\[ \newcommand{\Prob}[1]{\mathbb{P}\left( #1 \right)} \newcommand{\Probwrt}[2]{\mathbb{P}_{#1}\left( #2 \right)} \newcommand{\Var}[1]{\mathrm{Var}\left[ #1 \right]} \] Attention conservation notice: 4900+ words, plus two (ugly) pictures and many equations, on a common mis-understanding in statistics. Veers wildly between baby stats. and advanced probability theory, without explaining either. Its efficacy at remedying the confusion it attacks has not been evaluated by a randomized controlled trial. After ten years of teaching statistics, I feel pretty confident in saying…