Successes, runs and longest runs (Q5903000)

The probability distribution of the number of success runs of length k (\(\geq 1)\) in n (\(\geq 1)\) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let \(S_ n\) and \(L_ n\), respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability \(P(L_ n\leq k| S_ n=r)\) (0\(\leq k\leq r\leq n)\), which is alternative to those given by \textit{E. J. Burr} and \textit{G. Cane} [Biometrika 48, 461-465 (1961; Zbl 0134.150)] and \textit{J. D. Gibbons} [Nonparametric statistical inference. (1971; Zbl 0223.62050)]. Finally, the probability distribution of \(X^{(k)}_{n,L_ n}\) is established, where \(X^{(k)}_{n,L_ n}\) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.