Alternative proof for the bias of the hot hand statistic of streak length one

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].