Successes, runs and longest runs

The probability distribution of the number of success runs of length k ([greater-or-equal, slanted]1) in n ([greater-or-equal, slanted]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 Sn and Ln, 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(Ln [less-than-or-equals, slant] k Sn = r) (0 [less-than-or-equals, slant] k [less-than-or-equals, slant] r [less-than-or-equals, slant] n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.Bernoulli trials successes number of success runs of length k binomial distribution of order k length of the longest success run open problems