
The Time Everyone “Corrected” the World’s Smartest Woman (2015) - ingve
https://priceonomics.com/the-time-everyone-corrected-the-worlds-smartest/
======
gameswithgo
I recall this happening as a high school kid. I couldn't tell from the
arguments who was right or wrong, so I wrote a simulation, which was fairly
easy, and proved her right. I found it interesting that writing an simple
program in this case could confirm the right answer easily whereas having a
PHD in math was no guarantee that you would reason it out correctly

~~~
masswerk
This is actually a good example for a problem where simulation does
exceedingly help in gaining insight. I once did a webpage [1] comprising both
logic and simulation to sort out a dispute. (The debating parties each had a
PHD in philosophy, implying some courses in logic absolved – which apparently
didn't help much with the specific problem.)

[1]
[https://www.masswerk.at/misc/montyhall/](https://www.masswerk.at/misc/montyhall/)

~~~
MereInterest
The key aspect, I think, is in writing the simulation oneself. At one point, I
was attempting to explain the Monty Hall problem, and wrote a simulation out
of frustration. It failed to convince the other person, who accused me of
subverting the simulation such that it would give the answer that I wanted.

I'm wondering if gambling would be a good way to go about it. Start with $10
each, and alternate who plays the "host" of Monty Hall. One person is only
allowed to switch, and another is only allowed to stay. For each correct
answer, take $1 from the other person.

Hmm, maybe to sweeten the deal, the person who switches gets $2 for a win, but
the person who stays gets $3. That way, it would show that even with 50%
better payout, it doesn't overcome the 2x difference in odds.

~~~
gameswithgo
That is true, if is only convincing for people who can understand the code.

~~~
ci5er
I don't think I'm a sociopath, but often I use monte-carlo or agent based
simulations to help me understand what happens, because writing and running
the code is simpler (for me) than closed form analytic solutions. IF I then
need to convince others, I guess I can do the closed-form solution (based on
the prior intuition), but since I don't really care whether anyone agrees with
me or not (I go develop 'engines' in my corner of "the system"), I guess their
opinion is immaterial to my/group/company progress?

Do you think that agent-based computational simulations (say of micro-
economics bubbling into creating emergent macro-economics) are ever used in
debates?

------
dekhn
I recall this happening; in fact, I though she was wrong when I read the
original article.

After many years, I finally understood where I made the mistake. It's not
stated explicitly in the wording of the problem, but if you point this out
explicitly, nearly everybody changes their mind:

When Monty Hall opens the door, he has to pick one of them, and he has
knowledge of what's behind it.

The way the problem was described, I didn't appreciate that.

It wasn't until one of the groups that wrote in to support Vos Savant
explicitly said "since some of the students in the class were skeptical, we
wrote a simulation of the problem, which output the probabilities you would
expect if Vos Savant was right." I inspect the source code, and viola-
instantly understood my false assumption about the problem description.

It's an open problem whether the Monty Hall problem should explicitly state
the implication of which door Monty opens. To me, I think the whole point is
that it leaves one of the consequences unstated, and the reader is _expected_
to make that indirect inference.

~~~
lookACamel
>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, __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?

~~~
dekhn
There's actually some ambiguity here (I'm sorry, I'm a really pendantic
person). What it doesn't say is that he always picks the door with the goat.
What it says is that in a single trial, he picks the door with a goat. I
assumed that in half the trials, he'd pick the door without a goat. I see now
that was a wrong (and dumb) assumption.

This is why, rather than reading text, I think people should describe these
problems with code. Unambiguous code. With test cases. Then everybody can
inspect the unambiguous code rather than having to parse human text.

~~~
zzt
Where is the ambiguity in "opens another door which has a goat"?

~~~
dekhn
What happens in the next trial? Does MOnty always open a goat door, not a car
door?

~~~
ColinWright
We are told that the host knows what's behind the other doors, which implies
that the door he opens is deliberately chosen and is not random.

Additionally, to open the door with the prize would terminate the game, so to
open the door with the goat is the only action that makes sense given what we
know.

