In a high-profile legal case last week, the jury asked the judge to explain what was meant by the term “beyond reasonable doubt”. As is the custom of the English courts, the judge refused to clarify. 在近日一起备受瞩目的法律诉讼中，陪审团要求法官解释术语“排除合理怀疑”(beyond reasonable doubt)的含义。按照英国法庭的惯例，法官拒绝作出澄清。

But the jurors’ question is legitimate, and the attempt to answer it reveals much, not just about the law, but the analysis of complex problems. English law recognises two principal standards of proof. The criminal test is that a charge must be “beyond reasonable doubt”, while civil cases are decided on “the balance of probabilities”. 但陪审团成员的问题是合理的。尝试对这一问题作出解答，能够揭示出很多东西——不仅涉及法律，还涉及对复杂问题的分析。英国法律承认两种主要的证据标准。对于刑事案件，指控必须“排除合理怀疑”；民事案件则以“盖然性权衡”(the balance of probabilities)为裁定标准。

The meaning of these terms would seem obvious to anyone trained in basic statistics. Scientists think in terms of confidence intervals – they are inclined to accept a hypothesis if the probability that it is true exceeds 95 per cent. “Beyond reasonable doubt” appears to be a claim that there is a high probability that the hypothesis – the defendant’s guilt – is true. Perhaps criminal conviction requires a higher standard than the scientific norm – 99 per cent or even 99.9 per cent confidence is required to throw you in jail. “On the balance of probabilities” must surely mean that the probability the claim is well founded exceeds 50 per cent. 对任何接受过基本统计学培训的人而言，这些术语的含义似乎是明摆着的。科学家在思考问题时会基于置信区间(confidence interval)——如果某一假设为真的概率高于95%，他们便倾向于接受该假设。“排除合理怀疑”似乎是这样一种断言：它申明“被告有罪”这一假设有很高的概率为真。刑事定罪的裁定标准可能比科学研究的标准还要高：若要将某人投入监狱，置信水平需要达到99%乃至99.9%。“盖然性权衡”的含义则必然是，某项主张证据充分的概率高于50%。

And yet a brief conversation with experienced lawyers establishes that they do not interpret the terms in these ways. One famous illustration supposes you are knocked down by a bus, which you did not see (that is why it knocked you down). Say Company A operates more than half the buses in the town. Absent other evidence, the probability that your injuries were caused by a bus belonging to Company A is more than one half. But no court would determine that Company A was liable on that basis. 但与资深律师短暂交谈一番，便知他们并不这样解读这些术语。一个著名的例子是：假设你被一辆你没有看见的巴士撞倒（正因为你没有看见它，你才被撞倒），再假设甲公司运营着城里逾半数的巴士。若无其他证据，你被甲公司巴士撞伤的概率超过50%。但任何法庭都不会因此就判定甲公司负有法律责任。

A court approaches the issue in a different way. You must tell a story about yourself and the bus. Legal reasoning uses a narrative rather than a probabilistic approach, and when the courts are faced with probabilistic reasoning the result is often a damaging muddle – as, for example, the flawed testimony of Sir Roy Meadow on child deaths that led courts to wrongfully convict grieving parents of murdering their children. 法庭会以一直截然不同的方式处理此事。你必须叙述被撞经过。法律推理使用的是叙事方法，不是概率方法。如果法庭上出现概率推理，往往将造成破坏性的混乱。例如，罗伊•梅多爵士(Sir Roy Meadow)在儿童死亡问题上发表的漏洞百出的证言，导致法庭错误地把那些尚在悲痛中的死亡儿童的父母裁定为谋杀这些儿童的凶手。

The American legal system, tellingly, prefers the term “preponderance of the evidence” to “balance of probabilities”. To a statistician, it is not apparent that these terms have the same meaning. But lawyers take the view that broadly speaking they do. The reason is that the term “balance of probabilities” is not interpreted in the same way by lawyers and statisticians. 值得注意的是，比起“盖然性权衡”，美国法律体系更倾向使用术语“证据优势”(preponderance of the evidence)。统计学家似乎难以认同两者含义相同，但律师们却认为两者的含义大体上是相同的。原因是，律师和统计学家对术语“盖然性权衡”的解读方式不同。

When I have raised these issues with people with scientific training, they tend to reply that lawyers are mostly innumerate and with better education would learn to think in the same way as statisticians. Probabilistic reasoning has become the dominant method of structured thinking about problems involving risk and uncertainty – to such an extent that people who do not think this way are derided as incompetent and irrational. Yet this probabilistic approach, a recent intellectual development, was heavily implicated in the 2008 financial crisis. Legal systems have evolved over hundreds if not thousands of years, applying different modes of reasoning. 当我向接受过科学培训的人提起这些问题时，他们通常的回答是律师大多不懂数学，称如果律师接受更好的教育，便会掌握与统计学家相同的思考方式。概率推理已成为对涉及风险和不确定性的问题进行有条理思考的主流方式，以至于不使用概率推理来思考会被讥笑为无能、不理性。然而，直到近代才发展起来的概率方法与2008年金融危机难脱干系。而法律体系的发展即便没有几千年、也有几百年之久，它遵循的是另一种推理方式。

It is possible – common, even – to believe something is true without being confident in that belief. Or to be sure that, say, a housing bubble will burst without being able to attach a high probability to any specific event, such as “house prices will fall 20 per cent in the next year”. A court is concerned to establish the degree of confidence in a narrative, not to measure a probability in a model. 认为某事为真、而不对这一判断抱绝对把握，是可能乃至常见的。或者，我们可以确信房地产泡沫将会破裂，但不用认为某一特定事件——如“房价将在明年下跌20%”——具有较高的概率。法庭的责任是确定一个叙事的可信度，而不是衡量一个模型中的某种概率。

Such narrative reasoning is the most effective means humans have developed of handling complex and ill-defined problems. A court can rarely establish a complete account of the probabilities of the events on which it is required to adjudicate. Similarly, an individual cannot know how career and relationships will evolve. A business must be steered into a future of unknown and unknowable dimensions. 这种叙事推理是人类构想出来的应对复杂不清问题的最有效方式。法庭很难对其需要判决之事的概率给出完整的解释。与此类似的是，一个人是不可能知晓未来自己的事业和人际关系会如何发展的。一家企业驶向的未来，必将是未知且不可知的。

So while probabilistic thinking is indispensable when dealing with recurrent events or histories that repeat themselves, it often fails when we try to apply it to idiosyncratic events and open-ended problems. We cope with these situations by telling stories, and we base decisions on their persuasiveness. Not because we are stupid, but because experience has told us it is the best way to cope. That is why novels sell better than statistics texts. 因此，概率思维在应对复发事件或重演的历史时是不可或缺的，但面对特殊事件和没有确定答案的问题时，它却往往无能为力。我们应对这些情况的方式是讲故事，并依据故事的说服力做出判断。这么做不是因为我们笨，而是因为经验表明它是最佳的应对方式。正是因为如此，小说才比统计学课本卖得好。