约翰·凯=文 2005年8月20日


蒙提霍尔问题(Monty Hall problem)是以 20世纪70年代一套电视智力游戏节目“换换乐”(Let’s Make a Deal)主持人的名字命名的。胜出的参赛者从三个关闭的盒子中做出选择。一个盒子中装着汽车钥匙，另外两个是山羊的图片。选好之后，蒙提会打开剩下两个盒子中的一个，露出山羊图片。然后他就会撺掇参赛者改变选择。那么，参赛者应该转而选择另一个关闭的盒子吗？

当问题的答案刊登在一家美国杂志上时，成千的读者（包括统计学教授）断言这是个错误。据说著名数学家保罗·艾狄胥( Paul Erdös)临死时仍在思考蒙提霍尔问题。但答案却真的是：是的，你应该改变选择。

这并不是唯一的案例，说明直觉与数学概率不符。1000人中会有一人患上罕见疾病。一位朋友在该疾病的检测中显示为阳性，而这种检测的正确率为99％。那么这位朋友的患病几率有多大? 答案是：一点都不可能。在随机抽取的1000人中，平均有 10人被误诊为阳性，而只有一个人的疾病诊断是正确的。但大多数人，包括很多医生，却有不同的想法。科学作家斯蒂芬·杰伊·古尔德 (Stephen Jay Gould)说：“人的大脑还没有进化到能理解概率的程度。”

上月，医学委员会(General Medical Council)吊销了儿科医师罗伊·梅多爵士(Roy Meadow)的行医资格。他在对萨莉·克拉克(Sally Clarke) 的刑事诉讼案中给出了误导性证词。萨莉的两个幼子都在襁褓中夭折。当克拉克夫人被控谋杀时，罗伊爵士告诉陪审团，一个家庭中有两个婴儿相继猝死的概率为 “七千三百万分之一”。

但是尽管纪律委员会听取了著名统计学家的证词，但在理解概率理论在这类案件中的应用时，似乎并没有比罗伊爵士强多少。委员会认为他低估了婴儿猝死的机率，并且他没有把基因和环境因素考虑在内，因为发生一次婴儿猝死事件的家庭，要比一般家庭更有可能发生第二次。但是尽管认可这些影响，罗伊爵士的主要结论还是正确的。发生这样的事件本不可能。在同一个家庭中发生两次这样的事件则更加不可能。

这当然是不可能的。引发刑事诉讼的事件通常都不太可能发生，否则，法庭将无法处理大量积压案件。如果奥萨马·本·拉登(Osama bin Laden ) 有一天被送交法庭审判，问题将不是“两架飞机同时在9月11日撞上世界贸易中心，这可能吗？” （这个问题的答案是 “不可能”）；而是“假设两架飞机在9月11日撞上世界贸易中心，本·拉登有可能对此负责吗？”把这两个不同问题混为一谈就是大家所知的 “检察官之谬论”(prosecutor’s fallacy)。

一个家庭发生一次婴儿猝死事件，会增加另一起猝死案例的概率，但一个家庭内的谋杀将更有可能导致另一起谋杀：狠毒的父母会一直狠毒。罗伊爵士认为两个婴儿猝死比一个更加可疑，这可能没有过错。但上诉法院释放了克拉克夫人，认为仅凭这种统计学上的证据无法在排除合理怀疑(reasonable doubt)基础上定罪，这种理由也的确说得通。

谈到概率，你不应相信医生或律师，甚至连自己都不敢轻易相信。在剖析技术问题方面，法律诉讼辩护是一个糟糕的场所。而且我们不能因谴责某个特定的人而抹去集体在错误判断上的责任。

互联网泡沫的破灭应归咎于商业和财政系统，而远非伯尼·埃博斯 (Bernie Ebbers) 和柏亨利(Henry Blodget) 的过错。司法程序的错误比过于自信的教授更严重，导致了对萨莉·克拉克这类女性的不公判决。但替罪羊已有有很长历史—至少从《利未记》（基督教圣经《旧约全书》中的一卷）开始：“亚伦两手按在羊头上，承认以色列人诸般的罪孽、过犯…… 这羊要担当他们一切的罪孽，带到无人之地。”

问题

以下是蒙提霍尔问题的一个著名的叙述，来自 Craig F. Whitaker 于1990年寄给《展示杂志》（Parade Magazine）玛莉莲·莎凡（Marilyn vos Savant）专栏的信件：

