Check your intuition: The birthday problem - David Knuffke

翻译人员: Riley WANG 校对人员: Lipeng Chen

Imagine a group of people.

设想有一组人。

How big do you think the group would have to be

当人数达到多少时，

before there's more than a 50% chance that two people in the group

其中两人生日相同的概率

have the same birthday?

会超过50%？

Assume for the sake of argument that there are no twins,

假设这人群中没有双胞胎，

that every birthday is equally likely,

所有人的生日概率都相等，

and ignore leap years.

并且排除掉闰年的存在。

Take a moment to think about it.

花一点时间想想。

The answer may seem surprisingly low.

所需要的人数其实是相当少的。

In a group of 23 people,

若一组有23个人，

there's a 50.73% chance that two people will share the same birthday.

那么两个人有相同生日的概率是50.73%。

But with 365 days in a year,

但一年有365天这么多天，

how's it possible that you need such a small group

为什么只需要这么少的人

to get even odds of a shared birthday?

就可以使其中两人生日相同？

Why is our intuition so wrong?

为什么我们的直觉是如此错误的？

To figure out the answer,

为了找到答案，

let's look at one way a mathematician

让我们看看数学家

might calculate the odds of a birthday match.

是如何计算生日相同的概率的。

We can use a field of mathematics known as combinatorics,

我们可以使用组合数学的分析方法，

which deals with the likelihoods of different combinations.

它用来计算不同组合的概率。

The first step is to flip the problem.

首先我们需要转换问题，

Trying to calculate the odds of a match directly is challenging

直接计算生日相同的概率是很困难的，

因为二人相同的生日可能是在任一天。

because there are many ways you could get a birthday match in a group.

Instead, it's easier to calculate the odds that everyone's birthday is different.

相反，计算每个人生日不同的概率是更简单的。

How does that help?

这对于问题有什么帮助呢？

Either there's a birthday match in the group, or there isn't,

一组人中，要么有两人生日相同，要么没有，

so the odds of a match and the odds of no match

所以这两种情况的概率

must add up to 100%.

相加必然等于100%

That means we can find the probability of a match

这意味着我们可以通过

by subtracting the probability of no match from 100.

将无相同生日的概率从100%中减去 而得到有相同生日的概率。

To calculate the odds of no match, start small.

让我们先考虑简单情况。

Calculate the odds that just one pair of people have different birthdays.

先从只有一对人拥有不同生日开始。

One day of the year will be Person A's birthday,

假定对象A的生日是某天，

which leaves only 364 possible birthdays for Person B.

那么对象B的生日就是其余364天之一。

The probability of different birthdays for A and B, or any pair of people,

对象A和B或是任何两人生日不同的概率就是

is 364 out of 365,

364/365，

about 0.997, or 99.7%, pretty high.

大约是99.7%，非常高的概率。

Bring in Person C.

我们再引入对象C。

The probability that she has a unique birthday in this small group

她和其他人生日都不同的概率

is 363 out of 365

是363/365，

because there are two birthdates already accounted for by A and B.

因为已经有两个日期被对象A和B占用。

D's odds will be 362 out of 365, and so on,

对象D不与他人生日相同的概率是362/365，

all the way down to W's odds of 343 out of 365.

以此类推，到对象W是概率为343/365。

Multiply all of those terms together,

将这些数字相乘，

and you'll get the probability that no one shares a birthday.

就会得到没有人生日相同的概率。

This works out to 0.4927,

结果是0.4927，

so there's a 49.27% chance that no one in the group of 23 people shares a birthday.

所以23人当中，大家生日都不同的概率是49.27%。

When we subtract that from 100, we get a 50.73% chance

再用100%减去这个数值，就得到50.73%，

of at least one birthday match,

这就是至少有两个人生日相同的概率，

better than even odds.

已经超过了50%。

The key to such a high probability of a match in a relatively small group

在人数如此少的小组中出现高概率，

is the surprisingly large number of possible pairs.

其关键在于可能的组合数量巨大。

As a group grows, the number of possible combinations gets bigger much faster.

随着人数增加，可能出现的组合数量会快速增加。

A group of five people has ten possible pairs.

5人当中就有10种不同的组合方式。

Each of the five people can be paired with any of the other four.

五个人的任何一人都可以和其他四人构成组合。

Half of those combinations are redundant

其中一半是重复计算，

because pairing Person A with Person B is the same as pairing B with A,

因为对象A+对象B的组合与对象B+对象A相同，

so we divide by two.

所以我们将数字除以2。

By the same reasoning,

相同的道理，

a group of ten people has 45 pairs,

10个人当中就会出现45对组合，

and a group of 23 has 253.

23人当中会有253对组合。

The number of pairs grows quadratically,

组合的数量成倍增加，

meaning it's proportional to the square of the number of people in the group.

即以小组人数的基数的成比例增长。

Unfortunately, our brains are notoriously bad

不幸的是，我们的大脑非常不擅长

at intuitively grasping non-linear functions.

直接处理非线性函数。

So it seems improbable at first that 23 people could produce 253 possible pairs.

所以23人就有253种配对组合看上去是不可能的。

Once our brains accept that, the birthday problem makes more sense.

一旦明白了这个道理，生日问题就变得合理了。

Every one of those 253 pairs is a chance for a birthday match.

253对组合中的人都能找到生日相同的另一半。

For the same reason, in a group of 70 people,

同样的，在70个人构成的小组中，

there are 2,415 possible pairs,

组合数量能达到2415个，

and the probability that two people have the same birthday is more than 99.9%.

两个人拥有相同生日的概率超过了99.9%。

The birthday problem is just one example where math can show

生日问题是一个很好的例子

that things that seem impossible,

展示出数学可以解释看起来不可能的事情，

like the same person winning the lottery twice,

例如同一个人中两次彩票，

actually aren't unlikely at all.

不是几乎不可能的。

Sometimes coincidences aren't as coincidental as they seem.

有时巧合并没有看上去那么巧。