You must be careful when applying this sort of logic because it assumes that what happens to a plane after its shot is entirely deterministic, ie. if it gets shot in the tail fin it crashes, if it gets shot in the wing it doesn't. That may or may be accurate for planes (I really don't know) but may not for other things.
Suppose anywhere the plane was shot led to a 1/5 chance of the plane crashing, meaning that all places are equally deserving of armor. Suppose also that the wings comprised about 75% of the surface area of the plane. You'd see 3 planes returning with bullet holes in a wing for every 1 with a bullet hole somewhere else. That doesn't mean somewhere else is a better place to put the armor.
It's easy to see how it could be possible, if the odds of a crash weren't uniform (ie if the wings had a slightly higher than average chance of causing a crash if shot in my example above) you could easily come to the wrong conclusion.
Think of the scale. 40,000 US/British planes were lost or damaged beyond repair, nevermind the ones that got a few bullet/flack holes and survived - most planes every mission. Thousands of planes would meet for a single bombing run. One mission would give you a good enough sample.
Also note that between 5 and 20% of the planes didn't come back from each mission. How many more were damaged every run? Most? 15,000 flak cannons protected Europe, many of them radar guided/aimed. Strategic bombing was an incredibly crappy job: you died more often than a Marine in the Pacific.
Suppose anywhere the plane was shot led to a 1/5 chance of the plane crashing, meaning that all places are equally deserving of armor. Suppose also that the wings comprised about 75% of the surface area of the plane. You'd see 3 planes returning with bullet holes in a wing for every 1 with a bullet hole somewhere else.
So do your study with the plane divided up into equal-sized sections. Or look for clustering and divide it based on that.
Yea. The key assumption for the "put the armor where the bullets aren't" is that the initial bullets are spread evenly over the surface area (or, rather, that the bullets are spread evenly over the cross-section, which you then average over the typical orientation of the plane as it takes enemy fire). This turns out to be a good assumption because planes typically take highly dispersed enemy fire compared to the size of the plane.
On the other hand, if the enemy had extremely accurate guns and (say, for visibility reasons) always shot at and hit particular parts of the plane (say, the parts of the wings next to the necessarily shiny propellers) then the naive reasoning would be correct: the areas that were bullet-ridden on returning planes would be the ones that should be armored.
Suppose anywhere the plane was shot led to a 1/5 chance of the plane crashing, meaning that all places are equally deserving of armor. Suppose also that the wings comprised about 75% of the surface area of the plane. You'd see 3 planes returning with bullet holes in a wing for every 1 with a bullet hole somewhere else. That doesn't mean somewhere else is a better place to put the armor.
It's easy to see how it could be possible, if the odds of a crash weren't uniform (ie if the wings had a slightly higher than average chance of causing a crash if shot in my example above) you could easily come to the wrong conclusion.