Can you solve the frog riddle? - Derek Abbott

翻译人员: Jessie Lan 校对人员: Gabriella Hu

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

假如你被困在一个巨大的热带雨林里，然后你吃了一个毒蘑菇。

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

想要自救， 你需要由青蛙分泌的一种解药。

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

不幸的是，只有雌性青蛙才能分泌这种解药。

and to make matters worse,

然而更糟糕的是，

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

雌雄青蛙数目一样且长相一样。

with no way for you to tell them apart,

你没有办法区分他们。

except that the male has a distinctive croak.

除了雄性青蛙有不一样的叫声。

And it may just be your lucky day.

可能只是你运气好，

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

在你的左边， 你看到一只青蛙在树桩上。

but before you start running to it,

但在你冲向它之前，

you're startled by the croak of a male frog

你被一只雄性青蛙的叫声吓到了，

coming from a clearing in the opposite direction.

声音很明显来自相反的方向的空地。

There, you see two frogs,

在那里有两只青蛙，

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

但是你并不能分辨是哪一只青蛙叫的。

You feel yourself starting to lose consciousness,

你感觉到自己开始逐渐失去意识，

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

并且意识到在你晕倒之前你只有足够的时间往一个放向去。

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

你的生存几率是什么呢，如果你去空地

and lick both of the frogs there?

然后舔那里的两只青蛙？

What about if you go to the tree stump?

去树桩的生存几率又是多少呢？

Which way should you go?

你应该去哪一边？

Press pause now to calculate odds yourself.

请暂停视频思考两个选择的利与弊。

3

三

2

二

1

一

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

如果你选择去空地，你的选择是正确的。

but the hard part is correctly calculating your odds.

但是困难的是如何正确的计算几率。

There are two common incorrect ways of solving this problem.

有两个常见的不正确的解决办法

Wrong answer number one:

第一个：

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

假设雌雄青蛙数量大致相同，

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

一只青蛙是雌或是雄的概率

which is 0.5, or 50%.

是零点五或者百分之五十

And since all frogs are independent of each other,

因为所有的青蛙都是独立的个体

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

在你选择的时候，任意一只青蛙是雌性的可能性应该仍为百分之五十

This logic actually is correct for the tree stump,

对于树桩部分，这个逻辑其实是正确的

but not for the clearing.

但是对于空地来说不是

Wrong answer two:

第二个错误答案：

First, you saw two frogs in the clearing.

首先，你看到空地有两只青蛙，

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

现在你知道至少其中一只是雄性，

but what are the chances that both are?

但是两只都是的可能性是多少呢？

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

如果任意一只青蛙是雄性的几率是百分之五十，

then multiplying the two together will give you 0.25,

把两个可能性相乘，得到零点二五。

which is one in four, or 25%.

也就是四分之一或百分之二十五。

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

这么说，百分之七十五的可能性其中一只是雌性青蛙

and receiving the antidote.

然后你能获得解药。

So here's the right answer.

正确答案是：

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

去空地你能获得三分之二的生存几率，

or about 67%.

百分之六十七

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

如果你在想这怎么可能是正确答案，

it's because of something called conditional probability.

是因为条件可能性

Let's see how it unfolds.

让我们来看看为什么

When we first see the two frogs,

当我们一开始看到两只青蛙的时候，

there are several possible combinations of male and female.

有几个可能的雌雄组合，

If we write out the full list,

如果我们把所有可能性写出来，

we have what mathematicians call the sample space,

就有数学家所谓的样本空间

and as we can see,

就像我们看到的一样，

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

在四个可能性中，只有一个可能性有两只雄青蛙

So why was the answer of 75% wrong?

那为什么百分之七十五的答案是错误的呢？

Because the croak gives us additional information.

因为青蛙叫给了我们额外的信息

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

当我们知道了其中一只是雄性的时候，

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

这就告诉我们这其中不可能有两只雌性青蛙

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

也就是说我们可以从样本空间中删除那一个可能性。

leaving us with three possible combinations.

留给我们的是三个可能的组合

Of them, one still has two males,

其中一个是两个雄性

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

所以有雌性青蛙的概率是三分之二或者是百分之六十七

This is how conditional probability works.

这就是条件概率。

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

一开始， 你有一个很大的样本空间， 其中包含每一种可能性。

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

但是每一则额外的信息可以让你删除一些可能性。

shrinking the sample space

样本空间逐渐减小，

and increasing the probability of getting a particular combination.

获得一个特定的组合几率则增大。

The point is that information affects probability.

要点是：信息会影响可能性。

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

条件可能性并不只是抽象的数学游戏

It pops up in the real world, as well.

它在现实生活中会出现

Computers and other devices use conditional probability

电脑和其他的装置会使用条件可能性

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

在1 和 0 中 来侦测

that all our data consists of.

我们数据中的潜在错误

And in many of our own life decisions,

在我们的日常生活中，

we use information gained from past experience and our surroundings

我们会利用过去经历和周边环境获得的信息

to narrow down our choices to the best options

来缩小选择以获得最佳选项

so that maybe next time,

所以可能下次，

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

我们可以在最开始的时候避免吃那个有毒的蘑菇。