Peter Donnelly: How stats fool juries

譯者: Marie Wu 審譯者: Wang-Ju Tsai

As other speakers have said, it's a rather daunting experience --

我像其他講者一樣，覺得在各位面前演講，

a particularly daunting experience -- to be speaking in front of this audience.

是一件很令人害怕的事。

But unlike the other speakers, I'm not going to tell you about

但我不像其他演講者，我不會講述有關宇宙的奧妙，

the mysteries of the universe, or the wonders of evolution,

或是講述演化的神奇之處，

or the really clever, innovative ways people are attacking

我也不會講述那些人們用來對抗世上不公不義

the major inequalities in our world.

所採行的創新招術，

Or even the challenges of nation-states in the modern global economy.

甚至那些現代國家所需要面對的全球經濟問題，

My brief, as you've just heard, is to tell you about statistics --

我會講的就是剛才主持人所提到的：統計學，

and, to be more precise, to tell you some exciting things about statistics.

正確地說，我會告訴各位統計學有趣之處，

And that's --

那就是...

(Laughter)

(笑聲)

-- that's rather more challenging

這項挑戰可不亞於在我之前

than all the speakers before me and all the ones coming after me.

或在我之後出現的講者啊！

(Laughter)

(笑聲)

One of my senior colleagues told me, when I was a youngster in this profession,

有一位前輩在我剛加入這一行時很驕傲地告訴我，

rather proudly, that statisticians were people who liked figures

他說，統計學家是一群很喜歡數字的人，

but didn't have the personality skills to become accountants.

但卻不具備得以使他們成為會計師的人際關係技巧。

(Laughter)

(笑聲)

And there's another in-joke among statisticians, and that's,

還有另一個關於統計學家的笑話：

"How do you tell the introverted statistician from the extroverted statistician?"

「要怎麼分辨一個統計學家的個性是內向還是外向？」

To which the answer is,

答案是：

"The extroverted statistician's the one who looks at the other person's shoes."

「外向的統計學家會盯著別人的鞋子看。」

(Laughter)

(笑聲)

But I want to tell you something useful -- and here it is, so concentrate now.

我要告訴各位一些有用的資訊，所以請專心一點。

This evening, there's a reception in the University's Museum of Natural History.

今晚，在學校的自然歷史博物館裡有一場招待會，

And it's a wonderful setting, as I hope you'll find,

我希望各位覺得辦得還不錯，

and a great icon to the best of the Victorian tradition.

主題是維多利亞時期的優良傳統。

It's very unlikely -- in this special setting, and this collection of people --

在這場盛會裡，聚集了很多人，

but you might just find yourself talking to someone you'd rather wish that you weren't.

但你有可能會和一個你根本不想說話的人對談，

So here's what you do.

我給各位一點建議，

When they say to you, "What do you do?" -- you say, "I'm a statistician."

當他們問說：「你做哪一行？」，你就回答：「我是個統計學家。」

(Laughter)

(笑聲)

Well, except they've been pre-warned now, and they'll know you're making it up.

除非先前就有人告訴他們這個小伎倆，否則他們不會知道你在說謊。

And then one of two things will happen.

接下來就有二種可能，

They'll either discover their long-lost cousin in the other corner of the room

他們要不是會突然發現久未聯絡的表兄弟

and run over and talk to them.

出現在大廳另一頭而趕去找他說話，

Or they'll suddenly become parched and/or hungry -- and often both --

要不就會突然覺得很渴或很餓，或是又渴又餓，

and sprint off for a drink and some food.

不得不趕緊去找些東西來吃吃或喝喝。

And you'll be left in peace to talk to the person you really want to talk to.

這時你就獲得自由了，你可以找你想要說話的人聊天了。

It's one of the challenges in our profession to try and explain what we do.

做我們這一行的人，有時很難向別人解釋我們在做什麼，

We're not top on people's lists for dinner party guests and conversations and so on.

我們也不是別人晚宴賓客或是聊天的首選名單，

And it's something I've never really found a good way of doing.

甚至我自己也覺得很難説明我的工作内容。

But my wife -- who was then my girlfriend --

但我的太太，那時還是我的女友，

managed it much better than I've ever been able to.

倒是説明得比我還清楚。

Many years ago, when we first started going out, she was working for the BBC in Britain,

多年以前，當我們開始約會時，她那時在英國的BBC(英國廣播公司)工作，

and I was, at that stage, working in America.

而我那時則在美國工作，

I was coming back to visit her.

有一次我要回來英國跟她見面。

She told this to one of her colleagues, who said, "Well, what does your boyfriend do?"

她和一個同事有了這樣的對話，對方問：「你男朋友是做什麼的？」

Sarah thought quite hard about the things I'd explained --

於是莎拉把我之前對她解釋的工作內容

and she concentrated, in those days, on listening.

再仔細地想了一遍，她在那時候都還很認真地聽我說話。

(Laughter)

(笑聲)

Don't tell her I said that.

