Lessons Learned: Slow Thinking

By John Marhoefer Owner, Entech Innovative Engineering


Here is an excerpt from “How Not to Be Wrong, The Power of Mathematical Thinking” by Jordan Ellenberg (2014). In this excerpt, Mr. Ellenberg recounts a true story from World War II.


When American planes came back from engagements over Europe, they were covered in bullet holes. But the damage was not uniformly distributed across the aircraft. There were more bullet holes in the fuselage, not so many in the engines.

Table 1


“The officers saw an opportunity for efficiency; you can get the same protection with less armor if you concentrate the armor on the places with the greatest need, where the planes are getting hit the most. But exactly how much more armor belonged on those parts of the planes.”


Based upon this data, the military wanted to know where to add armor to the planes. As part of the story, Ellenberg recounts that this question was posed to a famous mathematician, Abraham Wald. Wald figured out the answer and it was unexpected.

Wald’s insight, according to Ellenberg, was to recognize that the bullet hole density on average for a plane could be expected to be THE SAME regardless of what or where it was on the plane. Remember that the data in Table 1 is for planes that returned from combat flights. Wald recognized that the “missing holes” for the lower density figures were on planes that never did return.


In other words, the reason planes were coming back with relatively fewer hits to the engine on average is that planes that got hit in the engines often did NOT return. Those planes, the ones that did not return, would have pulled the average up to “normal.” But since they never did return, they were not counted.


Wald’s recommendation for armoring was accordingly:

Table 2


This is a pretty powerful example of slow thinking and the benefits of it.

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