假设你正在参加一个游戏节目，你被要求在三扇门中选择一扇：其中一扇后面有一辆车；其余两扇后面则是山羊。你选择了一道门，假设是一号门，然后知道门后面有甚么的主持人，开启了另一扇后面有山羊的门，假设是三号门。他然后问你：“你想选择二号门吗？”转换你的选择对你来说是一种优势吗？

以上叙述是对 Steve Selvin 于1975年2月寄给 American Statistician 杂志的叙述的改编版本。如上文所述，蒙提霍尔问题是游戏节目环节的一个引申；蒙提·霍尔在节目中的确会开启一扇错误的门，以增加刺激感，但不会容许玩者更改他们的选择。如蒙提·霍尔寄给 Selvin 的信中所写：

　　——如果你上过我的节目的话，你会觉得游戏很快—选定以后就没有交换的机会。

Selvin 在随后寄给 American Statistician 的信件中(1975年8月) 首次使用了“蒙提霍尔问题”这个名称。

一个实质上完全相同的问题于1959年以“三囚犯问题”（three prisoners problem）的形式出现在马丁·葛登能（Martin Gardner）的《数学游戏》专栏中。葛登能版本的选择过程叙述得十分明确，避免了《展示杂志》版本里隐含的前提条件。

这条问题的首次出现，可能是在1889年约瑟夫·贝特朗所著的 Calcul des probabilités 一书中。 在这本书中，这条问题被称为“贝特朗箱子悖论”（Bertrand's Box Paradox）。

Mueser 和 Granberg 透过在主持人的行为身上加上明确的限制条件，提出了对这个问题的一种不含糊的陈述︰

[*]参赛者在三扇门中挑选一扇。他并不知道内里有甚么。
[*]主持人知道每扇门后面有什么。
[*]主持人必须开启剩下的其中一扇门，并且必须提供换门的机会。
[*]主持人永远都会挑一扇有山羊的门。
[*]如果参赛者挑了一扇有山羊的门，主持人必须挑另一扇有山羊的门。
[*]如果参赛者挑了一扇有汽车的门，主持人随机在另外两扇门中挑一扇有山羊的门。
[*]参赛者会被问是否保持他的原来选择，还是转而选择剩下的那一道门。

转换选择可以增加参赛者的机会吗？


解答

问题的答案是可以：当参赛者转向另一扇门而不是继续维持原先的选择时，赢得汽车的机会将会加倍。

有三种可能的情况，全部都有相等的可能性(1/3)︰

[*]参赛者挑山羊一号，主持人挑山羊二号。转换将赢得汽车。
[*]参赛者挑山羊二号，主持人挑山羊一号。转换将赢得汽车。
[*]参赛者挑汽车，主持人挑两头山羊的任何一头。转换将失败。

在头两种情况，参赛者可以透过转换选择而赢得汽车。第三种情况是唯一一种参赛者透过保持原来选择而赢的情况。因为三种情况中有两种是透过转换选择而赢的，所以透过转换选择而赢的概率是2/3。

如果没有最初选择，或者如果主持人随便打开一扇门，又或者如果主持人只会在参赛者作出某些选择时才会问是否转换选择的话，问题都将会变得不一样。例如，如果主持人先从两只山羊中剔除其中一只，然后才叫参赛者作出选择的话，选中的机会将会是 1/2。

另一种解答是假设你永远都会转换选择，这时赢的唯一可能性就是选一扇没有车的门，因为主持人其后必定会开启另外一扇有山羊的门，消除了转换选择后选到另外一只羊的可能性。因为门的总数是三扇，有山羊的门的总数是两扇，所以转换选择而赢得汽车的概率是2/3，与初次选择时选中有山羊的门的概率一样。



source: http://zh.wikipedia.org/wiki/蒙提霍爾問題

天才數學家艾狄胥對這問題就曾栽過觔斗，法桑尼有一次和艾狄胥一同造訪加州聖羅沙，法桑尼想測試當時已是機率大王的艾狄胥，對於機率的直覺。令法桑尼訝異的是艾狄胥竟堅持換與不換的機率都是1/2，甚至對法桑尼的解釋甚感不解而憤怒，直到法桑尼利用電腦模擬，艾狄胥才信服。可見連數學天才都有被機率法則戲弄的一天吧。

The prosecutor's fallacy is a fallacy of statistical reasoning that takes several forms.