不要告訴她我說過這件事。

And she was thinking about the work I did developing mathematical models

接著她想到我那時正在為解開演化與現代基因之謎

for understanding evolution and modern genetics.

建立一些數學模型，

So when her colleague said, "What does he do?"

所以當她的同事問道：「他是做什麼的？」

She paused and said, "He models things."

她停了好一會兒才說：「他是做模型的。」

(Laughter)

(笑聲)

Well, her colleague suddenly got much more interested than I had any right to expect

哇！她的同事突然對我所做的事感到高度興趣，

and went on and said, "What does he model?"

接著問：「他做什麼模型？」

Well, Sarah thought a little bit more about my work and said, "Genes."

莎拉想了一會兒，說：「基因。」

(Laughter)

(笑聲)

"He models genes."

「他為基因建立模型。」

That is my first love, and that's what I'll tell you a little bit about.

莎拉是我的初戀，只能說到這裡了。

What I want to do more generally is to get you thinking about

接下來，我想讓各位想想，我們所處的世界

the place of uncertainty and randomness and chance in our world,

是不是充滿了不確定性、各種隨機因素與機會？

and how we react to that, and how well we do or don't think about it.

我們的反應又是如何，有或沒有意識到這件事呢？

So you've had a pretty easy time up till now --

剛剛那幾分鐘各位都聽得很輕鬆，

a few laughs, and all that kind of thing -- in the talks to date.

有笑話還有一些別的事情，

You've got to think, and I'm going to ask you some questions.

但各位得動動腦，我要問各位幾個問題。

So here's the scene for the first question I'm going to ask you.

在我問各位第一個問題之前，

Can you imagine tossing a coin successively?

我要先請各位想像一下連續投擲幾次銅板的畫面。

And for some reason -- which shall remain rather vague --

基於一些我們還無法解釋的因素，

we're interested in a particular pattern.

統計學家對於銅板正反面出現的次序很感興趣，

Here's one -- a head, followed by a tail, followed by a tail.

例如：先是人頭、再來字、再來一次字。

So suppose we toss a coin repeatedly.

假設我們不斷重覆投擲一個銅板，

Then the pattern, head-tail-tail, that we've suddenly become fixated with happens here.

那麼「人頭、字、字」這個順序就是我們關注的重點，

And you can count: one, two, three, four, five, six, seven, eight, nine, 10 --

接下來你數：1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

it happens after the 10th toss.

在第10次投擲時才出現。

So you might think there are more interesting things to do, but humor me for the moment.

你一定在想這有什麼好玩的？但還是先遷就我一下。

Imagine this half of the audience each get out coins, and they toss them

想像一下，這一半的聽眾都拿到一個銅板，開始投擲，

until they first see the pattern head-tail-tail.

要一直投到看到「人頭、字、字」這個順序為止。

The first time they do it, maybe it happens after the 10th toss, as here.

第一輪，或許就像我剛才說的，到第十次才看到，

The second time, maybe it's after the fourth toss.

到了第二輪，或許在第四次會看到，

The next time, after the 15th toss.

第三輪，或許在第15次才看到。

So you do that lots and lots of times, and you average those numbers.

就這樣一直重覆做下去，然後把所有數字平均，

That's what I want this side to think about.

這是我要這一半聽眾去想的事情。

The other half of the audience doesn't like head-tail-tail --

另外這一半的聽眾，我就不要你們做「人頭、字、字」了，

they think, for deep cultural reasons, that's boring --

基於深厚的文化因素，你們一定覺得，那種順序太無聊了，

and they're much more interested in a different pattern -- head-tail-head.

我們想要有趣一點的順序：「人頭、字、人頭」

So, on this side, you get out your coins, and you toss and toss and toss.

所以，這邊的聽眾，你們拿起了銅板，投了再投

And you count the number of times until the pattern head-tail-head appears

把第一次出現「人頭、字、人頭」這個順序的次數記錄下來，

and you average them. OK?

再算出平均數，好嗎？

So on this side, you've got a number --

這一半的聽眾，你們有一個平均數，

you've done it lots of times, so you get it accurately --

你們投過很多次，所以一定很準確，

which is the average number of tosses until head-tail-tail.

一定可以得出一個第一次出現「人頭、字、字」的平均數。

On this side, you've got a number -- the average number of tosses until head-tail-head.

而這一半聽眾，你們也有一個關於「人頭、字、人頭」的平均數。

So here's a deep mathematical fact --

因此我們可以得出一個深奧的數學理論：

if you've got two numbers, one of three things must be true.

若你有二個數字，一定會有以下三種情形的其中之一，

Either they're the same, or this one's bigger than this one,

要不他們二個相等，要不就是這個數大於另一個數，

or this one's bigger than that one.

要不就是另一個數大於這個數。

So what's going on here?

你們覺得會是哪一種情形？

So you've all got to think about this, and you've all got to vote --

大家得好好想一想，然後我要你們投票，

and we're not moving on.

