Can you solve the false positive riddle

你能解決誤報(bào)之謎嗎

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Mining unobtainium is hard work.?

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

But your friend Tricky Joe has something up his sleeve.?

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

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

Otherwise, it’s still highly reliable, returning accurate readings 90% of the time.

挖掘稀有元素是很難的事。

稀有礦物僅僅存在于 礦井中那 1% 的石頭里。

但是你的朋友狡猾的喬 有他的秘密武器。

他改進(jìn)了近幾個(gè)月的 稀有元素檢測器終于準(zhǔn)備完畢。

如果有任何稀有元素存在， 這個(gè)設(shè)備則一定會檢測到。

在其他方面，它仍然是很可靠的：90%的幾率它會有準(zhǔn)確的讀數(shù)。

On his first day trying it out in the field, the device goes off, and Joe happily places the rock in his cart.

As the two of you head back to camp where the ore can be examined, Joe makes you an offer: he’ll sell you the ore for just $200.?

You know that a piece of unobtanium that size would easily be worth $1000, but any other minerals would be effectively worthless.?

Should you make the trade?

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

Answer in: 3 Answer in: 2 Answer in: 1

在場地進(jìn)行測試的第一天，設(shè)備響應(yīng)了， 喬高興地把石頭放在他的車?yán)铩?/p>

當(dāng)你們兩個(gè)返回營地， 在那里礦石能得到檢驗(yàn)，喬向你開了價(jià)：他可以將這塊礦石 以僅僅200美元賣給你。

你心里知道那種大小的稀有元素 最少價(jià)值1000美元。

但若是其他礦物，則一文不值。

你應(yīng)該做這筆交易嗎？

如果你想回答這個(gè)問題就暫停一下。

倒計(jì)時(shí)：321

Intuitively, it seems like a good deal.?

Since the detector is correct most of the time, shouldn’t you be able to trust its reading??

Unfortunately, no.?

Here’s why. Imagine the mine has exactly 1,000 pieces of ore.?

An unobtainium rarity of 1% means that there are only 10 rocks with the precious mineral inside.?

All 10 would set off the detector.

But what about the other 990 rocks without unobtainium??

Well, 90% of them, 891 rocks, to be exact, won’t set off anything.?

But 10%, or 99 rocks, will set off the detector despite not having unobtanium, a result known as a false positive.?

Why does that matter?

Because it means that all in all, 109 rocks will have triggered the detector.

直覺上，這似乎是個(gè)挺好的交易。

因?yàn)闄z測器大多時(shí)候都工作正常。

為什么不該相信它的讀數(shù)呢？

很抱歉，答案是不。

原因如下：假設(shè)你開采的礦 正好有1000塊礦石。

難得素的稀有度為1%意味著，其中只有10塊石頭 含有那種珍貴的礦物。

這10塊石頭都會 引起檢測器響應(yīng)。

但那其他990塊 不含稀有元素的石頭呢？

這些石頭中的90%， 確切來說就是891塊石頭，什么反應(yīng)都不會引起。

但10%（99塊）其他的石頭， 都會引起檢測器響應(yīng)。無論含不含稀有元素。這個(gè)結(jié)果就是假正。

為什么這個(gè)會有影響呢？

因?yàn)樗馕吨?總共有109塊石頭觸發(fā)了檢測器。

And Joe’s rock could be any one of them, from the 10 that contain the mineral to the 99 that don’t, which means the chances of it containing unobtainium are 10 out of 109 – about 9%.

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

So why is this result so unexpected, and why did Joe’s rock seem like such a sure bet??

The key is something called the base rate fallacy.?

While we’re focused on the relatively high accuracy of the detector, our intuition makes us forget to account for how rare the unobtanium was in the first place.

But because the device’s error rate of 10% is still higher than the mineral’s overall occurrence, any time it goes off is still more likely to be a false positive than a real finding.

而喬的那塊石頭 可能是它們當(dāng)中任意一個(gè)：可以是含有那種 礦物的10塊石頭，也可以是那 99塊不含有的，也就是說含有稀有元素的石頭 幾率是10/109，大約9%。

只有9%的幾率獲利1000， 而你要為此付200，這并不劃算。

為什么這個(gè)結(jié)果如此出乎意料？

為什么喬的石頭似乎 像一比劃算的交易呢？

答案在于“基本比率謬誤”。

當(dāng)我們聚焦于檢測器 有著相對較高的精度時(shí)，我們的直覺讓我們忘記了去思考，稀有元素本來就是非常稀有的。

但由于設(shè)備10%的錯(cuò)誤率，仍然高于礦物總體的出現(xiàn)率，只要設(shè)備響應(yīng)，仍有很大可能是假正，而不是真的發(fā)現(xiàn)了稀有元素。

This problem is an example of conditional probability.?

The answer lies neither in the overall chance of finding unobtainium, nor the overall chance of receiving a false positive reading.?

This kind of background information that we’re given before anything happens is known as unconditional, or prior probability.?

What we’re looking for, though, is the chance of finding unobtainium once we know that the device did return a positive reading.?

This is known as the conditional, or posterior probability, determined once the possibilities have been narrowed down through observation.

Many people are confused by the false positive paradox because we have a bias for focusing on specific information over the more general, especially when immediate decisions come into play.

這個(gè)問題是條件概率的一個(gè)例子。

答案既不在于發(fā)現(xiàn) 稀有元素的總體概率，也不在于收到誤報(bào)的總體概率。

在一切事情發(fā)生前， 先給我們的這種背景信息，就叫做無條件或者先驗(yàn)概率。

但我們尋求的是：在我們知道 設(shè)備返回了一個(gè)正確的讀數(shù)時(shí)，獲得稀有元素的概率。

這叫做條件或后驗(yàn)概率，是由一旦通過觀察將 可能性降低所決定的。

很多人困惑于假正悖論，因?yàn)槲覀兏蛴?將注意力集中在特定的信息，而不是更加一般的信息，特別是需要馬上做決定的時(shí)候。

And while in many cases it’s better to be safe than sorry, false positives can have real negative consequences.?

False positives in medical testing are preferable to false negatives, but they can still lead to stress or unnecessary treatment.

And false positives in mass surveillance can cause innocent people to be wrongfully arrested, jailed, or worse.?

As for this case, the one thing you can be positive about is that Tricky Joe is trying to take you for a ride.

不過大多數(shù)時(shí)候， 穩(wěn)妥總比遺憾要好。

假正會造成不良結(jié)果。在醫(yī)療測試中，假正比假負(fù)更加可靠，但是它們?nèi)匀荒軐?dǎo)致壓力 或者不必要的治療。

在大規(guī)模監(jiān)控下，誤報(bào)會導(dǎo)致無辜的人 被逮捕、入獄或更糟。

至于這種情況，你能肯定的一點(diǎn)是：狡猾的喬正試圖欺騙你。