~~~
dekhn
I agree, those are reasonable interpretations made from somebody who has done
a close reading of the text.

Or you know, somebody could just write a computer program that described the
rules unambiguously so everybody could inspect them and not have to make
reasonable interpretations.

------
alangpierce
The letters quoted in the article are frustrating to read. It doesn't really
matter that she was right, people shouldn't have sent those letters even if
she was wrong. Especially in probability and statistics, but also in pretty
much every area of knowledge, it is common for human beings to hold strong
beliefs that are completely wrong or seriously misguided, and we all need to
recognize that weakness in ourselves.

I think the world would be a much better place if we all dropped our egos and
approached these sorts of disagreements with kindness and benefit of the
doubt. Even if we're confident that we know better than the other person, we
should approach it as a genuine attempt to understand and clear up the
misconception as a way of improving the other person (and be willing to admit
that the "misconception" may end up being true after all, as in a case like
this). As far as I can think of, condescension is just some obnoxious human
habit and doesn't actually provide any value.

------
ColinWright
In presentations/workshops I now find vanilla-flavoured Monty Hall to be of no
real value - nearly everyone says _" SWITCH!!"_ without thinking. So I offer
this variant.

I have 10 cups, and under one is a prize. I get a victim - ahem - _volunteer_
to select two, and place markers on them.

I then say clearly that I will remove 6 of the other cups and, to retain the
mystery, will not reveal the prize. There are now four cups, the prize is
under one, and two of them are the volunteer's original choice.

And now the offer: They may retain their original two, or they may surrender
their two and select only one of the other, currently unselected two.

So:

* What would you do, and why?

* What interesting further variant can you think of?

~~~
yifanl
A very educational variant I've heard is to up the number of doors from 3 to
10000.

Now that Monty is closing off 9998 doors, it makes the leap of logic much less
difficult. Clearly the one door he didn't close off is special.

~~~
ColinWright
Personally, I find that completely uninformative. Others find it helpful and I
won't deny their experience, but I find that no more convincing that proper
arguments about the 3 door version.

My variant was not intended to assist explaining the original, it's a variant
intended to make people think again. It often exposes non-understanding in
those who will blindly and automatically say "Switch".

Here's another variant. 12 doors, I let you choose 3. I then open 4, leaving
your chosen 3 and 5 others.

I let you keep your original choice, or give them up to choose only one of the
others. Should you switch now?

~~~
grigri9
No, In that

P(3/12) = 25%

The remaining doors have 75% probability and choosing 1/5 only gives you 15%
chance of winning.

~~~
ColinWright
Correct.

Explaining that to muggles, though, is non-trivial.

------
mchannon
You can distill the disconnect down to one question.

Does Monty know where the car is? (The original article says he does, but this
often gets lost in the version people read.)

Suggest Monty doesn't, and he opens door #3 to reveal a goat only because it
wasn't the door you picked (and it was merely luck the car wasn't there). In
that case, it is genuinely a 50/50 shot between the remaining two.

Now if Monty did know where the car was, and he wouldn't have opened door #3
if the car had been there, then the 2/3 percentage to switch is intact.

To many this seems like a minor (or incorrect) distinction but it's little
assumptions (Monty knows) that underpin these gotcha questions. That's one
reason why Google-style interview questions irritate me so much. In many of
them, there's an implicit assumption that is necessary but never stated.

Many of the responses Marilyn got seemed to come from this form of irritation,
even if some of the people writing in couldn't express a logical or
justifiable basis for their irritation.

It's an interesting puzzle, but it's too easily rephrased like a con.

~~~
MilnerRoute
I always start explaining it like this. "There's a one in three chance you
guessed right on the first try."

And that just doesn't change, even if Monty Hall opens a door.

So not only is there a two-in-three chance you guessed _wrong_ on the first
try. Monty is helpfully offering you a chance to switch to the correct/winning
choice in the second round.