現在就想一想。

And I don't want to end up in the two-minute silence

我可不想讓接下來的二分鐘冷場，

to give you more time to think about it, until everyone's expressed a view. OK.

所以我要你們都好好想一想，每個人都得表達出自己的意見。

So what you want to do is compare the average number of tosses until we first see

我要你們比較一下，第一次出現「人頭、字、人頭」的平均投擲數，

head-tail-head with the average number of tosses until we first see head-tail-tail.

和第一次出現「人頭、字、字」的平均投擲數孰大孰小。

Who thinks that A is true --

認為A是正確的請舉手？

that, on average, it'll take longer to see head-tail-head than head-tail-tail?

也就是說，平均下來要花較多時間才會看到「人頭、字、人頭」這種順序？

Who thinks that B is true -- that on average, they're the same?

認為B是正確的請舉手？就是二者平均數相等？

Who thinks that C is true -- that, on average, it'll take less time

認為C是正確的請舉手？也就是說，平均下來，

to see head-tail-head than head-tail-tail?

要花較多時間才會看到「人頭、字、字」這種順序？

OK, who hasn't voted yet? Because that's really naughty -- I said you had to.

還有誰沒投票？你們真的很不乖哦！我說過你們都得投票啊！

(Laughter)

(笑聲)

OK. So most people think B is true.

好，大部分的人都認為B是正確的，

And you might be relieved to know even rather distinguished mathematicians think that.

如果你們知道最傑出的數學家也會這麼想，應該就會釋懷了吧！

It's not. A is true here.

事實上不是，A才是正確的，

It takes longer, on average.

平均來說會花比較多時間才會看到「人頭、字、人頭」這種順序。

In fact, the average number of tosses till head-tail-head is 10

「人頭、字、人頭」的平均投擲次數是10次，

and the average number of tosses until head-tail-tail is eight.

而「人頭、字、字」的平均投擲次數則是8次。

How could that be?

怎麼會這樣？

Anything different about the two patterns?

這二種順序有什麼不同？

There is. Head-tail-head overlaps itself.

的確有所不同，「人頭、字、人頭」的頭尾是重覆的，

If you went head-tail-head-tail-head, you can cunningly get two occurrences

所以如果你投出「人頭、字、人頭、字、人頭」，

of the pattern in only five tosses.

在這五次投擲裡你就會看到二次這種順序，

You can't do that with head-tail-tail.

「人頭、字、字」就沒有這種重覆性，

That turns out to be important.

這是很重要的一點，

There are two ways of thinking about this.

我們可以從二方面來思考這件事。

I'll give you one of them.

我們來看看其中一個面向，

So imagine -- let's suppose we're doing it.

先想像一下我們在投擲銅板，

On this side -- remember, you're excited about head-tail-tail;

記住，這一邊是支持「人頭、字、字」的，

you're excited about head-tail-head.

這一邊是支持「人頭、字、人頭」的。

We start tossing a coin, and we get a head --

我們來開始投吧！我們得到一個人頭，

and you start sitting on the edge of your seat

你緊張得坐不住了吧？

because something great and wonderful, or awesome, might be about to happen.

因為有件很神奇的事情就要發生了！

The next toss is a tail -- you get really excited.

接下來投出一個字，你真的很興奮，

The champagne's on ice just next to you; you've got the glasses chilled to celebrate.

似乎看到冰桶裡的香檳就在你身邊，只要拿起杯子就可以慶祝了！

You're waiting with bated breath for the final toss.

你現在不敢大口呼吸，

And if it comes down a head, that's great.

如果最後出現一個人頭，那就太棒了！

You're done, and you celebrate.

你成功了！你可以慶祝了！

If it's a tail -- well, rather disappointedly, you put the glasses away

但如果是字，嗯，你會很失望，只好把杯子放回去，

and put the champagne back.

把香檳退掉，

And you keep tossing, to wait for the next head, to get excited.

然後繼續投擲，等待下一個人頭出現。

On this side, there's a different experience.

而這一邊，則是完全不同的際遇，

It's the same for the first two parts of the sequence.

頭二次投擲的結果都一樣，

You're a little bit excited with the first head --

你對出現第一個人頭很興奮，

you get rather more excited with the next tail.

接下來出現一個字讓你更加興奮，

Then you toss the coin.

最後，你再投一次，

If it's a tail, you crack open the champagne.

如果是字，你就開香檳慶祝，

If it's a head you're disappointed,

如果是人頭，你就會很失望，

but you're still a third of the way to your pattern again.

但你至少不用再等下一個人頭，因為你已經投出下一輪的第一個人頭了。

And that's an informal way of presenting it -- that's why there's a difference.

這不是正規的解釋方法，但這確實是他們之間的差異所在。

Another way of thinking about it --

現在我用另一個思考面向來解釋，

if we tossed a coin eight million times,

如果我們投擲八百萬次，

then we'd expect a million head-tail-heads

「人頭、字、人頭」應該會出現一百萬次，

and a million head-tail-tails -- but the head-tail-heads could occur in clumps.

