How Much Does "Success Breeds Success" Actually Change a Free-Throw Sequence? Exact Results Under Illustrative, Literature-Calibrated, and Directly-Estimated State-Dependence, with a Precise Distinction from Dynamical Chaos

Teaching materials on stochastic dependence commonly illustrate the "hot hand" idea with a state-dependent free-throw model in which success probability rises after a make and falls after a miss, often using large, arbitrarily chosen swing magnitudes for pedagogical clarity. We ask how much of the resulting variance inflation and outcome sensitivity survives when the swing magnitude is instead anchored to actual estimates of hot-hand effect sizes. Using exact enumeration of all 2^10 throw sequences (cross-validated by Monte Carlo), we compare four parameterizations: an illustrative teaching version with an asymmetric +9 / -39 percentage-point swing; a version calibrated to Miller and Sanjurjo's (2018) bias-corrected pooled hot-hand estimate (±8 points, symmetric); a version calibrated to published real-game free-throw estimates from Lantis and Nesson (2021) (±3 points, symmetric); and — new in this version — a parameterization estimated directly by us from 18,681 two-shot free-throw trips in real 2011–12 NBA play-by-play data, finding a highly significant 6.03-percentage-point gap between second-shot success after a make versus after a miss (z = 8.40, p < 0.0001), independently corroborating the published real-game estimate. The illustrative version inflates outcome variance by 4.45× relative to the independent-trials binomial baseline and produces an average 4.33-make swing (out of 10) between a game that opens with a make versus a miss; all three literature- or data-calibrated versions agree closely with each other and show the same qualitative phenomena at 1.5–2× smaller variance inflation and 1.7–2.6× smaller sensitivity. We also show, via a common-random-numbers sensitivity test, that the model's outcome sensitivity to the first throw is monotone at every parameterization tested — a materially different, and much weaker, property than the non-monotone trajectory divergence required for deterministic chaos, contrary to an informal "butterfly effect" analogy sometimes invoked for this kind of model.