Finally a hot shower for the hot hand fallacy

We study whether basketball players exhibit a `hot hand' in their shooting behaviour, asking not only whether a hot hand exists but whether its magnitude is large enough to matter. We model each player's binary shooting sequence as a two-state kth order Markov chain governed by two parameters: a baseline log-odds eta and a hot hand log-odds shift delta. The joint probability mass function forms a curved exponential family with natural statistic (h_H, h, n_H), where h_H counts hits in the hot state, h counts total hits, and n_H counts hot trials. Conditioning on (h, n_H) removes the nuisance eta and yields a one-parameter regular exponential family in delta; we show that (h, n_H) is S-ancillary with respect to delta and is the coarsest statistic with this property. On the resulting conditional family we construct an e-value against any null of the form delta in D \subseteq \mathbb{R}, giving a continuous measure of evidence that can test existence, sign, or magnitude of the hot hand. Because each player's e-value is valid uniformly over their unknown skill eta, e-values from independent shooters multiply into a single pooled e-value, resolving both the incidental parameter problem and the low power of single-sequence tests documented by Ritzwoller and Romano (2022).