Success runs in series of Bernoulli trials

This work is focused on selected probability characteristics of runs in a sequence of Bernoulli trials and on some randomness tests based on these runs. Based on Markov chains, an explicit formula is derived for the probability that the first success run of a lenght $k$ in a sequence of independent Bernoulli trials occurs in the $n$-th trial and other formulas for this probability are mentioned. Furthermore, approximations of the exact value of this probability (particularly the Feller approximation), bounds of these approximations, and their numeric relations are examined. Lastly, a test of randomness based on the lenght of the longest run in a sequence of $n$ Bernoulli trials and a test based on the total amount of runs are derived