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.