Shortest and longest length of success runs in binary sequences

The shortest and the longest length of success runs statistics in binary sequences are considered. The sequences are arranged on a line or on a circle. Exact probabilities of these statistics are derived, both in closed formulae via combinatorial analysis, as well as recursively. Furthermore, their joint probability distribution function and cumulative distribution function are obtained. The results are developed first for Bernoulli trials (i.i.d. binary sequences), and then they are generalized to the Polya–Eggenberger sampling scheme. For the latter case, the length of the longest success run is related to other success runs statistics and to reliability of consecutive systems.