关于电影《决胜21点》中的一道数学题

这是很久以前看过的电影《决胜21点》中的一段对话，场景是在一节数学课上，教授问了Ben一道数学题，以下是摘取的电影台词：

Teacher: Ben, suppose you are on a game show.And you are given a chance to choose from three different doors, all right? Now, behind one of the doors is a new car. Behind the other two, goats. Which door would you choose, Ben?

Ben: Door number one?

Teacher: Door number one. Ben chooses door number one. All right, now, the game show host, who, by the way, knows what’s behind all the other doors, decides to open another door. Let’s say he chooses door number three. Behind which sits a goat. Now… He says, “Ben, do you want to stay with door number one or go with door number two?” Now, is it in your interest to switch your choice?

Ben: Yeah.

Teacher: Well, wait. Remember, the host knows where the car is, so how do you know he’s not playing a trick on you? Trying to use reverse psychology to get you to pick a goat?

Ben: Well, I wouldn’t really care. I mean, my answer’s based on statistic. Based on variable change.

Teacher: Variable change? But he just asked you a simple question.

Ben: Yeah, which changed everything.

Teacher: Enlighten us.

Ben: Well, when I was originally asked to choose a door, I had a 33.3% chance of choosing right. But after he opens one of the doors and then re-offers me the choice, it’s now 66.7% if I choose to switch. So, yeah, I’ll take door number two, and thank you for that extra 33.3%.

Teacher: Exactly. People, remember, if you don’t know which door to open, always account for variable change. Now, see, most people wouldn’t take the switch out of paranoia, fear, emotions. But Mr.Campbell, he kept emotions aside and let simple math get his ass into a brand-new car!

大意是三扇门，门后有一辆车和两只羊，当主人公选择一号门后，主持人将三号门打开并显示了门后是一只羊（主持人是知道门后都是什么的），主持人接下来问主人公是坚持选择一号门不变还是重新选择二号门。

请读者们先考虑一下你会如何选择，并给出合理的解释。

故事继续，主人公选择了二号门，他的解释是，初始选择一号门时，门后是车的概率是33.3%，当得知三号门后是一只羊后，自己选择二号门，则成功的概率是66.7%。他还要谢谢主持人给了他额外的33.3%。

咋听起来很难理解（也许你和电影中的天才一样秒懂），这里我还是想了有一段时间，还是再给读者们一段时间去考虑这解释背后的统计原理吧。

以下是我不成熟但自己认为说的通的解答（想到后感觉如此简单）：

主人公选择了一号门，门后是车的概率为33.3%。而二号门后是车也是33.3%，当然三号门也是。我们索性将三扇门分为两组，Ａ组和Ｂ组。Ａ组包含一号门，Ｂ组包含二号和三号门。那么车在Ａ组中的概率是33.3%，在Ｂ组则是66.7%（两扇门后有车的概率和）。当主持人将三号门排除后，本应落在二号门或三号门的概率则仅仅落在二号门，也就是说Ｂ组只有一辆车，概率仍然是66.7%。于是，选择二号门将会有66.7%的概率门后是车。

不知道我的解释是否可以使你信服，如果你有更好的解释，欢迎交流。

有关该问题的更多讨论，请访问果壳网并搜索“三扇门”。

这里也有国外关于这个问题的解答蒙提霍尔问题。