On the series of successes and failures in Bernoulli schemes

The classic scheme of independent trials by Bernoulli has been one of the most popular topics in probability theory for more than three centuries, starting with the work of Jacob Bernoulli. It is ideal for setting and solving various practical problems. Many results have been obtained related to modifications of this scheme, but new situations, new problems appear that require further progress in the study of various random variables related to one degree or another with independent Bernoulli tests. In this paper, we continue to study some problems related to the series of successes and failures in sequences of Bernoulli random variables. This work is a direct continuation of the authors' article “On some probability distributions related to the classical Bernoulli scheme”, published in 2022.