  For a sequence of $n$ random variables taking values $0$ or $1$, the hot hand
statistic of streak length $k$ counts what fraction of the streaks of length
$k$, that is, $k$ consecutive variables taking the value $1$, among the $n$
variables are followed by another $1$. Since this statistic does not use the
expected value of how many streaks of length $k$ are observed, but instead uses
the realization of the number of streaks present in the data, it may be a
biased estimator of the conditional probability of a fixed random variable
taking value $1$ if it is preceded by a streak of length $k$, as was first
studied and observed explicitly in [Miller and Sanjurjo, 2018]. In this short
note, we suggest an alternative proof for an explicit formula of the
expectation of the hot hand statistic for the case of streak length one. This
formula was obtained through a different argument in [Miller and Sanjurjo,
2018] and [Rinott and Bar-Hillel, 2015].
