World War II. Bombers return to base riddled with bullet holes: wings and fuselage densely peppered, the area around the engines relatively clean. Armor is limited — add too much and the plane can't fly. So where should it go?

The intuitive answer: where the holes are — that's where the planes get hit most. According to Wallis (1980), the military indeed leaned toward reinforcing the most-hit parts of the returning planes.

Abraham Wald of Columbia University's Statistical Research Group (SRG, founded in the summer of 1942) pointed out the fatal flaw in that reasoning: the only damage data available came from planes that returned (the survivors); the planes that were shot down were entirely unobserved — using the survivors' bullet-hole distribution directly to assess aircraft vulnerability produces a systematic bias. Under the assumption that hits were roughly uniformly distributed, he inverted the inference: planes hit in vital parts were less likely to make it back, so the survivor sample shows fewer holes exactly there — the area around the engines looks 'clean' not because it wasn't hit, but because the planes hit there never came home.

Wald's actual method is far more hardcore than the legend: he wrote 8 memoranda, 'A Method of Estimating Plane Vulnerability Based on Damage of Survivors,' introducing the conditional probability p_i = 'the probability of being downed by the i-th hit, given survival of the first i−1 hits,' and building equations under explicitly stated assumptions such as 'planes are downed only by enemy fire (L0=0 — an unhit plane doesn't fall),' to infer the hit distribution of the planes that never returned from the data of those that did — one memo specifically estimates the vulnerability of each section of the aircraft. The method remained in use through the Korean and Vietnam wars; the memoranda were only reprinted publicly in 1980 by the Center for Naval Analyses (CNA, report CRC 432).