~~~
mchannon
But Monty's _only_ being helpful if I guessed wrong. If it actually was behind
door #1, Monty's trying to entice me into picking a goat, and basically being
a dick. This is also assuming he could've just opened door #1 right away if it
was the correct answer and obviated this whole mess.

Monty's free will (an unknown) underpins this entire argument- what choices
does he have, and why does he even make me choose a door instead of giving me
the car (and the goats) outright?

Maybe from a statistical point of view it's better to have Monty pull this
shtick, but something that's helpful 2/3 of the time and harmful 1/3 of the
time doesn't make the cutoff for helpful.

~~~
ModernMech
I feel like you're still missing something. The probabilities come out of a
table. It's just math. There's no need to bring "free will" into the
explanation.

~~~
username90
No, it is you who are wrong. You can't write down a table if you don't have a
proper explanation of the possible event chains, and in this case you don't.
Instead when people write down these event chain they add information that
isn't there and then think they actually solved the problem. But all you
really know is that this one time the host opened a door with a goat, you have
no idea if the host always does this since the problem statement doesn't
actually say it outright. It might be a good assumption to make, but it isn't
in the description so it isn't a mathematically rigorous answer.

~~~
ModernMech
But once you do have a proper explanation of the possible event chains, free
will still doesn't play a part of it. It's a game with rules. You seem to be
keyed in on some aspect of the rules being unclear, but even with the rules
perfectly clear or not clear at all, the probabilities are the same regardless
of the psychological state of the contestant.

------
indigodaddy
Another way to think about it:

Your initital guess (which was 1/3 probability) would have had to have been
right, to make it not correct (or not the right choice) to move over to the
other door. So moving to the other side (essentially you are just moving over
to the 2/3 "probability block"; you were originally on the 1/3 "probability
block"), inherently moves your odds over to 2/3\. Staying only makes sense if
you think you guessed correctly initially, and what are those odds, well those
odds are 33%! And what are the odds of anything other than that initial guess
(eg moving over) ? Well those odds would be 2/3!

This is the only way that I was able to wrap my brain around it. We actually
did it by just doing an A B C guess three times in a row. My wife picked a
letter. Then I picked a letter. It worked three times in a row, because I
never initially picked the letter she did. So three times in a row, me moving
over to the remaining letter (after the letter that neither of us picked had
been eliminated; obviously we kind of got lucky in me not picking her letter
in three tries), ended up of course as being the letter she had picked. Moving
over doesn't work only when you actually choose correctly initially, which
would only be 1/3 of the time. So again, moving over makes your odds 2/3.

It's brillantly simple really, yet extremely difficult to get to a method of
actually understanding it, and this is the best way I've found.

------
mabbo
Ignore the fact that it was the Monty Hall problem- everyone on HN has played
that topic back and forth ad nauseum.

But look at the real story here: thousands of men going out of their way to
write letters- not quick 5 minute emails, but paper letters with envelopes and
stamps- to tell a woman she was wrong. If it had been a man writing the
article, would it have gotten the same reaction? I doubt it.

In my view, this is the crux of the gender problems we see in tech today.
Certainly not as strong (one hopes) but certainly the same weird psychological
problem that so many of us seem to have to some degree _requiring_ men to tell
women when they're wrong, but not care when men are. 'GamerGate' and the
various witch hunts around that topic is a great example. It's fine if a man
sucks at his job, but if a woman does and it's a role that society has labeled
'for men', suddenly it's an emotion-driven attack that must be defended
vigorously.

We need to see this and watch for it if we're ever going to end it.

~~~
slededit
> But look at the real story here: thousands of men going out of their way to
> write letters- not quick 5 minute emails, but paper letters with envelopes
> and stamps- to tell a woman she was wrong. If it had been a man writing the
> article, would it have gotten the same reaction? I doubt it.

