The Time Everyone “Corrected” the World’s Smartest Woman (2015)

I wrote a simulation as an exercise (not to convince myself, just to illuminate the logic). I recall becoming extra-convinced when I spotted an optimization that would have been paradoxical if the correct solution hadn't been correct. I don't remember exactly what it was; I think it had something to do with collapsing nested if() into a single if() based on p vs. (1 - p) for the original door choice.

edit: the more I think about it, the more I suspect that the actual optimization was not randomizing both the door choice and the location of the car, which then allowed simplifying the win logic. If door 0 is always considered the door with the car, then the whole loop body can be collapsed down to:

    if (rand() / (RAND_MAX / 3) == 0) noswitch_wins++;

No threat of a paradox as I was thinking before, but a fairly clear statement that the odds of winning by not switching are 1 in 3.

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/

I specifically changed my mind when one of the writers-in said they wrote a simulation and it matched what Vos Savant said.

That made me realize: code is a better way to describe problems than human text.

And I have written a simulation which proves her wrong, doesn't matter. I mean, when we deal with ambiguous questions like this a simulation doesn't really help since you had to add information not originally stated in order to write it. It just happens that most people make assumptions which aligns with this answer, but you could just as well make other assumptions and get another answer.

>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?

There's a 20% chance that the volunteer was initially correct. In the 80% case, one of the two remaining cups has the price beneath it, so choosing one of the remaining cups gets you a 40% chance of payout instead of your initial 20%. Switching doubles your chance of payout.

This should be straightforward to anyone who actually understands the Monty Hall problem.

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.

> I then say clearly that I will remove 6 of the other cups and, to retain the mystery, will not reveal the prize

Do you say 'I will now remove 6 empty cups?', or is it believed that you removed 6 unmarked cups at random?

I think I should still switch, and for the same reason as in the original problem. The fact that you chose not to remove those two cups makes it more likely that one of them contain the prize.

An interesting variation is "what if you toss a coin to decide whether to switch or not".

I find it weird that that resets the odds back to 50:50

I didn't think what you say is true, but it is. Here's what misled me:

Go back to the 100 door version. You start out by opening 1 door, so it's 99/100 that the car is behind another door. If monty just happens to open 98 of those doors and reveal all goats, then that 99% probability that you should switch to one of them combines with new knowledge of which to switch to. Its exceedingly rare, but if it happened, then you should still switch.

The above logic is wrong. I simulated and you're correct:

    n = 10000000
    n_doors = 3
    first_guess = np.random.randint(n_doors, size=n)
    car_behind = np.random.randint(n_doors, size=n)
    should_stay = first_guess == car_behind
    print('p(should stay)',
          np.mean(should_stay))
    # 0.3333381
    monty_opens_all_but = (first_guess + 1 + np.random.randint(n_doors-1, size=n)) % n_doors
    print('bad setup?: ',
          np.any(monty_opens_all_but == first_guess))
    # False
    monty_shows_only_goats = np.logical_or(monty_opens_all_but == car_behind,
                                           first_guess == car_behind)
    print('p(monty shows only goats)',
          np.mean(monty_shows_only_goats))
    # 0.6667607
    print('p(should stay| monty shows only goats)',
          np.mean(should_stay[monty_shows_only_goats]))
    # 0.499936633938

I'm kinda astounded. I do stats for a living, yet without writing out the math, my intuition misled me. I thought I had a framing of the problem that allowed me to use a quick shortcut in my thinking, and that framing was wrong. Two takeaways:

1) Probability is really hard to get the right intuition about. Reasoning by analogy/shortcut problem framing is dangerous. You have to write out the math.

2) The "100 doors" explanation for the usual monty hall problem is correct for subtler reasons than are immediately obvious. You could probably set up a counter monty hall problem to trick people where there are 100 doors and he just happens to show 98 goats.

You are right, to make this easier to understand, imagine replacing Monty Hall with a street peddler.

You are playing a game with a street peddler. There are 3 cards, two are duds and one is the prize. He lets you pick one card. When you picked a card he flips one of the other two revealing it as a dud, and gives you the option to switch. Do you switch?

A person who have heard the naive explanation to the Monty Hall problem would say yes. A smart person would say no. Why? Because it is not in the street peddlers interest to let you win, he has mouths to feed and need the money! So the only reason he reveals another card and asks you to switch is because you picked the right one from the beginning, so staying in this case means 100% chance of winning, and switching is 100% chance of losing.

For the naive Monty Hall interpretation to be correct we need two things to be true: Monty Hall always opens a door and the thing behind the door is always a goat. If any of those two are not true then the popular explanation is wrong, and if you look around most explanations of the problem leaves those facts out.

For example, in this article they forgot to say that the game host always opens a door and when he opens it it is always a goat. This shows that the author of the article (and almost everyone here at HN) doesn't understand Monty Hall. Instead they talk like non mathematicians like "Why would the game not open a door sometimes?" or "Why would he open a door with not a goat? It said he opened a door with a goat this one time!" etc. The problem with those things is that the explanation only says what happened this one time, so you can't possible write down the event tree without making a lot of extra assumptions about the game.

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.

If Monty would open the door where the car is, the decision whether to switch wouldn't be much of a gamble and there wouldn't be much entertainment in this part of the game show. So of course Monty knows where the car is. There is nothing mysterious or unclear about that.

This is just a hard question that is hard to get right and there is no shame in getting it wrong if you have to come up with the answer from scratch, never having heard such a puzzle before. There is no shame in not 'getting' it either. There is shame in allowing not 'getting' it to determine your conclusions.

> 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.

I remember this incident well.

Ascribing this all to gender seems like too broad of a brush, and with very little evidence to back it up either.

