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别相信直觉:蒙提霍尔问题

The Monty Hall Problem - Explained

所以 在你成功闯关到电视节目比赛最后环节
So, you’ve made it to the last round of a TV gameshow
有机会获得一辆名牌小轿车
and have the chance to win a brand new car.
它停在三扇门中的一扇门后
It sits behind one of these three doors,
而遗憾的是另两扇只有山羊在其后
but the other two have a sad little goat behind them.
你选中一扇门 主持人会为你打开一扇有山羊的门
You make your choice and the host decides to reveal where one of the goats is.
然后给你一个机会改选门
He then offers you a chance to change your door.
你会改选吗?
Do you do it?
改选门会让结果不一样吗?
Does changing your choice even make a difference?
答案短小精悍
The short answer is:
当然啦!
Yes!
虽然看起来违反直觉
Even though it seems counterintuitive,
改掉你原来选的门实际上是在质疑你自己赢得车的运气
changing you door choice actually doubles your odds of winning the car.
但这怎么可能呢?
But how is that possible?
这就是“蒙提霍尔问题”
This is the “MONTY HALL PROBLEM”.
一开始 绝大多数的人都正确的
At the start, most people correctly assume that
假定了有1/3的机会选到对的门
you have a 1/3 chance of choosing the correct door.
但在排除掉一扇门后 再假定每扇门有
But it would be incorrect to assume that when one door is removed,
各半成功的几率赢得轿车 就不正确了
each door now holds a 50/50 chance of having the car.
让我们用一沓纸牌来解释这是为什么
Let’s use a deck of cards to understand why.
不看花色随机取出一张
Pick a card from this deck without looking.
这张卡有1/52的机会可能是黑桃A
This card has a 1/52 chance of being the ace of spades.
但现在 我准备把另外的50张翻过来 剩下一张不翻
But now, I’m going to flip over all the other cards except one,
翻开的没有一张是黑桃A
none of which are the ace of spades
剩下的两张牌中 哪张更有可能是黑桃A呢?
Of the two cards left, which one seems more likely to be the ace of spades?
是在52张中随意取出来的那张
The one you chose randomly out of a deck of 52
还是我们故意留下来没有翻的那张
or the one I purposefully and suspiciously left turned down.
事实证明 你的那张牌有1/52的机会是黑桃A
It turns out your card remains at a chance of 1/52,
而我有51/52的概率是黑桃A
while my card now has 51/52 probability of being the ace of spades.
在这三扇门中也是同样的道理
The same principle is ture with the three doors.
看着 我排除一扇门意味着
You see, when I removed the door, I did so with motive,
我明确的被告知这扇门后面有一只山羊
knowing there was a goat behind it,
只有两种情况存在
The only two scenarios that exists are
A:你选了对的门 我随便选一扇错的门告诉你后面是山羊
A, you chose the correct door and I’m arbitrarily picking one of the wrong choices to show you,
这种情况下保留你的选择就赢了
in which case staying will make you win;
或者B:选了错的门 我会告诉你另一扇错的门
or B, you pick the wrong door and I show you the other incorrect answer.
这时候你改选门就会赢
in which case switching will make you win.
情况A在你一开始选对门的时候发生
Scenario A will always happen when you choose the winning door,
情况B在你选了错的门后发生
and B will always happen when you pick a losing door.
因此 情况A有1/3的发生概率
Therefore A will happen 1/3 of the time,
情况B有2/3的发生概率
and B will happen 2/3 of the time.
所以三次游戏中你换门就有可能赢两次
As such switching your door wins two out of the three times.
这个矛盾一直困扰许多人 包括科学家还有数学家
This paradox has perplexed many people, including scientists and mathematicians to this day,
因为我们的直觉告诉我们 换门不会改变结果
because our gut tells us that switching will have no consequence.
但当我们运用平常的思维或计算机模拟器时
But when using formal calculations or computer simulators,
答案就是错的了
the results don’t lie.
换门会增加你胜出的几率
Switching your door increases the probability of winning.
我们用表再验证一次
Let’s see it one more time using a chart.
这是所有可能的情况
Here are all the possible scenarios.
汽车在门1 2 3后 每扇门你都有机会选到
The car is behind door one, two or three, and you have the choice of each three doors.
就以为着有9种可能的结果
This means there are nine possible outcomes.
我快速地告诉你
Let’s tell you them up quickly.
如果在门1后 你又选了门1 那你就得保留选择
If it’s behind door one and you chose door one, you should stay.
但如果你选了门2或门3你就应该换门了
But if you chose door two or three, you should switch.
如果在门2后 你又选了门2 那你应该保留选择
If it’s behind door two and you chose door two, you should stay,
但如果选了另两扇门你就得换门了
but the other two you should switch.
情况统计起来 发现9次中你应6次需要换门
Add it all up and you should switch six out of nine times.
所以 现在你还相信你一开始的直觉吗?
So, do you still trust your gut feeling?
你有亟待解决的问题吗?
Got a burning question you want answered?
在评论区或者在脸书推特下提问吧
Ask it in the comments or on Facebook and Twitter
然后订阅更多我们周更的科普视频吧~
and subscribe for more weekly science videos.

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视频概述

正确答案三选一,如果可以改选,你是否会改选呢?对的几率会改变吗?小问题大奥妙,听我给你解释吧~

听录译者

韩箱子

翻译译者

CRAB

审核员

审核员 W

视频来源

https://www.youtube.com/watch?v=9vRUxbzJZ9Y

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