Can you solve the false positive riddle? - Alex Gendler

譯者: Lilian Chiu 審譯者: Helen Chang

Mining unobtainium is hard work.

挖掘「難得元素」是很艱辛的工作。

The rare mineral appears in only 1% of rocks in the mine.

在礦坑中，只有 1% 的石頭中 會出現這種稀有的礦物。

But your friend Tricky Joe has something up his sleeve.

但你的朋友「狡猾老喬」 有個秘密武器。

The unobtainium detector he’s been perfecting for months is finally ready.

他花了數個月把難得元素偵測器 做到完美，現在終於準備好了。

The device never fails to detect unobtainium if any is present.

只要有難得元素出現， 這個裝置一定會偵測到它。

Otherwise, it’s still highly reliable,

其他時候，它也仍是非常可靠的，

returning accurate readings 90% of the time.

90% 的時候呈現出的讀數都正確。

On his first day trying it out in the field,

在他做實地測試的第一天，

the device goes off, and Joe happily places the rock in his cart.

這個裝置響了，老喬很開心地 把石頭放到他的小推車上。

As the two of you head back to camp where the ore can be examined,

你和老喬兩人回到營地， 在那裡可以做礦石的檢驗，

Joe makes you an offer:

老喬向你提出了一項提議：

he’ll sell you the ore for just $200.

他把礦石賣給你，只要 $200。

You know that a piece of unobtanium that size would easily be worth $1000,

你知道那種尺寸的難得元素 很容易就能賣到 $1000，

but any other minerals would be effectively worthless.

但若是任何其他礦物質則毫無價值。

Should you make the trade?

你應該接受這項交易嗎？

Pause here if you want to figure it out for yourself.

你可以在這裡暫停， 自己試著解解看。

Answer in: 3

答案即將公佈，三。

Answer in: 2

二。

Answer in: 1

一。

Intuitively, it seems like a good deal.

直覺上，這似乎是個很好的交易。

Since the detector is correct most of the time,

既然偵測器大部分的時候是對的，

shouldn’t you be able to trust its reading?

你不該相信它的讀數嗎？

Unfortunately, no.

不幸的是，不。

Here’s why.

原因如下：

Imagine the mine has exactly 1,000 pieces of ore.

假設礦坑中有剛好 1000 個礦石。

An unobtainium rarity of 1%

難得元素的稀有度是 1%，

means that there are only 10 rocks with the precious mineral inside.

也就是說，只有 10 塊石頭的 內部會有這珍貴的礦物。

All 10 would set off the detector.

這 10 塊石頭都會 造成偵測器響起。

But what about the other 990 rocks without unobtainium?

但另外 990 塊沒有 難得元素的石頭呢？

Well, 90% of them, 891 rocks, to be exact,

其中有 90%，明確來說 就是 891 塊石頭，

won’t set off anything.

不會讓偵測器響起。

But 10%, or 99 rocks, will set off the detector

但 10%，也就是 99 塊石頭， 會讓偵測器誤響，

despite not having unobtanium,

儘管裡面沒有難得元素，

a result known as a false positive.

這個結果叫做「偽陽性」。

Why does that matter?

這有什麼重要？

Because it means that all in all,

因為它表示，從各方面來看，

109 rocks will have triggered the detector.

共有 109 塊石頭會讓偵測器響起。

And Joe’s rock could be any one of them,

而老喬的石頭可能是 其中的任何一塊，

from the 10 that contain the mineral

可能屬於 10 塊有礦物的石頭，

to the 99 that don’t,

也可能屬於 99 塊沒有礦物的，

which means the chances of it containing unobtainium are 10 out of 109 – about 9%.

這就表示，這塊石頭內含有難得 元素的機率是 10/109，大約 9%。

And paying $200 for a 9% chance of getting $1000 isn’t great odds.

為了 9% 能賺到 $1000 的機率 而花 $200，這賭注並不划算。

So why is this result so unexpected,

所以，為什麼這個結果 如此出乎預期？

and why did Joe’s rock seem like such a sure bet?

為什麼老喬的石頭看似 是個很可靠的賭注？

The key is something called the base rate fallacy.

關鍵就在於所謂的 「基本比率謬誤」。

While we’re focused on the relatively high accuracy of the detector,

雖然我們把焦點放在偵測器 算是相對很高的正確率上，

our intuition makes us forget to account

我們的直覺讓我們忘了去考量

for how rare the unobtanium was in the first place.

難得元素本來就非常稀有。

But because the device’s error rate of 10%

但，因為裝置的錯誤率是 10%，

is still higher than the mineral’s overall occurrence,

還是比那種礦物出現的機率要高，

any time it goes off is still more likely to be a false positive

偵測器響起時 仍然較有可能是偽陽性，

than a real finding.

而不是真正找到了。

This problem is an example of conditional probability.

這個問題是個「條件機率」的例子。

The answer lies neither in the overall chance of finding unobtainium,

答案並不在於找到 難得元素的整體機率，

nor the overall chance of receiving a false positive reading.

也不在於得到的讀數是 偽陽性的整體機率。

This kind of background information that we’re given before anything happens

這種在任何事發生之前 就先給我們的背景資訊，

is known as unconditional, or prior probability.

就是所謂的無條件機率或事先機率。

What we’re looking for, though, is the chance of finding unobtainium

不過，我們在找的是 得知這裝置回報找到了之後

once we know that the device did return a positive reading.

真正發現難得元素的機率。

This is known as the conditional, or posterior probability,

這就是條件機率或事後機率，

determined once the possibilities have been narrowed down through observation.

是在一旦透過觀察 將機率縮減之後確定下來的。

Many people are confused by the false positive paradox

許多人對偽陽性悖論感到困惑，

because we have a bias for focusing on specific information

因為我們會有偏見， 只把焦點放在特定資訊上，

over the more general,

而非更一般性的資訊上，

especially when immediate decisions come into play.

特別是在我們得要立即做出決策時。

And while in many cases it’s better to be safe than sorry,

雖然在許多情況下， 穩妥總比後悔好，

false positives can have real negative consequences.

但偽陽性有可能 造成很負面的結果。

False positives in medical testing are preferable to false negatives,

在醫療測試中，會比較偏好 偽陽性而非偽陰性，

but they can still lead to stress or unnecessary treatment.

但偽陽性仍然有可能造成 壓力或是不必要的治療。

And false positives in mass surveillance

在大眾監視中的偽陽性

can cause innocent people to be wrongfully arrested, jailed, or worse.

則可能造成無辜的人受冤屈， 被逮捕、坐牢，或更糟。

As for this case, the one thing you can be positive about

在這個案例中，有件事可以肯定，

is that Tricky Joe is trying to take you for a ride.

就是狡猾老喬試圖欺騙你。