哪些基本的数学技巧是每个人都应该掌握的?

What are the basic math skills that everyone should know?
哪些基本的数学技巧是每个人都应该掌握的?

Nipun Ramakrishnan | 丁仔 译
In the heat of World War II, a lot of American planes were coming back riddledwith bullet holes. One of the interesting things about this situation was that thebullet holes were concentrated in different regions of the plane. Some planescame back with bullet holes in the engine, some in the fuselage, etc.
在二战战况最激烈的时候,很多幸存飞回的美国飞机被子弹打得千疮百孔。有个很有意思的地方是,弹坑密集出现在飞机的几个区域,它们分别是引擎,机身等等。

The military saw an opportunity for efficiency. They realized that if they couldfocus on fortifying the areas of the plane that were the most prone to being hitand seriously damaged, they could not only save more planes, but also save onthe amount of armor they use. The military came to Abraham Wald and a fewother leading statistical researchers with the following data on the survivingplanes and where the bullet holes were concentrated:
于是,军队发现了提高效率的可能性。他们认为,如果着重强化飞机的几个容易被打击或者容易损坏区域,那么不仅能够拯救更多的飞机,而且还能节省装甲材料(抵御敌火的材料)的用量。军方找到了 Abraham Wald (一位数学家)还有几个顶尖的数据分析师,给他们提供了幸存飞机的数据,上面有弹坑分布的区域。

Based on this data, which part of the plane do you think the military should fortify?
基于这份数据,你觉得飞机的哪一部分是军方应该强化的呢?

Most people would answer that it should be the fuselage, and they would be wrong.
大多数人都觉得应该是机身,但你们很可能错了。

The more accurate answer is the engine. Wald realized that the data that he had been given was not an appropriate representation of the problem. The reason that there were fewer hits on the engine is that the planes that got hit in the engine were not coming back. And the fact that a majority of the surviving planes had bullet holes in the fuselage is pretty strong evidence that the planes could tolerate damage to the fuselage.
更准确的答案应该是引擎。Wald 注意到他们给的这份数据并没有很好的反应问题所在。其中,引擎受到的打击数比较少的原因是那些引擎被打爆的飞机已经飞不回来了。而大多数存活下来的飞机居然还有那么多弹坑足以说明,这些飞机的机身已经足以抵御攻击了。
Wald realized that there were only two explanations for the data:
Wald 认为对于这份数据只有两个解释:

1. Bullets just happened to hit every other part of the plane more often than ithit the engine
飞机的其它区域比引擎更容易被打到。
2. The engine is a point of vulnerability
引擎才是弱点。

And it doesn’t take a genius to see that the second explanation is a lot morereasonable than the first and after Wald’s recommendations were put intoeffect, a lot more American planes were saved.
一般人都可以看得出,第二个解释比第一个更合理。听取了 Wald 的建议重新改造之后,有更多的美国飞机得以幸存。

One of the most underrated mathematical skills is the ability to understand the underlying assumptions made by data. Abraham Wald was able to solve this problem because he realized the military made the assumption that the data represented all planes rather than just the surviving planes. A lot of the statistics that are thrown out in the world today tend to fool a lot of people into believing what they want to hear, but how you separate the facts from the bullshit is by understanding the assumptions made by the data.
能够读出数据背后隐藏的前提,是我们常常低估的数学能力。Abraham Wald 能够解决这个问题是因为他意识到军方假设数据能够代表所有飞机的情况,而不只是幸存的飞机。今天,我们在很多地方看到的数据都是有误导性的,它试图呈现你们想看到的东西。如何区分事实和误导就取决于你能否读懂数据假设的前提了。

If you want to see more of these types of examples, I highly recommend Jordan Ellenberg’s How Not to Be Wrong: The Power of Mathematical Thinking which does a great job explaining the Wald example as well as many other examples in which forgetting about the assumptions behind the data can lead to ridiculous conclusions.
如果你想看更多此类的例子,我强烈建议你去读 Jordan Ellenberg 的 《如何不犯错:数学思维的力量》。书里面很好地讲述了 Wald 的例子,以及许多忽略数据假设的前提而得出荒谬结论的案例。

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