Length of the minimum sequence containing repeats of success runs

Let a sequence of binary (zero–one or failure–success) trials ordered on a line. We consider runs of successes of length at least equal to a fixed number. The statistics denoting the size (length) as well as the starting and ending positions of the minimum subsequence containing all such runs are defined and studied. The study concerns with conditional probability distributions of these and other related statistics given that the number of such success runs in the sequence is at least equal to two. Numerical examples illustrate the theoretical results.