Can you solve the frog riddle? - Derek Abbott

譯者: Gentian Pan 審譯者: Max Chern

So you're stranded in a huge rainforest, and you've eaten a poisonous mushroom.

你受困於廣大的雨林中，誤食毒菇。

To save your life, you need the antidote excreted by a certain species of frog.

為了拯救自己， 需要某種青蛙的分泌物作為解藥。

Unfortunately, only the female of the species produces the antidote,

不幸的是，只有雌蛙分泌這種解藥。

and to make matters worse,

而更糟的是，

the male and female occur in equal numbers and look identical,

雌雄數量一樣，長相相同，

with no way for you to tell them apart,

你無法分辨，

except that the male has a distinctive croak.

唯一差異是雄性有獨特的蛙鳴聲。

And it may just be your lucky day.

今天可能是你的幸運日。

To your left, you've spotted a frog on a tree stump,

你發現左邊樹樁上有一隻青蛙，

but before you start running to it,

但在跑向牠之前，

you're startled by the croak of a male frog

你被雄蛙叫聲嚇一跳，

coming from a clearing in the opposite direction.

聲音來自相反方向的空地。

There, you see two frogs,

有兩隻青蛙在那裡，

but you can't tell which one made the sound.

只是分辨不出哪隻發出聲音。

You feel yourself starting to lose consciousness,

你覺得快失去知覺了，

and realize you only have time to go in one direction before you collapse.

意識到倒下之前， 時間只足夠讓你走往一個方向。

What are your chances of survival if you head for the clearing

你的生存機會有多少── 假如您跑向空地，

and lick both of the frogs there?

舔舐那兩隻青蛙？

What about if you go to the tree stump?

若到樹樁那頭呢？

Which way should you go?

該如何選擇？

Press pause now to calculate odds yourself.

先按暫停。 自己算一下概率。

3

3、

2

2、

1

1。

If you chose to go to the clearing, you're right,

如果你選擇到空地，就對了，

but the hard part is correctly calculating your odds.

但是，正確算出機率就困難了。

There are two common incorrect ways of solving this problem.

有兩種常見的錯誤計算方式。

Wrong answer number one:

錯誤算法一：

Assuming there's a roughly equal number of males and females,

假設雌雄數量大致相當，

the probability of any one frog being either sex is one in two,

任一隻青蛙是雌或雄的概率都是 1/2，

which is 0.5, or 50%.

即 0.5，或 50％。

And since all frogs are independent of each other,

由於青蛙都是獨立個體，

the chance of any one of them being female should still be 50% each time you choose.

每次任選一隻是雌性的機會還是 50％。

This logic actually is correct for the tree stump,

對於樹樁上的青蛙，這個邏輯正確。

but not for the clearing.

空地上的，並非此概率。

Wrong answer two:

錯誤算法二：

First, you saw two frogs in the clearing.

首先你看到空地上的兩隻青蛙，

Now you've learned that at least one of them is male,

你已經知道其中至少有一隻是雄性，

but what are the chances that both are?

但兩隻皆雄性的概率是多少？

If the probability of each individual frog being male is 0.5,

如果每隻蛙是雄性的概率是 0.5，

then multiplying the two together will give you 0.25,

兩者相乘得到 0.25，

which is one in four, or 25%.

概率就是 1/4，或 25％。

So, you have a 75% chance of getting at least one female

所以，你有 75％ 的機會 遇到至少一隻是雌蛙

and receiving the antidote.

而得到解藥──這也是錯。

So here's the right answer.

下面才是正確答案。

Going for the clearing gives you a two in three chance of survival,

到空地去，你的生存概率是 2/3，

or about 67%.

即約 67％。

If you're wondering how this could possibly be right,

如果你想知道這為什麼是對的，

it's because of something called conditional probability.

這是因為「條件概率」。

Let's see how it unfolds.

我們來看如何推導。

When we first see the two frogs,

最初看到兩隻青蛙時，

there are several possible combinations of male and female.

雌雄有幾種可能組合，

If we write out the full list,

可以完整列表，

we have what mathematicians call the sample space,

數學家稱它為「樣本空間」，

and as we can see,

我們可以看到，

out of the four possible combinations, only one has two males.

四種組合中， 只有一種是兩隻同為雄性。

So why was the answer of 75% wrong?

而 75％ 答案錯的理由是 －

Because the croak gives us additional information.

由於蛙鳴提供了額外資訊。

As soon as we know that one of the frogs is male,

我們一旦知道其中一隻是雄蛙，

that tells us there can't be a pair of females,

就知道「一對雌蛙」是不可能存在。

which means we can eliminate that possibility from the sample space,

因此，可以將它從樣本空間剔除。

leaving us with three possible combinations.

剩下三種可能組合。

Of them, one still has two males,

其中，仍有「一對雄蛙」的組合。

giving us our two in three, or 67% chance of getting a female.

所以，我們有 2/3 即 67％ 的機會遇到雌蛙。

This is how conditional probability works.

這是「條件概率」的計算方法。

You start off with a large sample space that includes every possibility.

從最大「樣本空間」開始 ── 包含各種可能性。

But every additional piece of information allows you to eliminate possibilities,

但是，每個額外資訊 都讓你可以剔除部分樣本，

shrinking the sample space

縮小樣本空間，

and increasing the probability of getting a particular combination.

為想得的特定組合增加概率。

The point is that information affects probability.

重點是資訊會影響概率。

And conditional probability isn't just the stuff of abstract mathematical games.

條件概率不只是抽象的數學遊戲，

It pops up in the real world, as well.

它在真實世界也時而可見。

Computers and other devices use conditional probability

電腦和其它裝置使用條件概率

to detect likely errors in the strings of 1's and 0's

來偵測 1 與 0 的字串中 可能的錯誤──而這些 1 與 0

that all our data consists of.

組合成電腦數據。

And in many of our own life decisions,

在我們人生抉擇中，

we use information gained from past experience and our surroundings

我們從過去經驗 和周邊環境中獲取資訊，

to narrow down our choices to the best options

以縮小選擇範圍，找到最佳選擇。

so that maybe next time,

所以也許下一次，

we can avoid eating that poisonous mushroom in the first place.

我們一開始就能避免吃到毒菇。

翻譯：Gentian Pan