Check your intuition: The birthday problem - David Knuffke

譯者: Crystal Yip 審譯者: nr chan

Imagine a group of people.

想像有一組人

How big do you think the group would have to be

你覺得組內要有多少人

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

其中二人生日相同的機率 才會超過 50%？

have the same birthday?

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

為了方便討論 假設組內沒有雙胞胎

that every birthday is equally likely,

每個生日的機率均等

and ignore leap years.

不計閏年

Take a moment to think about it.

現在試想一想

The answer may seem surprisingly low.

答案或許看來驚人地低

In a group of 23 people,

在 23 人的組內

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

有 50.3% 機率 二人會有相同的生日

But with 365 days in a year,

但一年 365 日

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

為何人數如此少的組內

to get even odds of a shared birthday?

會有過半機會有相同生日的人

Why is our intuition so wrong?

為什麼我們的直覺錯得這麼離譜？

To figure out the answer,

要找出答案

let's look at one way a mathematician

就讓我們看看數學家其中一種方法

might calculate the odds of a birthday match.

可用作計算二人擁有相同生日的機率

We can use a field of mathematics known as combinatorics,

我們可用一門數學領域 名為組合學

which deals with the likelihoods of different combinations.

處理不同組合的機率

The first step is to flip the problem.

第一步是反轉問題

Trying to calculate the odds of a match directly is challenging

嘗試直接計算相同生日的機率 是個挑戰

因為有相同生日的組合很多

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.

相反地，計算每人 都有不同生日就比較容易

How does that help?

這樣如何幫助我們解決問題呢？

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

組內的人不是有相同生日，就是沒有

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

所以有相同生日的人的機率 和沒有的機率

must add up to 100%.

加起來必然是 100%

That means we can find the probability of a match

從 100% 減去無相同生日機率 便是有相同生日的機率

by subtracting the probability of no match from 100.

To calculate the odds of no match, start small.

要計算沒有相同生日的機率 先考慮人數少的組

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

計算只有一對人有不同生日的機率

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

一年中的某日會是 A 君的生日

which leaves only 364 possible birthdays for Person B.

餘下的 364 天 皆有可能是 B 君的生日

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

A 和 B，或任意二人 有不同生日的機率

is 364 out of 365,

是 365 分之 364

about 0.997, or 99.7%, pretty high.

約 0.997 或 99.7% 這是相當高的機率

Bring in Person C.

再考慮 C 君

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

她在這小組內有不同生日的機率

is 363 out of 365

是 365 分之 363

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

因為 A 和 B 的生日 已佔兩個日子

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

D 的機率會是 365 分之 362 如此類推

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

一直至 W 的機率是 365 分之 343

Multiply all of those terms together,

把這些機率相乘

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

你會得出沒有人生日相同的機率

This works out to 0.4927,

得出 0.4927

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

因此在 23 人的組內 沒有人生日相同的機率是 49.27%

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

當我們從 100% 減去這機率 便得 50.73%

of at least one birthday match,

即至少有二人生日相同的機率

better than even odds.

這機率高於一半

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

人數相對少的組內有人生日相同的 機率如此高的關鍵在於

is the surprisingly large number of possible pairs.

相同生日的可能組合出人意料地多

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

當組內人數逐漸增加 可能組合的數目愈快速增加

A group of five people has ten possible pairs.

五人組內有十對可能組合

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

每人能與其餘四人各自組合

Half of those combinations are redundant

這些組合有一半是重複的

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

因為把 A 君配以 B 君 等同於把 B 君配以 A 君

so we divide by two.

所以我們將之除以二

By the same reasoning,

同樣道理

a group of ten people has 45 pairs,

十人組內有 45 對組合

and a group of 23 has 253.

而 23 人的組內有 253 對

The number of pairs grows quadratically,

組合的數量以平方關係增長

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

意即它按組內人數的平方比例增長

Unfortunately, our brains are notoriously bad

遺憾地，我們的腦袋不擅於

at intuitively grasping non-linear functions.

憑直覺即領會非線性函數

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

所以 23 人看來不大可能 產生出 253 對可能組合

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

當我們的腦袋接受這事實 生日問題變得容易理解

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

253 對組合皆可能有相同生日

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

同樣原因，在 70 人的組內

there are 2,415 possible pairs,

有 2,415 對可能組合

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

而有兩人有相同生日的機率 高於 99.9%

The birthday problem is just one example where math can show

生日問題只是其中一個例子 來藉由數學展示

that things that seem impossible,

看似不可能的事情

like the same person winning the lottery twice,

例如同一人中了兩次彩券

actually aren't unlikely at all.

事實上不是不大可能發生的

Sometimes coincidences aren't as coincidental as they seem.

有時巧合不如看似般巧合