Groups and Series of Successes and Failures in Bernoulli Schemes

Bernoulli sequences of independent random variables X1, X2, … taking the value 1 with some probability 0 < p < 1 and the value 0 with the probability q = 1 – p are considered. The events {Xn = 1} and {Xn = 0} are treated, respectively, as “success” and “failure” in the nth Bernoulli trial. In their earlier studies, the authors of this work obtained a number of new results for series of successes. Note that if the probabilities p and q are interchanged in the relations for the number of series of successes, similar results can be stated for series of failures. These earlier works studied properties of distributions of random variables such as the number of distinct success series among n variables, the number of distinct success series observed when the nth success takes place, the number of success series observed when the mth failure takes place, and the number of distinct success series observed when the first series of k successes takes place. This work studies the properties of new random variables associated with concepts of series and groups of successes and failures in Bernoulli sequences.