Abraham Wald and the bullet holes

So I’m constantly on the hunt for ways to relate maths to real-life, and to find hooks for topics. Over the summer I read Jordan Ellenberg’s book ‘How Not to Be Wrong’ and the following story is lifted straight from that book. It’s a great intro to a probability topic.

The premise
You’re in the air force during World War 2. Your job is to stop your planes getting shot down by enemy fighters, by armouring them. But armour makes a plane heavier, less manoeuvrable and uses more fuel. So you have to compromise between weight and protection from bullets.

The data
You are presented with the table below, which gives average bullet holes on planes returning from missions:

Section of plane Bullet holes per square metre
Engine 1.11
Fuselage 1.73
Fuel system 1.55
Rest of the plane 1.8

The question:

Where is the best place to put extra armour?

The answer?

On the engines.

Counterintuitive, isn’t it! But this is exactly what Abraham Wald, a famed statistician working with the United States Air Force, did in 1943. Why?! I hear you cry. It’s the place with the fewest bullet holes, so must be hardest to hit. Why would you waste armour there, when you could put it on the fuselage or the rest of the plane.

Seeing the bigger picture:

Wald’s genius was simply to ask: ‘Where are the missing holes?’ Where are the bullet holes that we aren’t counting? And the answer is on planes that have been shot down. The reason planes are coming back with fewer holes in their engines are because holes in engines tend to make planes crash. If a plane is able to get home with lots of holes in its fuselage, then those holes should be tolerated. Put the armour where holes can’t be tolerated – the engines.


Relevance to the classroom:

Maths without relevance to the real-world is boring, at least for some students. Stories like this are valuable for two reasons. Firstly they show a practical use for some of the maths that students are learning (probability) as well as highlighting the importance of critical thinking about that maths. But secondly, they’re just interesting stories, that help students to get hooked into the lesson. And that’s invaluable.


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