Fri, Jul 18, 2008 page20

• WEEKEND SCIENCE 週末科學版: And behind door number one is…a goat! 一號門後面是…一隻山羊！

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  Today we’re going to take a look at a mathematics problem that has embarrassed a number of famous mathematicians. To make sure we don’t suffer the same fate, first we’re going to do a probability experiment.

  What you will need:

  A friend

  

  Goats rest in the shade on Cimei Island, Penghu County, Taiwan. 澎湖七美島上一群山羊在涼亭內休息。照片：台北時報

  PHOTO: MICHAEL KEARNEY, TAIPEI TIMES

  Two cutout pictures of goats

  A cutout picture of a car

  Three “doors,” or any objects that can be used to hide the cutouts

  ▲You can be the host of the show, and have your friend play the unsuspecting contestant. Here’s the problem:

  There are three doors. A car is behind one of the doors, and a goat is behind each of the remaining doors. As the host, I’m going to let you pick one of the doors. But before you open it, I’m going to open another door to reveal a goat. You then get to choose if you want to open the door you originally picked or switch to the other door. Your goal is to open the door with the car behind it (Unless you really like goats!).

  First have your friend try the experiment 10 to 20 times without switching doors. Then have him try it another 10 to 20 times, making sure to switch doors each time. Record his ratio of wins to losses when he doesn’t switch doors and his ratio of wins to losses when he does switch doors. Is it to your friend’s advantage to switch doors or to stick with his original choice?

  (MICHAEL KEARNEY, STAFF WRITER)

  今天我們要來看一個讓很多知名數學家難堪的數學問題。為了確保我們不會陷入同樣的窘境，我們要先做一個機率的實驗。

  所需材料︰

  一個朋友

  兩張山羊圖案剪紙

  一張汽車圖案剪紙

  三扇「門」，或任何可以隱藏剪紙的物體

  ▲你可以當這個表演的主持人，然後請朋友當不知情的玩家，問題如下：

  這裡有三扇門，其中一扇門背後有一輛車，剩下兩扇門後各藏有一隻山羊。身為主持人，我要讓你挑一扇門，但在你打開之前，我會先打開一扇有山羊的門，然後你可以選擇要開原本選的那扇門，或換開另外那扇門。你要打開後面有車子的門才算成功(除非你真的很喜歡山羊！）。

  首先，請你的朋友嘗試十到二十次不換門的實驗。然後再請他試十到二十次，這回每次都要選擇另一扇門。記下他這兩項實驗的成功和失敗比，結果是換門、還是堅持原本的選擇對你朋友比較有利？

  （翻譯︰袁星塵）

  For your information

  VOCABULARY 今日單字

  1. embarrass v.t. 使難堪 (shi3 nan2 kan1) 2. probability n. 機率 (ji1 lü4) 3. unsuspecting adj. 不知情的 (bu4 zhi1 qing2 de5) 4. ratio n. 比率 (bi3 lü4) 5. sensation n. 轟動 (hong1 dong4) 6. prominent adj. 著名的 (zhu4 ming2 de5) 7. lament v.i./v.t. 痛惜 (tong4 xi2) 8. reputation n. 名聲 (ming2 sheng1)

  For your information

  BONUS 你知道嗎

  The Monty Hall Problem was a sensation in the US in the early 1990s after Marilyn vos Savant, who became famous for being listed in the Guinness Book of World Records Hall of Fame as having the highest IQ in the world, featured it in her weekly column in Parade magazine. A reader to Savant’s column wrote a letter asking her the question, and she answered it correctly (although there was some controversy over the wording of the reader’s question). About 10,000 people sent letters in response, most of them highly critical of Savant’s mathematical abilities. Many of the people who sent letters were prominent mathematicians lamenting the state of mathematics in the US. But Savant defended her original answer and encouraged teachers to try the experiment we did today in their classrooms. Most people came up with the same result as Savant, and in the end her reputation was saved. 因榮登《金氏世界紀錄》世界IQ最高的人而聞名的瑪莉蓮．福斯．薩凡特，一九九０年代初期在《展示雜誌》的每週專欄中刊載了蒙特霍爾問題後，蒙特霍爾問題便在美國造成一陣轟動。 一個薩凡特專欄的讀者寫信去問她這個問題，而她正確回答出問題（雖然這個讀者的問題在用字遣詞上有一些爭議）。 大約有一萬名民眾投書回應，其中多半大力批評薩凡特的數學能力。許多投書的民眾都是相當著名的數學家，他們為美國人的數理能力感到相當憂心。 但薩凡特堅持她的答案是對的，並且鼓勵老師在課堂上進行我們今天做的這個實驗。多數人得到和薩凡特一樣的結果，最後保住了她的聲譽。

  For your information

  WHAT’S THE SCIENCE? 科學原理

  The math problem you tried today is known as the Monty Hall Problem. It’s named after Monty Hall, who was the host of a TV show called Let’s Make a Deal. Hall presented contestants with a similar problem on his show. When your friend first picked a door, he had a one in three chance of picking the door with the car. As host you then had to open another door that hid a goat. That left two doors. Many people mistakenly think that with only two doors left, and a car behind one of them, the door they originally picked has a 50 percent chance of having a car behind it. But there’s still only a one in three chance it contains a car, while there’s a two out of three chance that the other door has a car behind it. Take a look at the illustrations to the right to get a better understanding. Once you understand how the problem works, see if you can fool your parents or teachers. 你今天試驗的這個數學問題是有名的「蒙特霍爾問題」，它是以電視節目《咱們打個商量》的主持人蒙特．霍爾命名。霍爾在節目中也以類似的方式請來賓做出選擇。 當你朋友一開始挑門時，他有三分之一的機會挑到有車子的門，你這個主持人接著得掀開一扇門後是山羊的門。於是還剩下兩扇門。許多人會誤以為現在只剩下兩扇門，其中一扇門後是汽車，所以他們原先選的那扇門後面有百分之五十的機率會是汽車。但其實結果那扇門後是汽車的機率仍然只有三分之一，另外那扇沒開的門後反而有三分之二的機率會是汽車。 看看右方的圖示，你可以看到更清楚的說明。一旦你了解這個問題的原理後，你可以試試看能不能騙過你的父母或老師。
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