「人頭、字、字」也應該會出現一百萬次，但是「人頭、字、人頭」卻會成群地出現。

So if you want to put a million things down amongst eight million positions

如果你要把一百萬件東西分散放在八百萬件東西裡面，

and you can have some of them overlapping, the clumps will be further apart.

而某些東西是可以重疊的話，群集間的距離會更遠，

It's another way of getting the intuition.

這就是另一種思考方式。

What's the point I want to make?

我到底想要說什麼？

It's a very, very simple example, an easily stated question in probability,

這是一個非常淺顯易懂的例子，很容易說明的機率問題，

which every -- you're in good company -- everybody gets wrong.

每一個人都會在這問題上犯錯，你們也不例外。

This is my little diversion into my real passion, which is genetics.

這是我的另一個嗜好，基因。

There's a connection between head-tail-heads and head-tail-tails in genetics,

「人頭、字、人頭」或「人頭、字、字」

and it's the following.

和基因有某種關聯，

When you toss a coin, you get a sequence of heads and tails.

當你投擲一個銅板，你會丟出一連串的人頭或字，

When you look at DNA, there's a sequence of not two things -- heads and tails --

而我們來看看DNA，它的組成就不是人頭或字，

but four letters -- As, Gs, Cs and Ts.

而是這四個字母：A, G, C, T。

And there are little chemical scissors, called restriction enzymes

有一種像是剪刀的化學成份，叫做限制酶，

which cut DNA whenever they see particular patterns.

會在他們看到某種特定順序組合出現時，將DNA切斷，

And they're an enormously useful tool in modern molecular biology.

這是現代分子生物學裡的一項強大工具。

And instead of asking the question, "How long until I see a head-tail-head?" --

除了問說：「多久才會看到一個人頭、字、人頭呢？」

you can ask, "How big will the chunks be when I use a restriction enzyme

你還可以問：「若限制酶在看到G-A-A-G出現時就切斷DNA，

which cuts whenever it sees G-A-A-G, for example?

那麼G-A-A-G出現前的那一段DNA

How long will those chunks be?"

會有多長呢？」

That's a rather trivial connection between probability and genetics.

這是機率與基因間淺顯的關聯性，

There's a much deeper connection, which I don't have time to go into

但他們之間還存在著很深的關係，今天我沒有足夠的時間可以說明，

and that is that modern genetics is a really exciting area of science.

但那卻是現代基因學最令人著迷之處，

And we'll hear some talks later in the conference specifically about that.

待會兒還會有其他講者就這個主題再詳細說明。

But it turns out that unlocking the secrets in the information generated by modern

我們發現，若要公開現代實驗科技產生的資訊的祕密，

experimental technologies, a key part of that has to do with fairly sophisticated --

就不得不提到一個很複雜的關鍵因素，

you'll be relieved to know that I do something useful in my day job,

各位會很高興知道我的工作還是有些用途的，

rather more sophisticated than the head-tail-head story --

這可比丟銅板複雜多了，

but quite sophisticated computer modelings and mathematical modelings

牽涉到複雜的電腦模型、數學模型

and modern statistical techniques.

和現代的統計技巧。

And I will give you two little snippets -- two examples --

我會給各位二個提示，也就是二個例子，

of projects we're involved in in my group in Oxford,

那是我在牛津的小組所參與的專案，

both of which I think are rather exciting.

這二個專案都很有趣。

You know about the Human Genome Project.

各位都知道人體基因元計畫，

That was a project which aimed to read one copy of the human genome.

這個專案的目標是要訂出人體的基因序列，

The natural thing to do after you've done that --

而接下來很自然就產生另一個專案，

and that's what this project, the International HapMap Project,

叫做國際單體型測繪計畫，

which is a collaboration between labs in five or six different countries.

由五、六個不同國家的實驗室共同合作執行。

Think of the Human Genome Project as learning what we've got in common,

人體基因計畫旨在瞭解人類基因的共通性，

and the HapMap Project is trying to understand

而國際單體型測繪計畫就是要去瞭解

where there are differences between different people.

不同人之間的基因有何相異之處。

Why do we care about that?

為什麼我們要知道這些？

Well, there are lots of reasons.

嗯，有許多原因，

The most pressing one is that we want to understand how some differences

最主要的原因是我們想要瞭解，為何基因的不同

make some people susceptible to one disease -- type-2 diabetes, for example --

會使某些人容易得某種疾病，例如第二型糖尿病，

and other differences make people more susceptible to heart disease,

而另一種基因的差異則會讓人容易產生心臟病，

or stroke, or autism and so on.

或是中風、自閉症等疾病。

That's one big project.

這是一項大型專案，

There's a second big project,

還有另一項大型專案，

recently funded by the Wellcome Trust in this country,

是由英國的衛爾康基金會出資運作，

involving very large studies --

要進行非常大規模的研究，

thousands of individuals, with each of eight different diseases,

針對數千人進行調查，主要研究八種不同的疾病，

common diseases like type-1 and type-2 diabetes, and coronary heart disease,

像是第一型與第二型糖尿病、冠狀動脈心臟病、

bipolar disease and so on -- to try and understand the genetics.