[*] One form of the fallacy results from neglecting the a priori odds of a defendant being guilty--i.e., the chance of an individual being guilty absenting specific evidence is the gross incident rate of perpetrators in the general population. When a prosecutor has collected some evidence (for instance a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. The probability of innocence would only necessarily be comparably tiny if the a priori odds of guilt were 1:1--that is, if the probability of innocence is computed with an a priori presumption of guilt, which violates the conventional presumption of innocence.

[*] Another form of the fallacy results when evidence is compared against a large database. The mere size of the database elevates the likelihood of finding a match by pure chance alone. i.e., DNA evidence is soundest when a match is found after a single directed comparison because the existence of matches against a large database where the test sample is of poor quality (common for recovered evidence) is very likely by mere chance.

[*] The terms "prosecutor's fallacy" and "defense attorney's fallacy" were originated by William C. Thompson and Edward Schumann in their classic article "Interpretation of statistical evidence in criminal trials: The prosecutor's fallacy and the defense attorney's fallacy." Law and Human Behavior, 1987, 11, 167-187.

Why this is fallacious: several examples

A concrete example can make it clear why this reasoning is fallacious. Suppose there is a one-in-a-million chance of a match given that the accused is innocent. The prosecutor says this means there is only a one-in-a-million chance of innocence. But in a community of 10 million people, one expects about 10 matches by pure chance, and the accused may be one of those ten. That would indicate only about one-in-ten chance of guilt, if no other evidence is available.

In another scenario, assume a rape has been committed and that a sample is compared against 20,000 men that have their DNA on record in a database. A match is found, and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000. This does not mean the probability that the suspect is innocent is 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance; the probability that there was at least one DNA match is



which is considerably more than 1 in 10,000. (The probability that exactly one of the 20,000 men has a match is about 27%, which is still rather high.)

Now consider this case: you win the lottery jackpot. You are then charged with having cheated, for instance with having bribed lottery officials. At the trial, the prosecutor points out that winning the lottery without cheating is extremely unlikely, and that therefore your being innocent must be comparably unlikely. This reasoning is clearly faulty: the prosecutor failed to mention that cheating lottery winners are much more rare than honest winners.


Mathematical analysis

We can view finding a person innocent or guilty in mathematical terms as a form of binary classification.

We start with a thought experiment. I have a big bowl with one thousand balls, some of them made of wood, some of them made of plastic. I know that 100% of the wooden balls are white, and only 1% of the plastic balls are white, the others being red. Now I pull a ball out at random, and observe that it is actually white. Given this information, how likely is it that the ball I pulled out is made of wood? Is it 99%? No! Maybe the bowl contains only 10 wooden and 990 plastic balls. Without that information (the a priori probability), we cannot make any statement. In this thought experiment, you should think of the wooden balls as "accused is guilty", the plastic balls as "accused is innocent", and the white balls as "the evidence is observed".

The fallacy can be analyzed using conditional probability: Suppose E is the observed evidence, and I stands for "accused is innocent". We know that P(E|I) (the probability that the evidence would be observed if the accused were innocent) is tiny. The prosecutor wrongly concludes that P(I|E) (the probability that the accused is innocent, given the evidence E) is comparatively tiny. However, P(E|I) and P(I|E) are quite different; using Bayes' theorem we see

　　P(I|E) = P(E|I) · P(I) / P(E)

So the a priori probability of innocence P(I) and the overall probability of the observed evidence P(E) need to be taken into account. If P(I) is much larger than P(E), then P(I|E) can be large as well.

We can also formulate Bayes' theorem with odds:

　　Odds(I|E) = Odds(I) · P(E|I)/P(E|~I)

Without knowledge of the a priori odds of I, the small value of P(E|I) does not necessarily imply that Odds(I|E) is small. (P(E|~I), the probability that the evidence is observed given the accused is guilty, is assumed to be high.)

The fallacy lies in the fact that the a priori probability of guilt is not taken into account. If this probability is small, then the only effect of the presented evidence is to increase that probability somewhat, but not necessarily dramatically. (In the earlier example of a 10 million city, the presented evidence raises the a priori probability of guilt of 1 in 10 million to an a posteriori probability of guilt of 1 in 10.)

The prosecutor's fallacy is therefore no fallacy if the a priori odds of guilt are assumed to be 1:1. In an Bayesian approach to personal probabilities, where probabilities represent degrees of belief of reasonable persons, this assumption can be justified as follows: a completely unbiased person, without having been shown any evidence and without any prior knowledge, will estimate the a priori odds of guilt as 1:1.

