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

為了拯救你的性命，你需要某種青蛙所生產的解毒劑。

Unfortunately, only the female of the species produces the antidote, and to make matters worse, the male and female occur in equal numbers and look identical, with no way for you to tell them apart, except that the male has a distinctive croak.

不幸的是，只有母蛙生產解毒劑，而更糟糕的是，公蛙和母蛙的數量不但一樣，而且還看起來一模一樣。你沒有辦法靠肉眼分辨它們，而他們之間唯一的差異是公蛙會發出獨特的鳴叫聲。

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, but before you start running to it, you're startled by the croak of a male frog coming from a clearing in the opposite direction.

你已經看到了左邊樹幹上有隻青蛙，但在你開始跑向他之前，你被來自反方向空地的公蛙鳴叫聲嚇到了。

There, you see two frogs, but you can't tell which one made the sound.

你看到那邊有兩隻青蛙，但是你無法分辨聲音是哪一隻發出來的。

You feel yourself starting to lose consciousness, and realize you only have time to go in one direction before you collapse.

你覺得意識開始逐漸遠去，並驚覺在到你倒下之前，時間只夠讓你往其中一個方向走去。

What are your chances of survival if you head for the clearing and lick both of the frogs there?

如果你走向空地然後舔那邊兩隻青蛙的話，會有多大的機會存活下來？

What about if you go to the tree stump?

那麼如果你去樹幹那邊呢？

Which way should you go?

你應該去哪邊呢？

[Press pause now to calculate odds yourself.]

[現在就按下暫停，自己計算機率看看吧。]

3, 2 ,1.

3、2、1。

If you chose to go to the clearing, you're right, but the hard part is correctly calculating your odds.

如果你選擇前往空地，那你答對了。但題目最難的部分在於如何正確地計算出機率。

There are two common incorrect ways of solving this problem.

有兩種常見錯誤的解題方式。

Wrong answer number one:

第一種錯誤方式：

Assuming there's a roughly equal number of males and females, the probability of any one frog being either sex is one in two, which is 0.5, or 50%.

假設有公蛙和母蛙數量差不多，而任何一隻青蛙性別的機率是二分之一，也就是 0.5 或 50% 。

And since all frogs are independent of each other, the chance of any one of them being female should still be 50% each time you choose.

由於每一隻青蛙都是獨立個體，因此每次你選擇的時候，該對象是母蛙的機率應該是 50%。

This logic actually is correct for the tree stump, but not for the clearing.

這樣的邏輯對於樹幹上的青蛙是正確的，但對於空地上的來說並不正確。

Wrong answer two:

第二種錯誤答案：

First, you saw two frogs in the clearing.

首先，你看到空地上有兩隻青蛙。

Now you've learned that at least one of them is male, but what are the chances that both are?

現在你已經知道至少有他們其中一隻是公的了，但是他們都是公蛙的機率是多少呢？

If the probability of each individual frog being male is 0.5, then multiplying the two together will give you 0.25, which is one in four, or 25%.

如果每隻青蛙是公蛙的機率為 0.5，兩隻青蛙皆為公蛙的機率就是相乘起來的 0.25，也就是四分之一或是 25%。

So, you have a 75% chance of getting at least one female and receiving the antidote.

所以你有 75% 的機會至少得到一隻母蛙並且取得解毒液。

So here's the right answer.

正確的答案是這樣的。

Going for the clearing gives you a two in three chance of survival, or about 67%.

去空地的話，你會有三分之二的機率存活也就大約是 67%。

If you're wondering how this could possibly be right, it's because of something called conditional probability.

或許你會覺得這怎麼可能是正確答案，但這都是因為有個叫做條件機率的東西。

Let's see how it unfolds.

讓我們來看看這個結果是怎麼堆導出來的。

When we first see the two frogs, there are several possible combinations of male and female.

當我們首先看到有兩隻青蛙時，有很多種公蛙和母蛙可能的組合方式。

If we write out the full list, we have what mathematicians call the sample space, and as we can see, out of the four possible combinations, only one has two males.

將它們全部寫出來後我們獲得了數學家們所稱的「樣本空間」，而我們能看到四種可能的組合當中，只有一種組合有兩隻公蛙。

So why was the answer of 75% wrong?

那麼為什麼 75% 的答案是錯的呢？

Because the croak gives us additional information.

因為鳴叫聲給了我們額外的資訊。

As soon as we know that one of the frogs is male, that tells us there can't be a pair of females, which means we can eliminate that possibility from the sample space, leaving us with three possible combinations.

當我們知道其中一隻青蛙是公蛙時，我們便能知道不可能有兩隻母蛙，也就是代表我們可以從樣本空間中消除掉這個可能性，並留下三個可能的組合。

Of them, one still has two males, giving us our two in three, or 67% chance of getting a female.

其中，一個組合仍然有兩隻公蛙，我們也就有三分之二或 67% 的機率得到母蛙。

This is how conditional probability works.

這就是條件機率的運作方式

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

你從包含了所有可能性，較大的樣本空間開始。

But every additional piece of information allows you to eliminate possibilities, shrinking the sample space and increasing the probability of getting a particular combination.

但是每個額外的資訊可以讓你消除掉一些可能性，縮小樣本空間，然後增加得到特定組合的可能性。

The point is that information affects probability.

重點就是資訊會影響機率。

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

而條件機率不僅僅是抽象的數學遊戲而已。

It pops up in the real world, as well.

它在生活中也會出現。

Computers and other devices use conditional probability to detect likely errors in the strings of 1's and 0's that all our data consists of.

電腦和其他裝置使用條件機率在 0 和 1 的字串中去發現可能的錯誤，而這些 0 和 1 組成了我們所有電腦資料的數據。

And in many of our own life decisions, we use information gained from past experience and our surroundings to narrow down our choices to the best options so that maybe next time, we can avoid eating that poisonous mushroom in the first place.

而在我們自己很多人生的抉擇中，我們也會運用從過去的經驗和環境所得到的資訊，去縮小我們的選擇，直到剩下最佳的選項。所以或許下一次我們便可以可以藉此避免吃到毒香菇。