躁鬱症等，要研究病患的基因序列，

To try and understand what it is about genetic differences that causes the diseases.

試圖找出病患的基因有何不同之處。

Why do we want to do that?

為什麼要做這個研究？

Because we understand very little about most human diseases.

因為我們對於大部分的疾病都瞭解不多，

We don't know what causes them.

我們不知道人們是怎麼染病的，

And if we can get in at the bottom and understand the genetics,

但如果我們能知道最基本的基因差異，

we'll have a window on the way the disease works,

我們或許可一窺疾病運作之祕密，

and a whole new way about thinking about disease therapies

並找出治療疾病的全新方法，

and preventative treatment and so on.

加以預防。

So that's, as I said, the little diversion on my main love.

這就是我所說的我的第二個嗜好。

Back to some of the more mundane issues of thinking about uncertainty.

現在我們回歸到現實面，來看看剛才我所說的不確定性，

Here's another quiz for you --

我要問各位另一個問題，

now suppose we've got a test for a disease

假設我們針對某項疾病研發了某種測試技術，

which isn't infallible, but it's pretty good.

雖然不是萬無一失，但尚稱良好，

It gets it right 99 percent of the time.

大約有99%的準確度。

And I take one of you, or I take someone off the street,

我請在座的一位或是街上隨便找個人，

and I test them for the disease in question.

來用這種技術檢驗是否得到了這種疾病，

Let's suppose there's a test for HIV -- the virus that causes AIDS --

假設是HIV病毒的檢驗試劑好了，就是愛滋病毒的檢驗試劑，

and the test says the person has the disease.

報告出來說這個人得病了。

What's the chance that they do?

那麼這個人真正得病的機率是多少？

The test gets it right 99 percent of the time.

試劑有99%的準確度，

So a natural answer is 99 percent.

大家自然會說這個人99%得了愛滋病，

Who likes that answer?

但誰會滿意這種答案？

Come on -- everyone's got to get involved.

拜託，每一個人都要參與啊...

Don't think you don't trust me anymore.

不要不信任我嘛...

(Laughter)

(笑聲)

Well, you're right to be a bit skeptical, because that's not the answer.

抱持懷疑態度是對的，因為這個答案不對，

That's what you might think.

你一定會這樣想。

It's not the answer, and it's not because it's only part of the story.

這個答案不對，但不是因為這個原因，

It actually depends on how common or how rare the disease is.

而是要看這種疾病的普遍程度來決定，

So let me try and illustrate that.

我來為各位解說一下。

Here's a little caricature of a million individuals.

假設這裡有一百萬人，

So let's think about a disease that affects --

我們來假設一種很罕見的疾病，

it's pretty rare, it affects one person in 10,000.

得病機率只有萬分之一，

Amongst these million individuals, most of them are healthy

所以在這一百萬人裡，大部分的人都是健康的，

and some of them will have the disease.

只有少數人會得病。

And in fact, if this is the prevalence of the disease,

如果這種疾病流行起來，

about 100 will have the disease and the rest won't.

也只有100個人會生病，其餘的人則不會生病。

So now suppose we test them all.

假設我們對全部的人做檢驗，

What happens?

會有什麼結果？

Well, amongst the 100 who do have the disease,

在這100個得病的人裡，

the test will get it right 99 percent of the time, and 99 will test positive.

以這99%準確度的試劑來檢驗，會有99個人呈陽性反應，

Amongst all these other people who don't have the disease,

而在其他沒有得病的人裡，

the test will get it right 99 percent of the time.

這個試劑的準確度還是99%，

It'll only get it wrong one percent of the time.

有1%的機會會出錯，

But there are so many of them that there'll be an enormous number of false positives.

但因為人數很多，所以假陽性的數量也就跟著變多。

Put that another way --

換個方式來說，

of all of them who test positive -- so here they are, the individuals involved --

在所有呈陽性反應的人裡，

less than one in 100 actually have the disease.

100個人裡只有不到一個人是真正染病的。

So even though we think the test is accurate, the important part of the story is

即使我們認為這種試劑很準確，

there's another bit of information we need.

但重點是我們還需要其他資訊來確認，

Here's the key intuition.

我們需要敏銳的洞察力。

What we have to do, once we know the test is positive,

一旦我們發現有人呈陽性反應，

is to weigh up the plausibility, or the likelihood, of two competing explanations.

我們就該去權衡二種不同解釋之間的可信度或可能性，

Each of those explanations has a likely bit and an unlikely bit.

每一種解釋都有可能的一面，也有不可能的一面。

One explanation is that the person doesn't have the disease --

你可以說這個人沒有染病，

that's overwhelmingly likely, if you pick someone at random --

這很有可能，因為你是隨機取樣的，

but the test gets it wrong, which is unlikely.

