Randomness describes outcomes that lack a deterministic pattern while still following probabilistic regularities. Streaks can appear in genuinely random sequences, so visible order does not establish causality or a hot-hand effect. section 16.3 section 16.6
A practical analysis of randomness separates simulation, dependence, and interpretation:
- Simulate independent coin flips by sampling 0 and 1 with equal probabilities and replacement; every flip remains unaffected by previous results. section 16.3
- Expect streaks even when outcomes are random: one 700-flip simulation produced a heads streak of 11. section 16.4.1
- Compare successive free-throw results within each game rather than treating a league-wide pattern as evidence about individual players. section 16.6 section 16.4.1
Testing the free-throw hot hand
Filter the free-throw data to one player, group attempts by date, and calculate consecutive makes with runner::streak_run(lag(points)). For Giannis Antetokounmpo, percentages vary from 61.7% after one make to 77.1% after four, then fall to 66.7% after five; sparse later streaks reach 100%. James Harden's corresponding plot shows randomness rather than a consistent Laplace curve, and the overall evidence favors coin-flip-like free throws over a hot hand. section 16.4.1 section 16.4.3 section 16.6
sample(x = c(0, 1), size = 10, prob = c(0.5, 0.5), replace = TRUE) ## [1] 0 1 1 1 1 1 1 0 1 0
Run the compact R simulation to see a fair coin generate a concrete sequence containing several consecutive identical outcomes. section 16.3