My experience with the internet is that people just can't help themselves to
point out when somebody is wrong - man or woman. Sometimes going as far to
write what could be considered full essays complete with citations. Its just
another anecdote but I don't think its at all obvious its because of gender.
People just like feeling smart.

~~~
ModernMech
Yes, but most people don't cite a man's gender or appearance as a reason for
his wrongness. Just from the sampling in the article we have two good examples
of criticisms a man will never see:

> Maybe women look at math problems differently than men.

> You are the goat!

I'm sure there were many others.

~~~
Confusion
We don't know the samples in the article are representative of the feedback.
In fact, I would expect the authors to cherry pick the feedback for
entertainment purposes.

I believe the OP is right, but providing uncontroversial evidence of that is
hard. You need a thorough classification of the feedback in a number of
sufficiently comparable cases, involving both men and women, to provide hard
evidence. And even with hard evidence in hand the conclusions could be
ignored; cf. climate change.

This isn't a debate that can be settled by rational evidence.

~~~
ModernMech
Have you ever seen a woman's inbox? This scientist got 10,000 pieces of hate
mail. Going by what I know about the kinds of e-mails women receive on the
daily, it's safe to assume a good 2000 of those messages were misogynistic.

It's just how the world works. We can all pretend like we don't have
uncontroversial evidence and that we'll never know if her identity as a female
really affected the response. That's pretty much the status quo. Or we can not
be blind and see what's happening right in front of us.

------
bayesian_horse
So, technically, the World's Smartest Woman could be wrong (not in this case,
because of the mathematical proof, but according to the title). Being smart
doesn't mean you're always right, especially when there is no good answer with
the available information.

But clearly, the responses show sexism at its finest, or worst. Partly,
though, many of the men refusing the argument probably honestly disagree with
the counter intuitive answer, but they seem to rationalize it with their
negative beliefs.

------
ameliaquining
Something similar happened a few years later with Fermat's last theorem,
except that time vos Savant was wrong. (She didn't believe Wiles's proof.)

~~~
mmusson
Which version? Because his first version of the proof had an error that he had
to go back and fix.

~~~
beder
She was claiming something more fundamental (and less sophisticated) than
that. Nigel Boston and Andrew Granville wrote a review [1] of her book that
distills some of the problems with her argument. A sample quote:

> In fact, her central theme is that non-Euclidean geometry, and indeed any
> mathematics related to non-Euclidean geometry, is nonsense.

[1]:
[http://www.dms.umontreal.ca/~andrew/PDF/VS.pdf](http://www.dms.umontreal.ca/~andrew/PDF/VS.pdf)

------
iamjdg
What I don’t get is in the first table, game 3 and game 5 no longer exist
after he opens doors 3. The auto can’t be behind door 3, because he revealed
there is a goat there. So yes, relative to the original choices, she is
correct. But we have been presented with new information, so if you reset the
problem based on the new information, don’t we now have a choice between 2
doors, one with a car and one with a goat?

~~~
ColinWright
The information you have gained is not about the door you've chosen. It
doesn't matter what door you choose, the host can always open a door to reveal
a goat. So there is no information given about the door you've chosen, so the
chances of that door containing the prize remain at 1/3.

But you have been given information about the door he didn't open, because he
didn't open it. That's why it's possible for the odds on that door of holding
the prize can change.

And yes, we do now have a choice between two doors, one with a goat and one
without. The error is in believing that these are equal choices.

I roll a die, and you can choose "1" or "not 1". You have two choices, but
they have unequal chances of being correct. Similarly with the doors. Just
because there are two choices, they may have different odds.

In the Monty Hall problem, they do.

~~~
aw3c2
So this is a psychology puzzle rather than pure math?

~~~
ColinWright
I'd be interested to know what it is about my reply that makes you think this
is a psychology puzzle. In particular, it seems to be people's psychology that
prevents them from understanding the mathematics underneath. People seem to
assume that if there are two choices then they must be equally likely. That's
a psychological thing, although to be honest, I don't understand it.