也就是說試劑出錯了，但這種機會不大。

The other explanation is that the person does have the disease -- that's unlikely --

你也可以說這個人確實是染病了，但這種疾病發生的機率很小，

but the test gets it right, which is likely.

試劑確實是準確的，這確實很有可能發生。

And the number we end up with --

最後我們得到的數據

that number which is a little bit less than one in 100 --

是比1%還稍小一點，

is to do with how likely one of those explanations is relative to the other.

也就是這二種解釋的發生的比例（幾乎是一比一百），

Each of them taken together is unlikely.

二者同時發生的可能性不高。

Here's a more topical example of exactly the same thing.

這裡還有一個很類似的例子，

Those of you in Britain will know about what's become rather a celebrated case

各位住在英國都知道一個很著名的案例，

of a woman called Sally Clark, who had two babies who died suddenly.

有個叫做莎莉．克拉克的婦人，她的二個嬰孩同時猝死，

And initially, it was thought that they died of what's known informally as "cot death,"

一開始大家都以為是猝死症，

and more formally as "Sudden Infant Death Syndrome."

正式名稱為嬰兒猝死症候群。

For various reasons, she was later charged with murder.

基於許多不同理由，莎莉被控謀殺，

And at the trial, her trial, a very distinguished pediatrician gave evidence

而在審判中，一位很知名的小兒科醫生做證說明，

that the chance of two cot deaths, innocent deaths, in a family like hers --

在他們這種家庭裡，也就是專業人士又不抽煙的家庭，

which was professional and non-smoking -- was one in 73 million.

二個嬰兒同時猝死的機率大約是7千3百萬分之一。

To cut a long story short, she was convicted at the time.

長話短說，她後來被定罪了。

Later, and fairly recently, acquitted on appeal -- in fact, on the second appeal.

但是後來，也就是最近的事，她在第二次上訴後獲判無罪。

And just to set it in context, you can imagine how awful it is for someone

請各位想想一下，如果有人失去了一個孩子，

to have lost one child, and then two, if they're innocent,

或甚至二個孩子，以清白之身卻被判謀殺定罪，

to be convicted of murdering them.

這是多麼殘忍的一件事。

To be put through the stress of the trial, convicted of murdering them --

就只為了紓解法庭所承擔的壓力，

and to spend time in a women's prison, where all the other prisoners

就把一個人以謀殺犯定罪，把她關進女子監獄，

think you killed your children -- is a really awful thing to happen to someone.

那裡的犯人都認為你殺了自己的小孩，這真是一件悲慘絕倫的事。

And it happened in large part here because the expert got the statistics

這個錯誤最主要是因為專家在二個不同的方面，

horribly wrong, in two different ways.

大錯特錯地引用了統計數據所造成。

So where did he get the one in 73 million number?

他怎麼得出7千3百萬分之一這個數據的？

He looked at some research, which said the chance of one cot death in a family

他看了某些研究文獻，裡頭說像莎莉這種家庭，

like Sally Clark's is about one in 8,500.

一個嬰孩猝死的機率約為8千5百分之一。

So he said, "I'll assume that if you have one cot death in a family,

他說：「先假設家裡已經有一個嬰孩猝死了，

the chance of a second child dying from cot death aren't changed."

第二個嬰孩猝死的機率與第一個相同。」

So that's what statisticians would call an assumption of independence.

這就是統計學所引用的獨立性假設，

It's like saying, "If you toss a coin and get a head the first time,

就好像是說：「若你第一次丟銅板得到一個人頭，

that won't affect the chance of getting a head the second time."

並不會影響你第二次再丟銅板，得到人頭的機率。」

So if you toss a coin twice, the chance of getting a head twice are a half --

所以，如果你丟一個銅板二次，那麼連丟二次都得到人頭的機率，

that's the chance the first time -- times a half -- the chance a second time.

就是第一次丟出銅板的機率，乘上第二次的機率（1/2＊1/2）。

So he said, "Here,

所以他才會說：「讓我們假設一下，

I'll assume that these events are independent.

假設這二個事件是獨立的，

When you multiply 8,500 together twice,

將8千5百乘二次，

you get about 73 million."

就會得到7千3百萬。」

And none of this was stated to the court as an assumption

但是這個前題假設並沒有在法庭上說明，

or presented to the jury that way.

也沒有對陪審團說明。

Unfortunately here -- and, really, regrettably --

很不幸也很遺憾的是，

first of all, in a situation like this you'd have to verify it empirically.

首先，像這種情形就該憑經驗先進行驗證，

And secondly, it's palpably false.

第二，這很明顯就是錯的。

There are lots and lots of things that we don't know about sudden infant deaths.

我們對於嬰兒猝死症所知真的不多，

It might well be that there are environmental factors that we're not aware of,

有可能是因為某些我們並不瞭解的環境因素所造成，

and it's pretty likely to be the case that there are

而這個個案更有可能是因為

genetic factors we're not aware of.

