Name: 三門問題：一道難倒無數人的數學題 (The math problem that stumped thousands of mansplainers)
Uploaded: 2020-01-21T21:01:10.000Z
Duration: 4 min 58 s

I hate math, but there's this statistics problem that has me geeking out.

我討厭數學，但有個統計問題讓我非常著迷。

程度副詞

It's a question that seems really, really simple, but it's stumped me.

這是個非常簡單的問題，但卻難倒了我。

And not just me, it has stumped thousands of people around the world including math professors and leading statisticians.

而且不只有我而已，全球數以千計的人，包含數學教授與統計學專家們都被難倒了。

But before we dive in, I'm going to introduce you to Zachery Crockett.

但在我們深入討論前，我想要介紹你認識 Zachery Crockett。

He first introduced me to the puzzle, and I called him up to talk about it.

他是告訴我這謎題的人，於是我打給他來討論。

My name is Zachery Crockett, I'm a writer for Priceonomics.

我是 Zachery Crockett，我是幫 Priceonomics 撰文的作家之一。

Zachery and his girlfriend were stumped by the problem too.

Zachery 和他女友也被這謎題難倒了。

We just sat there debating the answer to this problem for two hours, and I don't think any of us really understood it.

花了兩小時辯論這謎題的答案，但我覺得其實我們兩個都沒有真的理解它。

The puzzle we were all stumped by is called the Monty Hall Problem, named after the host of the game show that made it famous.

這道難倒所有人的謎題叫做蒙提霍爾問題，以讓這謎題出名的節目主持人而命名的。

You see the problem goes like this: there's a brand new car behind one of three doors.

題目是像這樣的，有三道門，其中一道的後方有一輛新車。

Monty then shows you the goat behind one of the doors you didn't choose - say, door three.

蒙提則會打開其中一道，後面是山羊，你沒選擇的門，假設是三號門。

Now here's the question: you're allowed to change your answer to Door two.

問題來了，你可以改變選擇到二號門。

Do you switch? Or do you stick with your original choice?

你會換嗎？還是你會堅持自己原本的選擇？

But yeah, I got interested in the Monty Hall Problem, did a little research, and then I found out there was this whole second angle to the story.

對蒙提霍爾問題起興趣後，我做了些研究，發現故事後還有一些秘辛。

You've never met a man who feared you a little bit because he thought you were much brighter than he was?

你有沒有遇過某個男人，因妳比他聰穎得多而對妳敬而遠之的？

That's Marilyn Vos Savant in 1988 being interviewed by Joe Franklyn.

這是 1988 年時，Marilyn Vos Savant 被 Joe Franklyn 訪問的片段。

Well, yeah maybe I've met a man or two, maybe a couple a hundred like that.

或許有一、兩個男人有這個情況，好啦，很多男人都這樣。

In fact, back when the Guinness World Records actually kept track of this, she was the world's highest IQ.

事實上，當金氏世界紀錄開始紀錄 IQ 時，她是世上 IQ 最高的人。

She now writes for Parade Magazine and has for the last 20 years.

到現在為止，她已經幫 Parade 雜誌寫作寫了 20 年。

So the premise of the column was of course, like, here is the person with the world's highest IQ, here to answer your challenging math questions.

這個專欄上頭寫著，世上 IQ 最高的人會在這裡回答你的數學問題。

This brings us to September 9th, 1990, when a reader submitted to Marilyn... the Monty Hall Problem.

這帶我們回到 1990 年 9 月 9 號，一名讀者向 Marilyn 提出蒙提霍爾的謎題。

Is it to your advantage to switch your choice of doors?

Now, if you're like me the obvious answer is no.

如果你跟我一樣，很明顯會選擇不換門。

There are two doors so the chance of getting a car and not a goat is 50/50.

剩下兩道門，所以得到車子的機率是一半一半。

But that's the wrong answer, and Marilyn knew that.