But the Monty Hall Problem as stated is about the probabilities, not on the
psychology. Computing the probabilities is simple math, once you understand
the situation. My explanation was to help the reader understand why the two
choices given don't have equal probability.

~~~
aw3c2
Hm, then I do not understand it. "But you have been given information about
the door he didn't open" was what made me comment.

If I chose a door before, then something happens that leads to only two doors
being left, both those doors have the same probability so I could just choose
the same door again.

~~~
ColinWright
aw3c2> _Hm, then I do not understand it. "But you have been given information
about the door he didn't open" was what made me comment._

aw3c2> _If I chose a door before, then something happens that leads to only
two doors being left, both those doors have the same probability so I could
just choose the same door again._

The original said this:

CW> _The information you have gained is not about the door you 've chosen. It
doesn't matter what door you choose, the host can always open a door to reveal
a goat. So there is no information given about the door you've chosen, so the
chances of that door containing the prize remain at 1/3._

CW> _But you have been given information about the door he didn 't open,
because he didn't open it. That's why it's possible for the odds on that door
of holding the prize can change._

So let's recap what's going on. There are three doors. For the sake of
concreteness let's call them A, B, and C. You choose one of them. For the sake
of concreteness let's suppose you choose A.

So now there are two doors, B and C, remaining unchosen by you. Currently
those two doors, B and C, each have probability 1/3 of having the prize. The
door you chose, door A, has probability 1/3 of holding the prize.

Now the host opens a door, taking care to open a door that does not hold the
prize. So the pair {B,C} still has total probability of holding the prize, but
you are being shown that one of them certainly does not. This doesn't affect
the probability that your chosen door, door A, holds the prize -- the
probability that the prize is behind door A is still 1/3.

The pair {B,C} still has total probability 2/3 of holding the prize. You're
now given the choice of staying with A, or switching.

Quoting you again, you said:

aw3c2> _something happens that leads to only two doors being left, both those
doors have the same probability ..._

That turns out not to be the case. Just because there are two doors they don't
have to have equal probability of holding the prize, and in this case they
don't. The probability that your door holds the prize has not changed and is
still 1/3\. The probability that the door neither chosen by you nor opened by
the host holds the prize is now 2/3.

Does that help?

~~~
aw3c2
Thanks!

------
nabla9
[http://marilynvossavant.com/game-show-
problem/](http://marilynvossavant.com/game-show-problem/)

It's interesting that kids in elementary schools were getting this right in
many schools. Probably because they worked it trough and did experiments to
verify it. Something that was beyond people with many PhD's.

It's the lesson for being intellectually humble.

------
bayesian_horse
Where does a sexist troll get his water? From a well, actually.

------
ModernMech
"You made a mistake, but look at the positive side. If all those Ph.D.’s were
wrong, the country would be in some very serious trouble."

So then we're in very serious trouble?

~~~
Sharlin
No, the person who said that was wrong on both counts. Turns out they were all
wrong but that doesn’t really matter.

~~~
DoctorOetker
>In the proceeding months, vos Savant received more than 10,000 letters --
_including a pair from the Deputy Director of the Center for Defense
Information_ , and a Research Mathematical Statistician from the National
Institutes of Health -- all of which contended that she was entirely
incompetent

So the Deputy Director of the Center for Defense Information failed to
estimate a concise problem in an ideal setting, but somehow the world is
supposed to believe in _Mutually "Assured" Destruction_ in a messy high
complexity real world setting?

~~~
Sharlin
Turns out people have evolved in an environment where it’s _really_ adaptive
to be able to manage messy ill-defined unquantifiable problems and not at all
adaptive to solve quantitative puzzles where you have to shut up and
calculate.

------
rachelbythebay
This sort of thing still happens. Right here on HN, even.

------
1in3tolose
Discussed to death:

1st explanation:

You have a 33% chance to pick car right away, if you switch then you lose.

In other words, you have a 33% chance to lose if you always switch.

2nd explanation:

You have a 66% chance to pick goat, game master will eliminate the other goat,
if you switch you win