我們所不知道的基因缺陷所造成，

So if a family suffers from one cot death, you'd put them in a high-risk group.

所以當某個家庭裡有一個嬰孩猝死時，他們就算是高風險的家庭，

They've probably got these environmental risk factors

有可能存在著某些環境風險因子，

and/or genetic risk factors we don't know about.

或是有我們不知道的基因缺陷，或是二者都有。

And to argue, then, that the chance of a second death is as if you didn't know

真要計較起來，若完全不考慮這些因素，

that information is really silly.

就來計算第二個嬰孩的猝死機率，是很可笑的。

It's worse than silly -- it's really bad science.

甚至比可笑還糟，簡直就是爛透了的科學證據。

Nonetheless, that's how it was presented, and at trial nobody even argued it.

但這個數據就這樣被當成呈堂證供，法庭上也沒有人懷疑，

That's the first problem.

這就是第一個問題。

The second problem is, what does the number of one in 73 million mean?

第二個問題是，7千3百萬分之一代表著什麼？

So after Sally Clark was convicted --

當莎拉．克拉克被定罪之後，

you can imagine, it made rather a splash in the press --

你可以想見又在媒體上掀起了多大的波瀾，

one of the journalists from one of Britain's more reputable newspapers wrote that

英國某家聲譽卓著的報社記者

what the expert had said was,

就引用專家的話說：

"The chance that she was innocent was one in 73 million."

「莎拉清白的機率是7千3百萬之一」

Now, that's a logical error.

這犯了邏輯上的錯誤，

It's exactly the same logical error as the logical error of thinking that

這個錯誤就和我們剛才所談到的疾病測試一樣，

after the disease test, which is 99 percent accurate,

同樣具有邏輯上的錯誤，有人會以為試劑有99%的準確度，

the chance of having the disease is 99 percent.

得到這種疾病的機率就是99%。

In the disease example, we had to bear in mind two things,

在疾病試劑的例子裡，我們得記住二件事，

one of which was the possibility that the test got it right or not.

其中之一是試劑的準確度，

And the other one was the chance, a priori, that the person had the disease or not.

另一個則是人們染病的先驗機率。

It's exactly the same in this context.

這和這個案子是一樣的情形，

There are two things involved -- two parts to the explanation.

這個案子也有二種解釋的方向，

We want to know how likely, or relatively how likely, two different explanations are.

我們得釐清這二種解釋發生的機率。

One of them is that Sally Clark was innocent --

第一種解釋是莎拉是清白的，

which is, a priori, overwhelmingly likely --

這在先驗機率上是很有可能的，

most mothers don't kill their children.

大部分的母親都不會殺害自己的小孩。

And the second part of the explanation

這種解釋的第二個部分是，

is that she suffered an incredibly unlikely event.

莎拉的遭遇真的是令人難以置信，

Not as unlikely as one in 73 million, but nonetheless rather unlikely.

雖然機率不像7千3百萬分之一那麼小，但確實是不太可能。

The other explanation is that she was guilty.

第二種解釋是莎拉確實是有罪的，

Now, we probably think a priori that's unlikely.

就先驗機率來說，這不太可能，

And we certainly should think in the context of a criminal trial

而且我們當然認為在這起犯罪的審判中，

that that's unlikely, because of the presumption of innocence.

一開始就要假設被告是無罪的，所以說莎拉有罪並不太可能。

And then if she were trying to kill the children, she succeeded.

但若她真的想要殺害小孩，她也成功了，

So the chance that she's innocent isn't one in 73 million.

所以她是清白的機率就不是7千3百萬分之一，

We don't know what it is.

沒人知道是多少，

It has to do with weighing up the strength of the other evidence against her

這個機率反而是和其他對她不利的證據和統計數據有關，

and the statistical evidence.

得視證據強度而定。

We know the children died.

我們只知道嬰孩死了，

What matters is how likely or unlikely, relative to each other,

重要的是要找出這二種解釋

the two explanations are.

之間的關聯性。

And they're both implausible.

這二種解釋都無法使人信服，

There's a situation where errors in statistics had really profound

有時統計上的錯誤所造成的影響，

and really unfortunate consequences.

是很深遠且會造成不幸的。

In fact, there are two other women who were convicted on the basis of the

事實上，還有有二位婦女因為這位小兒科醫生的證詞，

evidence of this pediatrician, who have subsequently been released on appeal.

而被判有罪，但在後來的上訴後又被無罪釋放。

Many cases were reviewed.

以往許多案子又被大家拿出來討論，

And it's particularly topical because he's currently facing a disrepute charge

因此又掀起一波話題，因為這個醫生正被英國醫藥委員會

at Britain's General Medical Council.

控以不名譽的罪名。

So just to conclude -- what are the take-home messages from this?

結論是，這個故事帶給我們什麼樣的啟示？

Well, we know that randomness and uncertainty and chance

我們知道隨機、不確定性及機率等，

are very much a part of our everyday life.