但這是錯的，Marilyn 當時也知道。

She replied: “Yes; you should switch.”

她當時回覆：「沒錯，你該換。」

There's a goat behind two and a car behind one.

兩道門後面是山羊，一道後面是車子。

In a blind test, you're more likely to pick a goat than a car.

在盲選的狀態下，比起車子，你更容易選到山羊。

In fact, you're two-thirds likely, so let's use that as our main scenario.

事實上，你有三分之二的機會選到山羊，那我們把這件事當成主要的情景。

So now, Monty Hall, who knows what's behind all the doors, is forced to reveal a goat regardless of the door you pick.

現在，知道所有門後面有什麼的蒙提霍爾，一定得要打開一道後面有山羊的門。

Since, in the most probable scenario you've also picked a goat, the only door left is the one with the car.

最可能的情況是，你一開始選了有山羊的門，所以剩下的那道門，就是有車子的那道。

So, now Monty Hall asks, "Would you rather keep the door you've picked, or would you switch?”

所以當蒙提霍爾問「你要保留你的選擇，還是改變？」

If you do you get the car two-thirds of the time.

如果你換，你會有三分之二的機會拿到車子。

Turns out when Marilyn correctly answered the Monty Hall Problem, she received thousands of letters from across the world telling her she was flat out wrong.

結果，當 Marilyn 正確地回答蒙提霍爾問題時，她收到數千封來自世界各地的信，說她完全錯了。

I think part of her was a little bit surprised that she received 10,000 letters calling her an idiot.

我想她當時應該有點驚訝，竟然會收到一堆信件說她是笨蛋。

There was, without a doubt, a little bit of sexism at play here.

Not only was her answer right, it wasn't anything new.

她不但答對了，而且這答案也不是什麼新穎的觀點。

The first time the Monty Hall problem was really conceived was in 1975.

蒙提霍爾問題第一次是在 1975 年出現的。

So, this guy named Steve Selvin at Berkeley presented this problem in The American Statistician.

柏克萊大學的 Steve Selvin 在美國統計學會會刊中提出這個問題。

He contested that the odds were two out of three, and no one argued with him.

他提出機率是三分之二，而沒人與他爭論。

You know, over the next 15 years multiple other academics reiterated the same problem, and no one ever told them that they were wrong.

接下來的 15 年間，許多學者也反覆提出這個結果，從來沒人站出來說他們是錯的。

Then in 1990 Marilyn answered the same question correctly, and people went bananas.

而 1990 年 Marilyn 答對這個問題時，人們卻抓狂了。

Marilyn ended up tallying up what percentage of the 10,000 responses claimed she was wrong.

Marilyn 後來計算了這 10,000 個說她錯了的回覆者佔了多少百分比。

Only 8% of readers actually agreed with her, and after subsequent columns, she was able to raise that to 56%.

只有 8% 的讀者同意她的觀點，而在之後的專欄，她讓同意度升高至 56%。

And among academics: It was 35% among academics initially supporting it.

而在學者界中，一開始時有 35% 的人支持此論點。

Around 70% of academics ultimately decided to agree with her.

最後，七成的學者都決定同意她。

The only way she managed to get people on her side was by asking them to do the experiment themselves.

她唯一使人們改變想法的方式，就是叫大家自己實驗看看。

Elementary, middle school, and high school teachers from all over the country wrote in, astounded that their students were able to prove her right.

全國的國小、中學甚至高中老師都投書，驚訝地說他們的學生證明了 Marilyn 是對的。

It's easy now to do a simple google search of this little sucker of a problem and get a million explanations on how to arrive at the right answer.

現在只要簡單的 Google 一下，就能找到 100 萬種解釋這機車問題的方式。

There's something about this problem that really strikes a chord with, not only statisticians, but just everyday problem solvers and people.

這問題有個神奇的魔力，不但讓統計學家們著迷，也讓所有人著迷。