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

在礦場裡的岩石堆中只會出現百分之一的稀有礦石

But your friend Tricky Joe has something up his sleeve.

但是你的朋友，狡猾老喬握有一手妙計

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

他花了數個月，終於將難得素偵測器調整至完美。

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

若是難得素出現，這個裝置就一定能偵測到

Otherwise, it's still highly reliable,

除此之外，它的可靠性非常高

returning accurate readings 90% of the time.

可以回傳百分之九十的正確讀數

On his first day trying it out in the field,

當他首次將偵測器應用於實務時

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

偵測器響了，於是老喬開心的將那塊岩石放入他的推車中

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

當你們倆出發前往可以檢驗岩石的營地時

Joe makes you an offer:

老喬向你提議

he'll sell you the ore for just $200.

他要用兩百元的價格賣你這塊礦石

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

而你知道這個大小的難得素一定可以輕輕鬆鬆賣到一千元

but any other minerals would be effectively worthless.

但是除此之外的礦石都毫無價值

Should you make the trade?

那你應該要跟他交易嗎?

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

(如果你想要自行尋找解答則在這裡暫停)

Answer in: 3

答案將在三秒後揭曉

Answer in: 2

答案將在兩秒後揭曉

Answer in: 1

答案將在一秒後揭曉

Intuitively, it seems like a good deal.

這個交易直覺上相當划算

Since the detector is correct most of the time,

因為偵測器在大多數時候是準確的

shouldn't you be able to trust its reading?

你有什麼理由不去相信它的反應呢

Unfortunately, no.

可惜的是，錯了

Here's why.

理由是這樣的

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

想像這個礦場實際上有一千塊礦石

An unobtainium rarity of 1%

而當中只存在著百分之一的難得素

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

也就是說只有十塊岩石含有珍貴的礦石

All 10 would set off the detector.

這十塊岩石都會觸發偵測器

But what about the other 990 rocks without unobtainium?

但是其他九百九十塊沒有難得素的岩石呢？

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

百分之九十的它們，準確地說，這八百九十一塊的岩石

won't set off anything.

不會觸發偵測器

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

但是，百分之十的它們，或其中九十九塊的岩石，將會觸發偵測器

despite not having unobtanium,

即便它們沒有難得素

a result known as a false positive.

這樣的結果就稱為偽陽性

Why does that matter?

為什麼這樣重要呢?

Because it means that all in all,

因為整體來說

109 rocks will have triggered the detector.

將會有一百零九塊的岩石會觸發偵測器

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

而老喬的岩石可能會是其中之一

from the 10 that contain the mineral

從包含稀有礦石的那十塊

to the 99 that don't,

到不包含的那九十九塊岩石

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

代表一百零九塊岩石中只會有十塊包含有難得素，也就是大概百分之九

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

而為了那百分之九之獲得一千元的機率付出兩百元不太合理

So why is this result so unexpected,

那麼，這樣的結果為什麼會難以預料呢

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

又為什麼老喬的石頭看似是穩贏的賭注呢?

The key is something called the base rate fallacy.

關鍵就是基本率謬誤

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

當我們專注於偵測器的相對高準確度時

our intuition makes us forget to account

我們的直覺使我們忘記

for how rare the unobtanium was in the first place.

將難得素的稀有程度放在第一考量上

But because the device's error rate of 10%

但因為偵測器百分之十的錯誤率

is still higher than the mineral's overall occurrence,

依舊高於稀有礦石的整體出現率

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

不管什麼時候觸發看起來都會像是偽陽性

than a real finding.

而非實際的發現

This problem is an example of conditional probability.

這個問題就是條件機率的一個範例

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

答案不在發現難得素的整體機會

nor the overall chance of receiving a false positive reading.

也不在接收到偽陽性的整體機會中

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

像這樣，我們在任何情況發生前所被給予的背景資訊

is known as unconditional, or prior probability.

被稱為無條件機率，或稱為事前機率

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

不過，我們所追尋的是

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

一旦偵測器會回傳正讀數而找到難得素的機會

This is known as the conditional, or posterior probability,

這就是條件機率，或稱為事後機率

determined once the possibilities have been narrowed down through observation.

是一旦透過觀察機率而縮小機率後所決定的

Many people are confused by the false positive paradox

許多人因偽陽性悖論而困惑

because we have a bias for focusing on specific information

因為我們會有專注於特定資訊上的偏見

over the more general,

而非專注於一般的資訊上

especially when immediate decisions come into play.

特別是發生需要立即做反應的決定時

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

而當許多需要安全為上的案例發生時

false positives can have real negative consequences.

偽陽性可以帶來非常不好的後果

False positives in medical testing are preferable to false negatives,

在醫學試驗上，比較偏好偽陽性而非偽陰性

but they can still lead to stress or unnecessary treatment.

惟其依舊可以導致壓力或非必要的治療

And false positives in mass surveillance

而在大規模監控的偽陽性

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

則可以造成無辜的人被不正當的逮捕、監禁，甚至更糟

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

在這個案例裡，你可以確定的是

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

狡猾老喬試著要騙你囉