1/3-1/3, or 50:50.
The solution is trivial. The problem is effectively asking: Do you want to open the door you picked or the two doors you didn’t pick? Clearly you double your chances of winning by opening 2 doors. You know that at least one of the doors you didn’t pick has a goat behind it. So all the host is doing is showing which one it is. It does not change choice being given of opening the two doors you didn’t pick, because the host is not making a random choice, when he opens one of the doors you didn’t pick.
I had read that Priceonmics article this morning, and have spent the rest of the day puzzling over it. Now I think I understand, but it is a trick, really. I was analogizing it to a situation where someone gives you a fair deck of cards, face down, and says he will pay you $50,000 if you pick the ace of hearts on your one try. Then after you pick, he turns over 50 of the remaining 51 cards, leaving yours and one more. It would seem that your odds of winning are now one in two–but not if the other person knows where the ace of hearts is (it’s marked, in a way that he can tell but you can’t). Then his turning over 50 cards is a dupe, because he’s obviously not going to turn over the ace of hearts.
And that appears to me to be the “trick’ in this rather unfair question. It is a matter of Monty knowing where the cadillac is, so he obviously will not open the door which has it, if it is one of the other two; and he won’t open yours. The first door he opens is meaningless. But if it is a “fair” game, and he opens randomly, then your odds have improved to one in two. I did not read Marilyn’s explanations, but I doubt that she explained this aspect. This is really not an exercise in probability, it is about a game which appears “fair,” but is meant for TV drama, and is controlled by the host.
Now, actually, having watched that show many times as a boy, I always felt that the show liked to give away big prizes, it made for a better show. And Monty liked people to win. So I always felt that in such a situation, he would open one door which had decent prizes, leaving the non-prize behind one door and the big prize benind another. Oh, sometimes someone would end up with the bag of dirt, but not often. He sort of set it up so that if the contestant had the good door picked, he would not offer too much to try to dissuade him/her from it. Thus the entire problem, predicated on the motivations of the host and the people who pay for the show, is a psychological problem masquerading as a probability problem. At least that is how I see it.