> We observe that the canonical study in the influential hot hand fallacy literature [2],Gilovich, Vallone, and Tversky(1985), along with replications, have mistakenly employed a biased selection procedure that is analogous to Jack’s. Upon conducting a de-biased analysis, we find that the longstanding conclusions of the canonical study are reversed.
> [2] The hot hand fallacy has been given considerable weight as a candidate explanation for various puzzles and behavioral anomalies identified in the domains of financial markets, sports wagering, casino gambling,and lotteries (Arkes(2011),Avery and Chevalier(1999),Barberis and Thaler(2003),Brown and Sauer(1993),Camerer(1989),Croson and Sundali(2005),De Bondt(1993),Long et al.(1991),Durham, Hertzel, and Martin(2005),Galbo-Jørgensen, Suetens, and Tyran(2016),Guryan and Kearney(2008),Kahneman andRiepe(1998),Lee and Smith(2002),Loh and Warachka(2012),Malkiel(2011),Narayanan and Manchanda(2012),Paul and Weinbach(2005),Rabin and Vayanos(2010),Sinkey and Logan(2013),Smith, Levere, and Kurtzman(2009),Sundali and Croson(2006),Xu and Harvey(2014),Yuan, Sun, and Siu(2014))
Their claim is that people do apply "arbitrary preconditions" in the analysis of various situations.
You get exactly 50%, as expected because the probabilities are independent.
But when you only look at streaks, you find that the 1s are clustered in the streaks! Thus, if you average across many sequences (aka games) and weight per-series instead of per-1, you underweight the successes that occur in sequences with longer streaks and more 1s, giving an average-of-averages that is less than 0.5.
Why the researchers insist on averaging across sequences, I don't know.
Unsurprising, as the 50% expectation is only obvious for infinite sequences. All kinds of statistical correlations happen if you start adding arbitrary preconditions to finite sequences.