In this picture then, the fallacy consists in the fact that the prosecutor claims an absolutely low probability of innocence, without mentioning that the information he conveniently omitted would have led to a different estimate.

In legal terms, the prosecutor is operating in terms of a presumption of guilt, something which is contrary to the normal presumption of innocence where a person is assumed to be innocent unless found guilty. A more reasonable value for the prior odds of guilt might be a value estimated from the overall frequency of the given crime in the general population.


Defendant's fallacy

The defendant's fallacy (taking the earlier example) would be to say, "We would expect 10 matches in this city of 10 million people, so this particular piece of evidence suggests there is 90% chance that the accused is innocent. So this evidence cannot be used to point to a conclusion of guilt, and should be excluded."

The problem with the defendant's argument is that there may be other available evidence which on its own is also not conclusive. For example if CCTV cameras surrounding the scene of the crime spotted one hundred people there at the relevant time, one of which was the accused, then the defendant could claim: "The video suggests a 99% chance that the defendant is innocent. The match suggested a 90% chance of innocence. So the conclusion should be a finding of innocence."

When the photographic evidence is combined with the match, the two together point strongly towards guilt, since (assuming the chance of being in the photograph and having the match are independent) the chance that the accused is innocent falls to about 0.01%. This low probability of innocence is not proof of guilt.


The Sally Clark case

Consider for instance the case of Sally Clark, who was accused in 1998 of having killed her first child at 11 weeks of age, then conceived another child and allegedly killed it at 8 weeks of age. The defense claimed that these were two cases of sudden infant death syndrome; neither prosecution nor defense offered any other explanations for the deaths. The prosecution had expert witness Sir Roy Meadow testify that the probability of two children in the same family dying from sudden infant death syndrome is about 1 in 73 million. But based on this alone, it is likely that there would be at least one person in the country to whom this has occurred. To provide proper context for this number, the probability of a mother killing one child, conceiving another and killing that one too, should have been estimated and compared to the 1 in 73 million figure, but it was not. Ms. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistake. (See link at end of article.) A higher court later quashed Sally Clark's conviction, on other grounds, on 29 January 2003.



source: http://en.wikipedia.org/wiki/Prosecutor's_fallacy

Re.: http://en.wikipedia.org/wiki/Talk:Prosecutor's_fallacy
http://www.conceptstew.co.uk/PAGES/prosecutors_fallacy.html

伯尼·埃博斯 (Bernie Ebbers)，美国电信运营商世界通信公司(WorldCom)前首席执行官。世通公司是一家经营长途电话业务的小公司，通过60多笔成功的生意运作，它在15年时间里迅速成长为仅次于美国电话电报公司(AT&&T)的电信业巨头。 2000年，网络经济泡沫破碎，世通公司的股票市值从巅峰时期的1500亿美元缩水至不足500亿美元。2002年3月，美国证券交易委员会发现世通公司向伯尼·埃博斯提供了4亿多美元的违规贷款，以及客户帐单、销售佣金等诸多问题。迫于调查压力，伯尼·埃博斯于4月辞职。6月25日，世通公司承认它在过去几年中通过做假账虚报了38亿美元利润。7月21日，世通公司向美国纽约南区破产法院申请破产。

柏亨利(Henry Blodget) ，美林银行著名网络投资分析师。1998年，他准确预测240美元的Amazon股票将飙升到400美元，从而赢得广泛声望。2000年，他被《金融机构投资者》杂志评选为当年的头牌分析师。众多投资者受他的预言影响将资金投入互联网行业，形成了美国历史上最大的互联网金融泡沫。2001年互联网泡沫破灭时，柏亨利依然对网络股发表利好评价，从而导致投资者蒙受重大损失。投资者与政府机构开始怀疑包括美林银行在内的投资公司所发表观点的客观公正性，分析师被要求公布其个人持股状况，并不得购买自己公司跟踪的股票。12月，柏亨利被迫从美林辞职，获得大约200万美元的离职金。2003年1月，纳斯达克市场起诉柏亨利曾上调一些互联网类股的评级，以便赢得利润丰厚的投资银行业务，同时却隐瞒了他自己对于这些公司的疑问。4月，柏亨利和纳斯达克达成和解协议，柏亨利支付200万美元罚金，并被没收200万美元非法所得，终生不得从事经纪业务，换取纳斯达克对其免于起诉。