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• 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。

• 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。

• Behind the other two are goats.

另外兩道門後則是山羊。

• Say you pick door number one.

假設你選了一號門。

• 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。

• Marilyn is very intelligent.

Marilyn 非常聰明。

• 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.

• 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。

到底更換選擇，是好是壞？

• 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。

• Bing. Bang. Boom.

乒砰蹦！

• 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。」

• And here's why.

原因如下。

• Here are three doors.

有三道門。

• 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。

• You pick door one.

你選了一號門。

• 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？」

• Well, you should most definitely switch.

這個嘛，你當然要換。

• 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.

當然，當時有點性別歧視。

• 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.

• 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。

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

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# 三門問題：一道難倒無數人的數學題 (The math problem that stumped thousands of mansplainers)

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Mackenzie 發佈於 2020 年 01 月 21 日