【英语生活】直觉比概率论更有用？

Our ancestors gambled around camp-fires on the savannahs thousands of years ago. The ancient Greeks had games of chance, embryonic financial markets and some of the finest mathematicians who have ever lived. But they did not discover the rather simple arithmetic of probability and the methods we use today are no more than 300 years old.

Even today, many people struggle to learn probability theory and experts make mistakes in applying it. This can have disastrous results, as in the imprisonment of innocent mothers whose children had been victims of unlikely, but not that unlikely, sequences of accidental death. Stephen Jay Gould is only one of many science writers to have observed that human minds are not well adapted to dealing with issues of probability.



The two-box problem, which I described last week, is one in which a knowledge of probabilities seems to be a hindrance rather than a help. The puzzle offers you the choice of two boxes, one containing more money than the other. Once you have made a decision, you are shown what is in your preferred box. Do you stick with your original choice, or switch?



You have so little information that there seems no rational basis for decision. Yet the dilemma is real. In our personal lives, in business and finance, in employment and in house-hunting, we repeatedly encounter two-box problems – should we stick with what we know or switch to something about which we know much less?



The extraordinary feature of the two-box problem is that there is a strategy that seems better than always switching or always sticking, one that beats random choice even in a situation of almost total ignorance.



Before the game starts, focus on a sum of money. It does not matter what the amount is – say, £100. The “threshold strategy” is to switch if the box you choose contains less than £100 and to stick if it contains more. The threshold strategy gives you a better-than-even chance of getting the larger sum. It does so for any value of the threshold you choose.



If both boxes have less than £100, or more than £100, then the probability that you get the larger sum from your random choice remains one-half. But if one box has less than £100 and the other has more than £100, adopting the threshold strategy makes sure you get the larger sum. Since there is at least a possibility that the amounts in the boxes lie in this range, the threshold strategy must increase your chance of winning.



So how should you set the threshold? There are two criteria. How likely is it that the box with the smaller amount of money has an amount below, but not too much below, the figure you set? There may not be much benefit in setting a very high or very low threshold. So if you have some idea, however vague, about the range of possible contents, you can tweak the threshold strategy to your advantage. Second, choose a threshold in the range of sums of money that would make a real difference to you: if £20,000 would not transform your life but £50,000 would, then go for £50,000. The range from £25,000 to £50,000 is the range in which the benefit from switching might be greatest.



The curious feature of the threshold strategy is that the maths is surprising but the intuition is familiar. In real-life search problems the principle of “be realistic but look for something that will make a difference” represents typical behaviour. In the version of the two-box problem I described last week, where you had the additional information that one box contained twice as much money as the other, there seemed always to be an argument for switching; the potential gain is always twice the seemingly equally likely potential loss. But this conclusion is wrong. Once more, our intuition runs ahead of our mathematical understanding.



Probability theory works well for a limited class of – mostly artificial – problems, such as coin tossing and roulette. But the real world is much more open-ended and there is usually fundamental uncertainty about both the nature of the outcomes and the process that gives rise to them. Perhaps the reason we do not use probability theory much is that it is not all that useful

数万年前，我们的祖先就在大草原上围着营火赌博。古希腊人有博弈游戏、萌芽状态的金融市场以及一些有史以来最出色的数学家。但古人们没有发现十分简单的概率算法，如今我们运用的这些方法，其历史不过300年。



即便在今天，许多人对学习概率论感到费力，专家在应用概率论时也会犯错。这可能导致灾难性的后果，就像无辜母亲被判入狱，只因为她们的子女接连意外死亡，这种意外概率很小，但并非那么没有可能。诸多科学作家中，只有斯蒂芬•杰伊•古尔德(Stephen Jay Gould)一人观察到人类头脑不适合处理概率问题。



在“两个盒子问题”（我上周对此进行了描述）中，概率方面的知识似乎成了妨碍而不是帮助。这个难题给出两个盒子供你选择，其中一个盒子里的钱比另外一个多。一旦做出决定，别人会向你展示选中盒子里的东西。这时候你是坚持原来的选择，还是换一个呢？



你得到的信息如此之少，似乎不具备理性的决策基础。然而这种困境却是实实在在的。在我们的个人生活中，在商业和金融界，在就业与选购房屋时，我们不断遇到“两个盒子问题”——我们应坚持已知选择，还是转向所知甚少的东西呢？



两个盒子问题的一个不同寻常的特点是，有一种策略似乎优于总是转变选择或坚持选择，即使在几乎对所选对象一无所知的情况下，这种策略也胜过随机选择。





在游戏开始前，确定一个目标金额。这个金额的大小没有关系——比如100英镑。这种“门槛策略”是，如果你选择的盒子里不到100英镑就转变选择，而如果超过100英镑就坚持选择。门槛策略使你有超过一半的机会获得较大金额。不管你所选择的门槛金额大小，它都会起到这个效果。

如果两个盒子里的钱都不到100英镑或都超过100英镑，那么通过随机选择获得更大金额的概率仍然是一半。但如果一个盒子里不到100英镑而另一个超过100英镑，采取门槛策略能确保获得较大金额。既然两个盒子里的钱至少有可能在这个范围之内，门槛策略就必然能增大获胜机会。



那么你应当如何设定门槛呢？有两条标准。钱数较少的盒子里低于（但不过分低于）设定金额的可能性有多大？门槛定得过高或过低都没有多少好处。因此，如果对可能结果的区间有些概念，即使很模糊，你也能利用门槛策略产生对自己有利的结果。其次，在真正能对你产生影响的金额区间中选择一个门槛：如果2万英镑不能改变你的生活而5万英镑可以，那就选择5万英镑。2.5万到5万英镑是从转变选择中可能获益最大的区间。







门槛策略的奇妙特点在于，数学结果令人惊讶，而直觉则很熟悉。在现实生活的探索问题中，代表典型行为的原则是，“保持现实心态，但同时寻求足以带来不同的选项”。在我上周描述的两个盒子问题中，你所知道的额外信息是，一个盒子里的钱是另外一个的两倍。因此改变选择似乎总是明智的做法，因为潜在收益总是潜在损失的两倍，而两者发生的概率似乎相同。但这个结论是错误的。直觉再一次胜过了数学理解。





概率论只适用于有限类别的问题——这些问题大多是人为设定的，如抛硬币和轮盘赌。但现实世界存在更多可能性，结果的性质乃至产生结果的过程，通常都存在根本的不确定性。也许，我们之所以不常运用概率论，就是因为它不那么有用。



译者/张征