# Uncertainty in the Hot Hand Fallacy: Detecting Streaky Alternatives to Random Bernoulli Sequences

## TL;DR

This paper develops permutation tests to analyze streaks in Bernoulli sequences, applying them to basketball shooting data to evaluate the hot hand fallacy and establishing the statistical foundation for such analyses.

## Contribution

It introduces new permutation test methods for detecting non-random streaks in Bernoulli sequences and characterizes their asymptotic behavior under various processes.

## Key findings

- Larger datasets are needed for reliable deviation measurement.
- One shooter showed statistically significant non-random shooting patterns.
- Limited evidence of the hot hand fallacy across all shooters.

## Abstract

We study a class of permutation tests of the randomness of a collection of Bernoulli sequences and their application to analyses of the human tendency to perceive streaks of consecutive successes as overly representative of positive dependence - the hot hand fallacy. In particular, we study permutation tests of the null hypothesis of randomness (i.e., that trials are i.i.d.) based on test statistics that compare the proportion of successes that directly follow k consecutive successes with either the overall proportion of successes or the proportion of successes that directly follow k consecutive failures. We characterize the asymptotic distributions of these test statistics and their permutation distributions under randomness, under a set of general stationary processes, and under a class of Markov chain alternatives, which allow us to derive their local asymptotic power. The results are applied to evaluate the empirical support for the hot hand fallacy provided by four controlled basketball shooting experiments. We establish that substantially larger data sets are required to derive an informative measurement of the deviation from randomness in basketball shooting. In one experiment, for which we were able to obtain data, multiple testing procedures reveal that one shooter exhibits a shooting pattern significantly inconsistent with randomness - supplying strong evidence that basketball shooting is not random for all shooters all of the time. However, we find that the evidence against randomness in this experiment is limited to this shooter. Our results provide a mathematical and statistical foundation for the design and validation of experiments that directly compare deviations from randomness with human beliefs about deviations from randomness, and thereby constitute a direct test of the hot hand fallacy.