A Fresh Look at the "Hot Hand" Paradox

TL;DR

This work reframes the hot hand paradox as a first-passage problem in a fair coin-toss process and employs the backward Kolmogorov equation to compute mean waiting times for fixed-length head/tail sequences. It derives exact results up to length 5, develops moment generating functions for these sequences, and extends the analysis to arbitrarily long sequences, including $n$ consecutive heads and $n$ consecutive (HT)'s. A key finding is that sequences of equal length can have substantially different mean waiting times, with asymptotic behavior $T_{2nH} \,\sim\, 3\,T_{n(HT)}$, illustrating a robust separation of outcomes despite identical long-run frequencies. The approach also yields higher-order statistics and MGFs in a unified, tractable framework, offering a clear, calculational path for a broad class of first-passage problems in binary processes.

Abstract

We use the backward Kolmogorov equation approach to understand the apparently paradoxical feature that the mean waiting time to encounter distinct fixed-length sequences of heads and tails upon repeated fair coin flips can be different. For sequences of length 2, the mean time until the sequence HH (heads-heads) appears equals 6, while the waiting time for the sequence HT (heads-tails) equals 4. We give complete results for the waiting times of sequences of lengths 3, 4, and 5; the extension to longer sequences is straightforward (albeit more tedious). We also derive the moment generating functions, from which any moment of the mean waiting time for specific sequences can be found. Finally, we compute the mean waiting times $T_{2n\rm H}$ for $2n$ heads in a row and $T_{n\rm(HT)}$ for $n$ alternating heads and tails. For large $n$, $T_{2n\rm H}\sim 3 T_{n\rm(HT)}$. Thus distinct sequences of coin flips of the same length can have very different mean waiting times.