都是我們日常生活的一部分，

It's also true -- and, although, you, as a collective, are very special in many ways,

而雖然我們每一個人都與眾不同，

you're completely typical in not getting the examples I gave right.

但就我所提出的問題沒有做出正確的回答這件事，這也是常態

It's very well documented that people get things wrong.

很多過去的記錄顯示人們確實有時會做出錯誤判斷。

They make errors of logic in reasoning with uncertainty.

在不確定的情況下，人們會犯下合理的邏輯錯誤。

We can cope with the subtleties of language brilliantly --

人類可以運用精巧的語言，

and there are interesting evolutionary questions about how we got here.

也能對人類本身的進化提出有趣的問題，

We are not good at reasoning with uncertainty.

但我們就是不擅長預測不確定性，

That's an issue in our everyday lives.

這是我們每天都必須面對的問題。

As you've heard from many of the talks, statistics underpins an enormous amount

如同其他講者所提到的，統計學是其他許多科學研究的基礎，

of research in science -- in social science, in medicine

不管是社會科學還是醫學都一樣，

and indeed, quite a lot of industry.

還包括大部分的工業，

All of quality control, which has had a major impact on industrial processing,

那些品質控制理論，對於工業流程管制具有重大的影響，

is underpinned by statistics.

都是靠統計學做基礎。

It's something we're bad at doing.

但這卻是我們所不擅長的事，

At the very least, we should recognize that, and we tend not to.

至少我們該承認這一點，但我們卻沒人願意承認。

To go back to the legal context, at the Sally Clark trial

回到法律層面，回到莎拉的案子上，

all of the lawyers just accepted what the expert said.

所有的律師都接受這位專家的說法，

So if a pediatrician had come out and said to a jury,

所以如果有一位小兒科醫生站出來對陪審團說，

"I know how to build bridges. I've built one down the road.

「我知道如何建造橋樑，我已經在這條路上蓋了一座橋，

Please drive your car home over it,"

請把你的車開上橋回家吧！」

they would have said, "Well, pediatricians don't know how to build bridges.

陪審團會說：「小兒科醫生不是建造橋樑的專家，

That's what engineers do."

這是工程師該做的事。」

On the other hand, he came out and effectively said, or implied,

而在另一方面，這位醫師卻站出來發表專業意見，甚至暗示：

"I know how to reason with uncertainty. I know how to do statistics."

「我知道如何解釋不確定性，我瞭解統計方法。」

And everyone said, "Well, that's fine. He's an expert."

然後大家附和：「對，他是專家。」

So we need to understand where our competence is and isn't.

我們必須瞭解每一個人的專長為何，

Exactly the same kinds of issues arose in the early days of DNA profiling,

就像早期我們在描繪DNA時所引發的爭議一樣，

when scientists, and lawyers and in some cases judges,

有些科學家、律師，或甚至法官，

routinely misrepresented evidence.

都曾不斷地錯誤解讀他們所看到的證據。

Usually -- one hopes -- innocently, but misrepresented evidence.

他們通常不是故意的，我們也衷心希望不是，但卻還是扭曲了證據的本質。

Forensic scientists said, "The chance that this guy's innocent is one in three million."

鑑識專家說：「這傢伙清白的機率是三百萬分之一。」

Even if you believe the number, just like the 73 million to one,

即使各位相信這個數據，就像先前提到的7千3百萬分之一那樣，

that's not what it meant.

但這數據的意義並非如此，

And there have been celebrated appeal cases

在英國和其他地方，

in Britain and elsewhere because of that.

都有因為誤解數據而誤判的有名案例。

And just to finish in the context of the legal system.

再讓我們回過頭來看看我們的法庭，

It's all very well to say, "Let's do our best to present the evidence."

你大可以說：「我們得盡力將證據的原貌呈現出來。」

But more and more, in cases of DNA profiling -- this is another one --

但是在DNA描繪的案例裡，一次又一次我們看到，這是另一個案例，

we expect juries, who are ordinary people --

我們期望陪審團這些一般大眾，

and it's documented they're very bad at this --

這些本來就對統計不甚在行在大眾，

we expect juries to be able to cope with the sorts of reasoning that goes on.

我們竟然期望他們能解讀這些統計數據。

In other spheres of life, if people argued -- well, except possibly for politics --

但在現實生活裡，如果有人爭論...嗯，除了政治話題之外，

but in other spheres of life, if people argued illogically,

在現實生活裡，如果有人不合邏輯地爭論，

we'd say that's not a good thing.

我們會說這樣做不好，

We sort of expect it of politicians and don't hope for much more.

我們會認為這是政客做的事，因為我們對政客沒什麽太大的期望。

In the case of uncertainty, we get it wrong all the time --

在面對不確定的事情時，我們總是犯錯，

and at the very least, we should be aware of that,

但是至少我們應該知道我們會犯錯。

and ideally, we might try and do something about it.

並希望我們能嘗試去減少錯誤的發生。

Thanks very much.

謝謝各位！