There were likely a multitude of motivations for people to write in and correct her, but isn't it natural for us to want to take someone down a peg when they are advertised as "all that" (in this case - smartest person alive), and they seem obviously wrong about something?

This is a statistics problem that many people who have had statistics training have been wrong about.

Seems like the setup for a perfect storm.

Personally, I don't think the larger response was because of gender. Well, a couple clearly were, but there's always outliers in any group. I think the vast majority were due to her status... the smartest person in the world. That leaves people chomping at the bit to disprove it and correct any little mistake.

That exists in any field, sadly. Top draft picks are scrutinized every year in sports. Every presidential gaffe gets its own 'mightier than thou' correction article. Anecdotally, I knew a mechanic who couldn't wait to read ClickNClack to find any little mistake and claim superiority.

> If it had been a man writing the article, would it have gotten the same reaction? I doubt it.

Based on what? I hear this only brought up in the media reaction of "men try and prove smart woman wrong!"

From the article I think I only saw one comment that involved gender - I think it’s pretty unfair to ascribe gender motivation to many of the responses (as embarrassing to read as they are).

Secondly, what is the implied norm that the author wants - you can’t disagree with someone of a different gender who has a higher IQ than you?

Yes, this was quintessentially sexist bullshit. It's so ubiquitous that there's a word for it -- mansplaining.

But it's not just about gender. Consider the racism faced by so many Black mathematicians and scientists. And Indians, such as Jagadish Chandra Bose. Even Srinivasa Ramanujan, before his genius was recognized in the UK.

But recall that Black men got human rights in the US before women did. And that corporations also got them before women did. So it's arguable that sexism runs deeper than racism.

On the other hand, I cringe at "smartest woman". What does that really mean? It's like describing some hugely multidimensional thing with a few numbers. All we know is that she did extremely well at whatever tests were used. And we also know that many such tests are culturally biased.

Anecdotally, I suspect this pattern extends all the way back to childhood. The tendency I've seen, repeatedly, is that misbehavior and horseplay is tolerated in boys, or even encouraged, whereas retribution is swift when little girls step out of line. I've found this tendency in myself, sadly. When my girls misbehave, take stupid risks, boss other kids around, tear around the house and cause general mayhem, I have to consciously skew my responses back to the mean. Not that I think the world would be a better place if we raised everyone up as misbehaving man-children, but at least I want to level the playing field somewhat for my own girls.

It wasn't just any woman, it was a threatening woman -- the "smartest" woman. They were so eager to prove her wrong, they didn't stop to even spend 5 minutes to brute force the problem. She seemed intuitively wrong, they couldn't even consider that the "world's smartest woman" saw something they, exalted PhDs, did not.

Paper letters with envelopes and stamps were as ubiquitous in 1990 as email is today, and there have been plenty of articles in The Straight Dope where Cecil Adams - a fictional, but decidedly male identity - claims to have received letters telling him he's wrong "from every unemployed Ph.D. in North America, plus a few who aren’t unemployed."

I think plenty of people would have still written in but almost certainly the tone of the letters would have been different.

Also, more people probably would have second-guessed their intuition, at least for a moment, and perhaps there would have been a few less letters.

Actually yes, "a man" would. This is the same damned groupthink ignorance anybody intelligent gets when dealing with the less intelligent. It's fundamentally not about getting the right answer, but preserving monkeysphere group cohesion and hierarchy.

In fact, I'd say the zeitgeist of casting everything in an identity politics narrative is in the same exact vein. It's a nice, simple, and wrong answer, but it does sort out who has committed to the group (ie religion).

I'm certainly willing to posit that women are quite often on the receiving end due to many men immediately writing them off as lower status (likely in an attempt to keep from sliding further down the scale themselves), but I'm not signing up to drink divisive kool-aid that ultimately obscures the problem.

Gamergate wasn't about telling a women that she was wrong. It was gamers reacting to disdain and collusion from the journalists that were covering their industry.

The "Gamers are Dead" articles were authored by many different journalists, not just women. Perpetuating the narrative that it was an attack on women is exactly they type of behavior that keeps Gamergate alive.

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

well thats not really similar, since in the first case, a large portion of the respondents turned out to be mansplaining (referencing her sex for example).

I don't think the situations are similar...

Also, her problem and solution are concise to verify, I have yet to see a mechanized version of Wiles' proof. (I am willing to believe it when I see that, but until then I too have a hard time accepting the proof... but of course if an oracle put a gun to my temple and I had to guess, I'd be with Wiles...)

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.

I think the easiest way to grasp this without getting into the math is to make such a counter intuitive problem intuitive again.

Let's change the number of doors !

First, I'll rephrase the original problem ================================================

You have 3 doors, 1 hides a car, all others hide goats.

You pick one door, then __ALL doors you have NOT selected AND that hide goats are revealed, except one__

Here it's only one door because we have a total of 3. But it's one door that is indeed, ALL except one, for this particular situation.

Do the same, but with 100 doors ====================================================

You have 100 doors, 1 has a car, all others have goats.

You pick one, then __ALL doors you have NOT selected AND hide goats are revealed, except one__

Now you get:

- 1 selected, 99 not selected => the car is most probably among the 99 ones.

- Monty reveals 98 goats

- 1 selected, 98 goats, 1 not selected => the car is most probably in the later one.

I think the second table is more clear than the first, since it shows that Monty Hall's choice depends on your choice. The main nuance to this problem is that Monty Hall is not picking randomly. There's always some unpicked door with a goat, and he'll pick that one to reveal. So if your original choice is door #1, then when deciding to stay or switch, you're really deciding whether go with door #1 or with the better of doors #2 and #3 (since the worse of the two doors has already been shown to you).

Yes, if you selected one of the remaining doors at random you have a 50% chance of winning.

If you always switch to the door you didn't initially choose your odds of winning rise to 2/3.

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